Calculus: Early Transcendentals (3rd Edition)
3rd Edition
ISBN: 9780134763644
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Textbook Question
Chapter 3.11, Problem 4QC
In Example 4, notice that as the balloon rises (as θ increases), the rate of change of the angle of elevation decreases to zero. When does the maximum value of
Example 4 Observing a Launch
An observer stands 200 meters from the launch site of a hot-air balloon at an elevation equal to the elevation of the launch site. The balloon rises vertically at a constant rate of 4 m/s. How fast is the angle of elevation of the balloon increasing 30 seconds after the launch? (The angle of elevation is the angle between the ground and the observer’s line of sight to the balloon.)
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Use the information to find and compare Δy and dy. (Round your answers to four decimal places.)
y = x4 + 7 x = −3 Δx = dx = 0.01
Δy =
dy =
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in the table. For each problem, approximate the distance the car traveled (in miles) using the given method,
on the provided interval, and with the given number of rectangles or trapezoids, n.
Time (min) 0 6 12 18|24|30|36|42|48|54|60
Speed (mph) 0 10 20 40 60 50 40 30 40 40 65
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c.) Midpoint Rectangles, [24, 60] n=3
d.) Trapezoids, [0, 24] n=4
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0.24 m
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D
Chapter 3 Solutions
Calculus: Early Transcendentals (3rd Edition)
Ch. 3.1 - Quick Check 1
In Example 1, is the slope of the...Ch. 3.1 - Sketch the graph of a function f near a point a....Ch. 3.1 - Set up the calculation in Example 3 using...Ch. 3.1 - Verify that the derivative of the function f in...Ch. 3.1 - Use definition (1) (p. 127) for the slope of a...Ch. 3.1 - Explain why the slope of a secant line can be...Ch. 3.1 - Explain why the slope of the tangent line can be...Ch. 3.1 - Explain the relationships among the slope of a...Ch. 3.1 - Given a function f and a point a in its domain,...Ch. 3.1 - The following figure shows the graph of f and a...
Ch. 3.1 - An equation of the line tangent to the graph of f...Ch. 3.1 - An equation of the line tangent to the graph of g...Ch. 3.1 - If h(1) = 2 and h'(1) = 3, find an equation of the...Ch. 3.1 - If f(2)=7, find an equation of the line tangent to...Ch. 3.1 - Use definition (1) (p. 133) to find the slope of...Ch. 3.1 - Use definition (2) (p. 135) to find the slope of...Ch. 3.1 - Velocity functions A projectile is fired...Ch. 3.1 - Velocity functions A projectile is fired...Ch. 3.1 - Equations of tangent lines by definition (1) a....Ch. 3.1 - Equations of tangent lines by definition (1) a....Ch. 3.1 - Equations of tangent lines by definition (1) a....Ch. 3.1 - Equations of tangent lines by definition (1) a....Ch. 3.1 - Equations of tangent lines by definition (1) a....Ch. 3.1 - Equations of tangent lines by definition (1) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Equations of tangent lines by definition (2) a....Ch. 3.1 - Prob. 32ECh. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivatives and tangent lines a. For the following...Ch. 3.1 - Derivative calculations Evaluate the derivative of...Ch. 3.1 - Derivative calculations Evaluate the derivative of...Ch. 3.1 - Derivative calculations Evaluate the derivative of...Ch. 3.1 - Derivative calculations Evaluate the derivative of...Ch. 3.1 - Explain why or why not Determine whether the...Ch. 3.1 - Interpreting the derivative Find the derivative of...Ch. 3.1 - Interpreting the derivative Find the derivative of...Ch. 3.1 - Interpreting the derivative Find the derivative of...Ch. 3.1 - Interpreting the derivative Find the derivative of...Ch. 3.1 - Population of Las Vegas Let p(t) represent the...Ch. 3.1 - Owlet talons Let L(t) equal the average length (in...Ch. 3.1 - Caffeine levels Let 4(f) be the amount of caffeine...Ch. 3.1 - Let D(t) equal the number of daylight hours at a...Ch. 3.1 - Find the function The following limits represent...Ch. 3.1 - Find the function The following limits represent...Ch. 3.1 - Find the function The following limits represent...Ch. 3.1 - Find the function The following limits represent...Ch. 3.1 - Find the function The following limits represent...Ch. 3.1 - Find the function The following limits represent...Ch. 3.1 - Approximating derivatives Assuming the limit...Ch. 3.1 - Another way to approximate derivatives is to use...Ch. 3.1 - Prob. 64ECh. 3.1 - Approximating derivatives Assuming the limit...Ch. 3.2 - In Example 1, determine the slope of the tangent...Ch. 3.2 - What are some other ways to write f(3), where y =...Ch. 3.2 - In Example 2, do the slopes of the tangent lines...Ch. 3.2 - Express the derivative of p = q(r) in three ways.Ch. 3.2 - Is it true that if f(x) 0 at a point, than f(x) ...Ch. 3.2 - Verify tha1 the right-hand side of (1) equals f(x)...Ch. 3.2 - For a given function f what does frepresent?Ch. 3.2 - If f(x)=3x+2, find the slope of the line tangent...Ch. 3.2 - Why is the notation dydx used to represent the...Ch. 3.2 - Give three different notations for the derivative...Ch. 3.2 - Sketch a graph of a function f, where f(x) 0 and...Ch. 3.2 - Sketch a graph of a function f, where f(x) 0 and...Ch. 3.2 - If f is differentiable at a, must f be continuous...Ch. 3.2 - If f is continuous at a, must f be differentiable...Ch. 3.2 - Describe the graph of f if f(0)=1 and f(x)=3, for...Ch. 3.2 - Use the graph of f(x)=|x| to find f(x).Ch. 3.2 - Use limits to find f(x) if f(x)=7x.Ch. 3.2 - Use limits to find f(x) if f(x)=3x.Ch. 3.2 - Prob. 13ECh. 3.2 - The weight w(x) (in pounds) of an Atlantic salmon...Ch. 3.2 - Matching functions with derivatives Match graphs...Ch. 3.2 - Matching derivatives with functions Match graphs...Ch. 3.2 - Sketching derivatives Reproduce the graph of f and...Ch. 3.2 - Sketching derivatives Reproduce the graph of f and...Ch. 3.2 - Use the graph of f in the figure to do the...Ch. 3.2 - Use the graph of g in the figure to do the...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Derivatives a.Use limits to find the derivative...Ch. 3.2 - Velocity functions A projectile is fired...Ch. 3.2 - Velocity functions A projectile is fired...Ch. 3.2 - Evaluate dydx and dydx|x=2 if y=x+1x+2.Ch. 3.2 - Evaluate dydx and dydx|x=2 if s=11t3+t+1.Ch. 3.2 - Tangent lines a.Find the derivative function f for...Ch. 3.2 - Tangent lines a.Find the derivative function f for...Ch. 3.2 - Calculating derivatives a. For the following...Ch. 3.2 - Calculating derivatives a. For the following...Ch. 3.2 - Calculating derivatives a. For the following...Ch. 3.2 - Calculating derivatives a. For the following...Ch. 3.2 - Power and energy Energy is the capacity to do...Ch. 3.2 - Slope of a line Consider the line f(x) = mx + b,...Ch. 3.2 - A derivative formula a. Use the definition of the...Ch. 3.2 - A derivative formula a. Use the definition of the...Ch. 3.2 - Analyzing slopes Use the points A, B, C, D, and E...Ch. 3.2 - Analyzing slopes Use the points A, B, C, D, and E...Ch. 3.2 - Matching functions with derivatives Match the...Ch. 3.2 - Sketching derivatives Reproduce the graph of f and...Ch. 3.2 - Sketching derivatives Reproduce the graph of f and...Ch. 3.2 - Sketching derivatives Reproduce the graph of f and...Ch. 3.2 - Graphing the derivative with asymptotes Sketch a...Ch. 3.2 - Graphing the derivative with asymptotes Sketch a...Ch. 3.2 - Where is the function continuous? Differentiable?...Ch. 3.2 - Where is the function continuous? Differentiable?...Ch. 3.2 - Voltage on a capacitor A capacitor is a device in...Ch. 3.2 - Logistic growth A common model for population...Ch. 3.2 - Explain why or why not Determine whether the...Ch. 3.2 - Looking ahead: Derivative of xn Use the definition...Ch. 3.2 - Prob. 59ECh. 3.2 - Prob. 60ECh. 3.2 - Finding f from f Sketch the graph of f(x) = x....Ch. 3.2 - Prob. 62ECh. 3.2 - Normal lines A line perpendicular to another line...Ch. 3.2 - Normal lines A line perpendicular to another line...Ch. 3.2 - Normal lines A line perpendicular to another line...Ch. 3.2 - Prob. 66ECh. 3.2 - Aiming a tangent line Given the function f and the...Ch. 3.2 - Aiming a tangent line Given the function f and the...Ch. 3.2 - Prob. 69ECh. 3.2 - Aiming a tangent line Given the function f and the...Ch. 3.2 - One-sided derivatives The right-sided and...Ch. 3.2 - One-sided derivatives The right-sided and...Ch. 3.2 - Prob. 73ECh. 3.2 - Prob. 74ECh. 3.2 - Prob. 75ECh. 3.2 - Vertical tangent lines If a function f is...Ch. 3.2 - Continuity is necessary for differentiability a....Ch. 3.2 - Prob. 78ECh. 3.3 - Find the values of ddx(11) and ddx()Ch. 3.3 - Use the graph of y = x to give a geometric...Ch. 3.3 - Prob. 3QCCh. 3.3 - Find the derivative of f(x)=4ex3x2.Ch. 3.3 - Determine the point(s) at which f(x) = x3 12x has...Ch. 3.3 - Prob. 6QCCh. 3.3 - Assume the derivatives of f and g exist in...Ch. 3.3 - Assume the derivatives of f and g exist in...Ch. 3.3 - Assume the derivatives of f and g exist in...Ch. 3.3 - Prob. 4ECh. 3.3 - Assume the derivatives of f and g exist in...Ch. 3.3 - Prob. 6ECh. 3.3 - Given that f(3) = 6 and g(3) = 2, find (f + g)(3).Ch. 3.3 - If f(0)=6 and g(x)=f(x)+ex+1, find g(0).Ch. 3.3 - Let F(x)=f(x)+g(x),G(x)=f(x)g(x), and...Ch. 3.3 - Let F(x)=f(x)+g(x),G(x)=f(x)g(x), and...Ch. 3.3 - Let F(x)=f(x)+g(x),G(x)=f(x)g(x), and...Ch. 3.3 - Derivatives from a table Use the table to find the...Ch. 3.3 - Derivatives from a table Use the table to find the...Ch. 3.3 - Derivatives from a table Use the table to find the...Ch. 3.3 - If f(t)=t10, find f(t),f(t), and f(t).Ch. 3.3 - Find an equation of the line tangent to the graph...Ch. 3.3 - The line tangent to the graph of f at x = 5 is...Ch. 3.3 - The line tangent to the graph of f at x = 3 is y =...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Derivatives Find the derivative of the following...Ch. 3.3 - Height estimate The distance an object falls (when...Ch. 3.3 - Projectile trajectory The position of a small...Ch. 3.3 - City urbanization City planners model the size of...Ch. 3.3 - Cell growth When observations begin at t = 0, a...Ch. 3.3 - Weight of Atlantic salmon The weight w(x) (in...Ch. 3.3 - Derivatives of products and quotients Find the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products and quotients Find the...Ch. 3.3 - Derivatives of products and quotients Find the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products Find the derivative of the...Ch. 3.3 - Derivatives of products and quotients Find the...Ch. 3.3 - Derivatives of products and quotients Find the...Ch. 3.3 - Equations of tangent lines a. Find an equation of...Ch. 3.3 - Equations of tangent lines a. Find an equation of...Ch. 3.3 - Equations of tangent lines a. Find an equation of...Ch. 3.3 - Equations of tangent lines a. Find an equation of...Ch. 3.3 - Finding slope locations Let f(x) = x3 6x + 5. a....Ch. 3.3 - Finding slope locations Let f(t) = t3 27t + 5. a....Ch. 3.3 - Finding slope locations Let f(x) = 2x3 3x2 12x +...Ch. 3.3 - Finding slope locations Let f(x) = 2ex 6x. a....Ch. 3.3 - Finding slope locations Let f(x)=4xx. a. Find all...Ch. 3.3 - Higher-order derivatives Find f(x), f(x), and f(x)...Ch. 3.3 - Higher-order derivatives Find f(x), f(x), and f(x)...Ch. 3.3 - Higher-order derivatives Find f(x), f(x), and f(x)...Ch. 3.3 - Higher-order derivatives Find f(x), f(x), and f(x)...Ch. 3.3 - Higher-order derivatives Find f(x), f(x), and f(x)...Ch. 3.3 - Explain why or why not Determine whether the...Ch. 3.3 - Tangent lines Suppose f(3) = 1 and f(3) = 4. Let...Ch. 3.3 - Derivatives from tangent lines Suppose the line...Ch. 3.3 - Tangent line Find the equation of the line tangent...Ch. 3.3 - Tangent line given Determine the constants b and c...Ch. 3.3 - Derivatives from a graph Let F = f + g and G = 3f ...Ch. 3.3 - Derivatives from a graph Let F = f + g and G = 3f ...Ch. 3.3 - Derivatives from a graph Let F = f + g and G = 3f ...Ch. 3.3 - Derivatives from a graph Let F = f + g and G = 3f ...Ch. 3.3 - Derivatives from limits The following limits...Ch. 3.3 - Derivatives from limits The following limits...Ch. 3.3 - Derivatives from limits The following limits...Ch. 3.3 - Derivatives from limits The following limits...Ch. 3.3 - Derivatives from limits The following limits...Ch. 3.3 - Derivatives from limits The following limits...Ch. 3.3 - Prob. 88ECh. 3.3 - Prob. 89ECh. 3.3 - Calculator limits Use a calculator to approximate...Ch. 3.3 - Prob. 91ECh. 3.3 - Constant Rule proof For the constant function f(x)...Ch. 3.3 - Prob. 93ECh. 3.3 - Looking ahead: Power Rule for negative integers...Ch. 3.3 - Prob. 95ECh. 3.3 - Computing the derivative of f(x) = ex a. Use the...Ch. 3.3 - Prob. 97ECh. 3.3 - Computing the derivative of f(x) = x2ex a. Use the...Ch. 3.4 - Find the derivative of f(x) = x5. Then find the...Ch. 3.4 - Find the derivative of f(x)=x5 Then find the same...Ch. 3.4 - Find the derivative of f(x)=1/x5 in two different...Ch. 3.4 - How do you find the derivative of the product of...Ch. 3.4 - How do you find the derivative of the quotient of...Ch. 3.4 - Use the Product Rule to evaluate and simplify...Ch. 3.4 - Use the Product Rule to find f(1) given that...Ch. 3.4 - Use the Quotient Rule to evaluate and simplify...Ch. 3.4 - Use the Quotient Rule to find g(1) given that...Ch. 3.4 - Find the derivative the following ways: a. Using...Ch. 3.4 - Find the derivative the following ways: a. Using...Ch. 3.4 - Find the derivative the following ways: a. Using...Ch. 3.4 - Derivatives by two different methods a. Use the...Ch. 3.4 - Derivatives by two different methods a. Use the...Ch. 3.4 - Find the derivative the following ways: a. Using...Ch. 3.4 - Derivatives by two different methods a. Use the...Ch. 3.4 - Derivatives by two different methods a. Use the...Ch. 3.4 - Given that f(1)=5,f(1)=4,g(1)=2, and g(1)=3, find...Ch. 3.4 - Show two ways to differentiate f(x) = 1/x10.Ch. 3.4 - Find the slope of the line tangent to the graph of...Ch. 3.4 - Find the slope of the graph of f(x)=2+xex at the...Ch. 3.4 - Derivatives of products Find the derivative of the...Ch. 3.4 - Derivatives of products Find the derivative of the...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives of products Find the derivative of the...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives of products Find the derivative of the...Ch. 3.4 - Derivatives of products Find the derivative of the...Ch. 3.4 - Derivatives of products Find the derivative of the...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Derivatives of quotients Find the derivative of...Ch. 3.4 - Extended Power Rule Find the derivative of the...Ch. 3.4 - Extended Power Rule Find the derivative of the...Ch. 3.4 - Extended Power Rule Find the derivative of the...Ch. 3.4 - Extended Power Rule Find the derivative of the...Ch. 3.4 - Extended Power Rule Find the derivative of the...Ch. 3.4 - Extended Power Rule Find the derivative of the...Ch. 3.4 - Combining rules Compute the derivative of the...Ch. 3.4 - Combining rules Compute the derivative of the...Ch. 3.4 - Combining rules Compute the derivative of the...Ch. 3.4 - Combining rules Compute the derivative of the...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Choose your method Use any method to evaluate the...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Choose your method Use any method to evaluate the...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Choose your method Use any method to evaluate the...Ch. 3.4 - Choose your method Use any method to evaluate the...Ch. 3.4 - Choose your method Use any method to evaluate the...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Derivatives Find and simplify the derivative of...Ch. 3.4 - Equations of tangent lines a. Find an equation of...Ch. 3.4 - Equations of tangent lines a. Find an equation of...Ch. 3.4 - Equations of tangent lines a. Find an equation of...Ch. 3.4 - Equations of tangent lines a. Find an equation of...Ch. 3.4 - Population growth Consider the following...Ch. 3.4 - Population growth Consider the following...Ch. 3.4 - Electrostatic force The magnitude of the...Ch. 3.4 - Gravitational force The magnitude of the...Ch. 3.4 - Explain why or why not Determine whether the...Ch. 3.4 - Higher-order derivatives Find f(x), f(x), and...Ch. 3.4 - Higher-order derivatives Find f(x),f(x), f(x), and...Ch. 3.4 - First and second derivatives Find f(x) and f(x)....Ch. 3.4 - First and second derivatives Find f(x) and f(x)....Ch. 3.4 - Tangent lines Suppose f(2) = 2 and f(2) = 3. Let...Ch. 3.4 - The Witch of Agnesi The graph of y=a3x2+a2, where...Ch. 3.4 - Derivatives from a table Use the following table...Ch. 3.4 - Derivatives from a table Use the following table...Ch. 3.4 - Derivatives from a table Use the following table...Ch. 3.4 - Derivatives from a table Use the following table...Ch. 3.4 - Derivatives from a table Use the following table...Ch. 3.4 - Derivatives from a table Use the following table...Ch. 3.4 - Flight formula for Indian spotted owlets The...Ch. 3.4 - Flight formula for Indian spotted owlets The...Ch. 3.4 - Prob. 84ECh. 3.4 - Assume both the graphs of f and g pass through the...Ch. 3.4 - Prob. 86ECh. 3.4 - Derivatives from graphs Use the figure to find the...Ch. 3.4 - Prob. 88ECh. 3.4 - Derivatives from graphs Use the figure to find the...Ch. 3.4 - Derivatives from graphs Use the figure to find the...Ch. 3.4 - Prob. 91ECh. 3.4 - Derivatives from tangent lines Suppose the line...Ch. 3.4 - Explorations and Challenges Avoiding tedious work...Ch. 3.4 - Given that...Ch. 3.4 - Means and tangents Suppose f is differentiable on...Ch. 3.4 - Proof of the Quotient Rule Let F = f/g be the...Ch. 3.4 - Product Rule for the second derivative Assuming...Ch. 3.4 - One of the Leibniz Rules One of several Leibniz...Ch. 3.4 - Product Rule for three functions Assume that f, g,...Ch. 3.5 - Evaluate limx0tan2xxCh. 3.5 - At what point on the interval [0,2] does the graph...Ch. 3.5 - The formulas for ddx(cotx),ddx(secx), and...Ch. 3.5 - Prob. 4QCCh. 3.5 - Why is it not possible to evaluate limx0sinxx by...Ch. 3.5 - How is limx0sinxx used in this section?Ch. 3.5 - Explain why the Quotient Rule is used to determine...Ch. 3.5 - How can you use the derivatives ddx(sinx)=cosx,...Ch. 3.5 - Let f(x) = sin x. What is the value of f()?Ch. 3.5 - Find the value of ddx(tanx)|x=3Ch. 3.5 - Find an equation of the line tangent to the curve...Ch. 3.5 - Where does the graph of sin x have a horizontal...Ch. 3.5 - Find d2dx2(sinx+cosx)Ch. 3.5 - Find d2dx2(secx).Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Use Theorem 3.11 to evaluate...Ch. 3.5 - Trigonometric limits Evaluate the following limits...Ch. 3.5 - Trigonometric limits Evaluate the following limits...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find the derivative of the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find the derivative of the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find the derivative of the...Ch. 3.5 - Calculating derivatives Find the derivative of the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Calculating derivatives Find the derivative of the...Ch. 3.5 - Calculating derivatives Find dy/dx for the...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Calculating derivatives Find the derivative of the...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Derivatives involving other trigonometric...Ch. 3.5 - Calculating derivatives Find the derivative of the...Ch. 3.5 - Derivatives of other trigonometric functions...Ch. 3.5 - Derivatives of other trigonometric functions...Ch. 3.5 - Derivatives of other trigonometric functions...Ch. 3.5 - Velocity of an oscillator An object oscillates...Ch. 3.5 - Damped sine wave The graph of f(t)=etsin t is an...Ch. 3.5 - Second-order derivatives Find y for the following...Ch. 3.5 - Second derivatives Find y for the functions....Ch. 3.5 - Second-order derivatives Find y for the following...Ch. 3.5 - Second-order derivatives Find y for the following...Ch. 3.5 - Second-order derivatives Find y for the following...Ch. 3.5 - Second-order derivatives Find y for the following...Ch. 3.5 - Second-order derivatives Find y for the following...Ch. 3.5 - Second-order derivatives Find y for the following...Ch. 3.5 - Explain why or why not Determine whether the...Ch. 3.5 - Trigonometric limits Evaluate the following limits...Ch. 3.5 - Trigonometric limits Evaluate the following limits...Ch. 3.5 - Trigonometric limits Evaluate the following limits...Ch. 3.5 - Trigonometric limits Evaluate the following limits...Ch. 3.5 - Prob. 70ECh. 3.5 - Trigonometric limits Evaluate the following limits...Ch. 3.5 - Equations of tangent lines a. Find an equation of...Ch. 3.5 - Equations of tangent lines a. Find an equation of...Ch. 3.5 - Equations of tangent lines a. Find an equation of...Ch. 3.5 - Equations of tangent lines a. Find an equation of...Ch. 3.5 - Locations of tangent lines a. For what values of x...Ch. 3.5 - Locations of horizontal tangent lines For what...Ch. 3.5 - Matching Match the graphs of the functions in ad...Ch. 3.5 - A differential equation A differential equation is...Ch. 3.5 - Using identities Use the identity sin 2x = 2 sin x...Ch. 3.5 - Prob. 81ECh. 3.5 - Prob. 82ECh. 3.5 - Proof of ddx(cosx)=sinx Use the definition of the...Ch. 3.5 - Continuity of a piecewise function Let...Ch. 3.5 - Continuity of a piecewise function Let...Ch. 3.5 - Prob. 86ECh. 3.5 - Prob. 87ECh. 3.5 - Prob. 88ECh. 3.5 - Prob. 89ECh. 3.5 - Prob. 90ECh. 3.6 - Does the speedometer in your car measure average...Ch. 3.6 - For an object moving along a line, is it possible...Ch. 3.6 - Describe the velocity of an object that has a...Ch. 3.6 - In Example 3, does the rock have a greater speed...Ch. 3.6 - Prob. 5QCCh. 3.6 - In Example 5, what happens to the average cost as...Ch. 3.6 - Explain the difference between the average rate of...Ch. 3.6 - Complete the following statement. If dydx is...Ch. 3.6 - Complete the following statement: If dydx is...Ch. 3.6 - Suppose the function s(t) represents the position...Ch. 3.6 - Suppose w(t) is the weight (in pounds) of a golden...Ch. 3.6 - What is the difference between the velocity and...Ch. 3.6 - Define the acceleration of an object moving in a...Ch. 3.6 - An object moving along a line has a constant...Ch. 3.6 - The speed of sound (in m/s) in dry air is...Ch. 3.6 - At noon, a city park manager starts filling a...Ch. 3.6 - Highway travel A state patrol station is located...Ch. 3.6 - Airline travel The following figure shows the...Ch. 3.6 - Prob. 13ECh. 3.6 - Explain why a decreasing demand function has a...Ch. 3.6 - Position, velocity, and acceleration Suppose the...Ch. 3.6 - Position, velocity, and acceleration Suppose the...Ch. 3.6 - Position, velocity, and acceleration Suppose the...Ch. 3.6 - Position, velocity, and acceleration Suppose the...Ch. 3.6 - Position, velocity, and acceleration Suppose the...Ch. 3.6 - Position, velocity, and acceleration Suppose the...Ch. 3.6 - A dropped stone on Earth The height (in feet) of a...Ch. 3.6 - A dropped stone on Mars A stone is dropped off the...Ch. 3.6 - Throwing a stone Suppose a stone is thrown...Ch. 3.6 - Suppose a stone is thrown vertically upward from...Ch. 3.6 - A stone thrown vertically on Mars Suppose a stone...Ch. 3.6 - Maximum height Suppose a baseball is thrown...Ch. 3.6 - Initial velocity Suppose a baseball is thrown...Ch. 3.6 - Population growth in Washington The population of...Ch. 3.6 - Average and marginal cost Consider the following...Ch. 3.6 - Average and marginal cost Consider the following...Ch. 3.6 - Average and marginal cost Consider the following...Ch. 3.6 - Average and marginal cost Consider the following...Ch. 3.6 - Demand and elasticity Based on sales data over the...Ch. 3.6 - Demand and elasticity The economic advisor of a...Ch. 3.6 - Explain why or why not Determine whether the...Ch. 3.6 - A feather dropped on the moon On the moon, a...Ch. 3.6 - Comparing velocities A stone is thrown vertically...Ch. 3.6 - Comparing velocities Two stones are thrown...Ch. 3.6 - Matching heights A stone is thrown from the edge...Ch. 3.6 - Velocity of a car The graph shows the position s =...Ch. 3.6 - Velocity from position The graph of s = f(t)...Ch. 3.6 - Fish length Assume the length L (in cm) of a...Ch. 3.6 - Average and marginal profit Let C(x) represent the...Ch. 3.6 - Average and marginal profit Let C(x) represent the...Ch. 3.6 - Average and marginal profit Let C(x) represent the...Ch. 3.6 - Average and marginal profit Let C(x) represent the...Ch. 3.6 - U.S. population growth The population p(t) (in...Ch. 3.6 - Average and marginal production Economists use...Ch. 3.6 - Velocity of a marble The position (in meters) of a...Ch. 3.6 - Tree growth Let b represent the base diameter of a...Ch. 3.6 - Prob. 51ECh. 3.6 - Diminishing returns A cost function of the form...Ch. 3.6 - Revenue function A store manager estimates that...Ch. 3.6 - Fuel economy Suppose you own a fuel-efficient...Ch. 3.6 - Spring oscillations A spring hangs from the...Ch. 3.6 - Power and energy Power and energy are often used...Ch. 3.6 - A race Jean and Juan run a one-lap race on a...Ch. 3.6 - Flow from a tank A cylindrical tank is full at...Ch. 3.6 - Bungee jumper A woman attached to a bungee cord...Ch. 3.6 - Spring runoff The flow of a small stream is...Ch. 3.6 - Temperature distribution A thin copper rod, 4...Ch. 3.7 - Explain why it is not practical to calculate...Ch. 3.7 - Identify an inner function (call it g) of y = (5x...Ch. 3.7 - Let y = tan10 (x6). Find f, g, and h such that y =...Ch. 3.7 - Two equivalent forms of the Chain Rule for...Ch. 3.7 - Identify the inner and outer functions in the...Ch. 3.7 - Identify an inner function u = g(x) and an outer...Ch. 3.7 - Identify an inner function u = g(x) and an outer...Ch. 3.7 - The two composite functions y = cos3 x and y = cos...Ch. 3.7 - Let h(x) = f(g(x)), where f and g are...Ch. 3.7 - Fill in the blanks. The derivative of f(g(x))...Ch. 3.7 - Evaluate the derivative of y=(x2+2x+1)2 using...Ch. 3.7 - Evaluate the derivative of y=4x+1 using...Ch. 3.7 - Express Q(x) = cos4 (x2 + 1) as the composition of...Ch. 3.7 - Given that h(x)=f(g(x)), find h(3) if...Ch. 3.7 - Given that h(x) = f(g(x)), use the graphs of f and...Ch. 3.7 - What is the derivative of y=ekx?Ch. 3.7 - Find f(x) if f(x)=15e3x.Ch. 3.7 - Version 1 of the Chain Rule Use Version 1 of the...Ch. 3.7 - For each of the following composite functions,...Ch. 3.7 - Version 1 of the Chain Rule Use Version 1 of the...Ch. 3.7 - For each of the following composite functions,...Ch. 3.7 - Version 1 of the Chain Rule Use Version 1 of the...Ch. 3.7 - Version 1 of the Chain Rule Use Version 1 of the...Ch. 3.7 - For each of the following composite functions,...Ch. 3.7 - Version 1 of the Chain Rule Use Version 1 of the...Ch. 3.7 - Version 1 of the Chain Rule Use Version 1 of the...Ch. 3.7 - Version 1 of the Chain Rule Use Version 1 of the...Ch. 3.7 - Chain Rule using a table Let h(x)= f(g(x)) and...Ch. 3.7 - Prob. 26ECh. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Version 2 of the Chain Rule Use Version 2 of the...Ch. 3.7 - Chain Rule for powers Use the Chain Rule to find...Ch. 3.7 - Chain Rule for powers Use the Chain Rule to find...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Chain Rule for powers Use the Chain Rule to find...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Repeated use of the Chain Rule Calculate the...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Combining rules Use the Chain Rule combined with...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Square root derivatives Find the derivative of the...Ch. 3.7 - Calculate the derivative of the following...Ch. 3.7 - Explain why or why not Determine whether the...Ch. 3.7 - Smartphones From 2007 to 2014, there was a...Ch. 3.7 - Applying the Chain Rule Use the data in Tables 3.4...Ch. 3.7 - Mass of Juvenile desert tortoises A study...Ch. 3.7 - Cell population The population of a culture of...Ch. 3.7 - Bank account A 200 investment in a savings account...Ch. 3.7 - Pressure and altitude Earths atmospheric pressure...Ch. 3.7 - Finding slope locations Let f(x) = xe2x. a. Find...Ch. 3.7 - Second derivatives Find d2ydx2 for the following...Ch. 3.7 - Second derivatives Find d2ydx2 for the following...Ch. 3.7 - Second derivatives Find d2ydx2 for the following...Ch. 3.7 - Second derivatives Find d2ydx2 for the following...Ch. 3.7 - Prob. 90ECh. 3.7 - Tangent lines Determine an equation of the line...Ch. 3.7 - Tangent lines Determine equations of the lines...Ch. 3.7 - Tangent lines Assume f and g are differentiable on...Ch. 3.7 - Prob. 94ECh. 3.7 - Tangent lines Find the equation of the line...Ch. 3.7 - Prob. 96ECh. 3.7 - Composition containing sin x Suppose f is...Ch. 3.7 - Vibrations of a spring Suppose an object of mass m...Ch. 3.7 - Vibrations of a spring Suppose an object of mass m...Ch. 3.7 - Vibrations of a spring Suppose an object of mass m...Ch. 3.7 - A damped oscillator The displacement of a mass on...Ch. 3.7 - Oscillator equation A mechanical oscillator (such...Ch. 3.7 - Prob. 103ECh. 3.7 - Prob. 104ECh. 3.7 - Prob. 105ECh. 3.7 - Deriving trigonometric identities a. Differentiate...Ch. 3.7 - Prob. 107ECh. 3.7 - Prob. 108ECh. 3.7 - Prob. 109ECh. 3.7 - Prob. 110ECh. 3.7 - Prob. 111ECh. 3.7 - Prob. 112ECh. 3.7 - Prob. 113ECh. 3.7 - Prob. 114ECh. 3.7 - Prob. 115ECh. 3.8 - The equation x y2 = 0 implicitly defines what two...Ch. 3.8 - Use implicit differentiation to find dydx for x ...Ch. 3.8 - If a function is defined explicitly in the form y...Ch. 3.8 - For some equations, such as x2 + y2 = l or x y2 =...Ch. 3.8 - Explain the differences between computing the...Ch. 3.8 - Why are both the x-coordinate and the y-coordinate...Ch. 3.8 - Identify and correct the error in the following...Ch. 3.8 - Calculate dydx using implicit differentiation....Ch. 3.8 - Calculate dydx using implicit differentiation. 6....Ch. 3.8 - Calculate dydx using implicit differentiation. 7....Ch. 3.8 - Calculate dydx using implicit differentiation....Ch. 3.8 - Consider the curve defined by 2x y + y3 = 0 (see...Ch. 3.8 - Find the slope of the curve x2 + y3 = 2 at each...Ch. 3.8 - Consider the curve x=y3. Use implicit...Ch. 3.8 - Consider the curve x=ey. Use implicit...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Carry out the following...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Implicit differentiation Use implicit...Ch. 3.8 - Cobb-Douglas production function The output of an...Ch. 3.8 - Surface area of a cone The lateral surface area of...Ch. 3.8 - Volume of a spherical cap Imagine slicing through...Ch. 3.8 - Volume of a torus The volume of a torus (doughnut...Ch. 3.8 - Tangent lines Carry out the following steps....Ch. 3.8 - Tangent lines Carry out the following steps. a....Ch. 3.8 - Tangent lines Carry out the following steps. a....Ch. 3.8 - Tangent lines Carry out the following steps. a....Ch. 3.8 - Tangent lines Carry out the following steps. a....Ch. 3.8 - Tangent lines Carry out the following steps. a....Ch. 3.8 - Second derivatives Find d2ydx2. 31. x + y2 = 1Ch. 3.8 - Second derivatives Find d2ydx2. 32. 2x2 + y2 = 4Ch. 3.8 - Second derivatives Find d2ydx2. 33. x + y = sin yCh. 3.8 - Second derivatives Find d2ydx2. 34. x4 + y4 = 64Ch. 3.8 - Second derivatives Find d2ydx2. 35. e2y + x = yCh. 3.8 - Second derivatives Find d2ydx2 36. sin x + x2y =...Ch. 3.8 - Explain why or why not Determine whether the...Ch. 3.8 - Carry out the following steps. a.Use implicit...Ch. 3.8 - Carry out the following steps. a.Use implicit...Ch. 3.8 - Multiple tangent lines Complete the following...Ch. 3.8 - Multiple tangent lines Complete the following...Ch. 3.8 - Multiple tangent lines Complete the following...Ch. 3.8 - Witch of Agnesi Let y(x2 + 4) = 8 (see figure). a....Ch. 3.8 - Vertical tangent lines a. Determine the points at...Ch. 3.8 - Vertical tangent lines a. Determine the points...Ch. 3.8 - Tangent lines for ellipses Find the equations of...Ch. 3.8 - Tangent lines for ellipses Find the equations of...Ch. 3.8 - Prob. 68ECh. 3.8 - Identifying functions from an equation The...Ch. 3.8 - Prob. 70ECh. 3.8 - Prob. 71ECh. 3.8 - Prob. 72ECh. 3.8 - Normal lines A normal line at a point P on a curve...Ch. 3.8 - Normal lines A normal line at a point P on a curve...Ch. 3.8 - Normal lines A normal line at a point P on a curve...Ch. 3.8 - Normal lines A normal line at a point P on a curve...Ch. 3.8 - Prob. 77ECh. 3.8 - Normal lines A normal line at a point P on a curve...Ch. 3.8 - Prob. 79ECh. 3.8 - Visualizing tangent and normal lines a. Determine...Ch. 3.8 - Visualizing tangent and normal lines a. Determine...Ch. 3.8 - Prob. 82ECh. 3.8 - Orthogonal trajectories Two curves are orthogonal...Ch. 3.8 - Orthogonal trajectories Two curves are orthogonal...Ch. 3.8 - Orthogonal trajectories Two curves are orthogonal...Ch. 3.8 - Finding slope Find the slope of the curve...Ch. 3.8 - A challenging derivative Find dydx, where (x2 +...Ch. 3.8 - Prob. 88ECh. 3.8 - A challenging derivative Find d2ydx2, where...Ch. 3.8 - Work carefully Proceed with caution when using...Ch. 3.8 - Work carefully Proceed with caution when using...Ch. 3.8 - Work carefully Proceed with caution when using...Ch. 3.8 - Work carefully Proceed with caution when using...Ch. 3.9 - Quick Check 1
Simplify e2 ln x. Express 5x using...Ch. 3.9 - Find ddx(lnxp), where x 0 and p is a real number...Ch. 3.9 - Prob. 3QCCh. 3.9 - Prob. 4QCCh. 3.9 - Show that the derivative computed in Example 7b...Ch. 3.9 - Use x = ey to explain why ddx(lnx)=1x, for x 0.Ch. 3.9 - Prob. 2ECh. 3.9 - Show that ddx(lnkx)=ddx(lnx), where x 0 and k is...Ch. 3.9 - State the derivative rule for the exponential...Ch. 3.9 - State the derivative rule for the logarithmic...Ch. 3.9 - Explain why bx = ex ln bCh. 3.9 - Simplify the expression exln(x2+1).Ch. 3.9 - Prob. 8ECh. 3.9 - Find ddx(lnx2+1).Ch. 3.9 - Evaluate ddx(xe+ex)Ch. 3.9 - Express the function f(x)=f(x)h(x) in terms of the...Ch. 3.9 - Prob. 12ECh. 3.9 - Find ddx(ln(xex)) without using the Chain Rule and...Ch. 3.9 - Find ddx(lnx101) without using the Chain Rule.Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives involving ln x Find the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - General Power Rule Use the General Power Rule...Ch. 3.9 - General Power Rule Use the General Power Rule...Ch. 3.9 - Derivatives of bx Find the derivatives of the...Ch. 3.9 - Derivatives of bx Find the derivatives of the...Ch. 3.9 - Derivatives of bx Find the derivatives of the...Ch. 3.9 - Derivatives of bx Find the derivatives of the...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - Derivatives of bx Find the derivatives of the...Ch. 3.9 - Derivatives of bx Find the derivatives of the...Ch. 3.9 - Derivatives of bx Find the derivatives of the...Ch. 3.9 - Derivatives Find the derivative of the following...Ch. 3.9 - General Power Rule Use the General Power Rule...Ch. 3.9 - General Power Rule Use the General Power Rule...Ch. 3.9 - Derivatives of Tower Functions (or gh) Find the...Ch. 3.9 - Derivatives of Tower Functions (or gh) Find the...Ch. 3.9 - Derivatives of Tower Functions (or gh) Find the...Ch. 3.9 - Derivatives of Tower Functions (or gh) Find the...Ch. 3.9 - Derivatives of Tower Functions (or gh) Find the...Ch. 3.9 - Derivatives of Tower Functions (or gh) Find the...Ch. 3.9 - Derivatives of lower function(or gh) Find the...Ch. 3.9 - Magnitude of an earthquake The energy (in joules)...Ch. 3.9 - Exponential model The following table shows the...Ch. 3.9 - Diagnostic scanning Iodine-123 is a radioactive...Ch. 3.9 - Find an equation of the line tangent to y = xsin x...Ch. 3.9 - Determine whether the graph of y=xx has any...Ch. 3.9 - The graph of y = (x2)x has two horizontal tangent...Ch. 3.9 - The graph of y = xln x has one horizontal tangent...Ch. 3.9 - Derivatives of logarithmic functions Calculate the...Ch. 3.9 - Derivatives of logarithmic functions Calculate the...Ch. 3.9 - Derivatives of logarithmic functions Calculate the...Ch. 3.9 - Derivatives of logarithmic functions Calculate the...Ch. 3.9 - Derivatives of logarithmic functions Calculate the...Ch. 3.9 - Derivatives of logarithmic functions Calculate the...Ch. 3.9 - Derivatives of logarithmic functions Use the...Ch. 3.9 - Derivatives of logarithmic functions Use the...Ch. 3.9 - Derivatives of logarithmic functions Use the...Ch. 3.9 - Derivatives of logarithmic functions Use the...Ch. 3.9 - Derivatives of logarithmic functions Use the...Ch. 3.9 - Derivatives of logarithmic functions Use the...Ch. 3.9 - General logarithmic and exponential derivatives...Ch. 3.9 - General logarithmic and exponential derivatives...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - General logarithmic and exponential derivatives...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Logarithmic differentiation Use logarithmic...Ch. 3.9 - Explain why or why not Determine whether the...Ch. 3.9 - Higher-order derivatives Find the following...Ch. 3.9 - Higher-order derivatives Find the following...Ch. 3.9 - Higher-order derivatives Find the following...Ch. 3.9 - Higher-order derivatives Find the following...Ch. 3.9 - Derivatives by different methods Calculate the...Ch. 3.9 - Derivatives by different methods Calculate the...Ch. 3.9 - Derivatives by different methods Calculate the...Ch. 3.9 - Tangent lines Find the equation of the line...Ch. 3.9 - Horizontal tangents The graph of y = cos x ln...Ch. 3.9 - Logistic growth Scientists often use the logistic...Ch. 3.9 - Logistic growth Scientists often use the logistic...Ch. 3.9 - World population (part 2) The relative growth rate...Ch. 3.9 - Logistic growth Scientists often use the logistic...Ch. 3.9 - Savings plan Beginning at age 30, a self-employed...Ch. 3.9 - Tangency question It is easily verified that the...Ch. 3.9 - Tangency question It is easily verified that the...Ch. 3.9 - Triple intersection Graph the functions f(x) = x3,...Ch. 3.9 - Calculating limits exactly Use the definition of...Ch. 3.9 - Calculating limits exactly Use the definition of...Ch. 3.9 - Calculating limits exactly Use the definition of...Ch. 3.9 - Calculating limits exactly Use the definition of...Ch. 3.9 - Derivative of u(x)v(x) Use logarithmic...Ch. 3.9 - Tangent lines and exponentials. Assume b is given...Ch. 3.10 - Is f(x) = sin1x an even or odd function? Is f(x)...Ch. 3.10 - How do the slopes of the lines tangent to the...Ch. 3.10 - Summarize how the derivatives of inverse...Ch. 3.10 - Example 3 makes the claim that d/ds = 0.0024...Ch. 3.10 - Sketch the graphs of y = sin x and y = sin1x. Then...Ch. 3.10 - State the derivative formulas for sin1 x, tan1 x,...Ch. 3.10 - What is the slope of the line tangent to the graph...Ch. 3.10 - What is the slope of the line tangent to the graph...Ch. 3.10 - How are the derivatives of sin1 x and cos1 x...Ch. 3.10 - Suppose f is a one-to-one function with f(2) = 8...Ch. 3.10 - Explain how to find (f1)(y0), given that y0 =...Ch. 3.10 - Derivatives of inverse functions from a table Use...Ch. 3.10 - Derivatives of inverse functions from a table Use...Ch. 3.10 - If f is a one-to-one function with f(3) = 8 and...Ch. 3.10 - The line tangent to the graph of the one-to-one...Ch. 3.10 - Find the slope of the curve y = sin1x at...Ch. 3.10 - Prob. 12ECh. 3.10 - Derivatives of inverse sine Evaluate the...Ch. 3.10 - Derivatives of inverse sine Evaluate the...Ch. 3.10 - Derivatives of inverse sine Evaluate the...Ch. 3.10 - Derivatives of inverse sine Evaluate the...Ch. 3.10 - Derivatives of inverse sine Evaluate the...Ch. 3.10 - Derivatives of inverse sine Evaluate the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Evaluate the derivative of the following...Ch. 3.10 - Prob. 21ECh. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Evaluate the derivative of the following...Ch. 3.10 - Prob. 26ECh. 3.10 - Evaluate the derivative of the following...Ch. 3.10 - Evaluate the derivative of the following...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Prob. 30ECh. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Derivatives Evaluate the derivatives of the...Ch. 3.10 - Tangent lines Find an equation of the line tangent...Ch. 3.10 - Tangent lines Find an equation of the line tangent...Ch. 3.10 - Tangent lines Find an equation of the line tangent...Ch. 3.10 - Tangent lines Find an equation of the line tangent...Ch. 3.10 - Angular size A boat sails directly toward a...Ch. 3.10 - Prob. 46ECh. 3.10 - Derivatives of inverse functions at a point Find...Ch. 3.10 - Derivatives of inverse functions at a point Find...Ch. 3.10 - Derivatives of inverse functions at a point...Ch. 3.10 - Derivatives of inverse functions at a point Find...Ch. 3.10 - Derivatives of inverse functions at a point Find...Ch. 3.10 - Derivatives of inverse functions at a point Find...Ch. 3.10 - Derivatives of inverse functions at a point...Ch. 3.10 - Derivatives of inverse functions at a point...Ch. 3.10 - Derivatives of inverse functions at a point...Ch. 3.10 - Derivatives of inverse functions at a point...Ch. 3.10 - Find (f1)(3), where f(3)=x3+x+1.Ch. 3.10 - Suppose the slope of the curve y = f(x) at (7, 0)...Ch. 3.10 - Suppose the slope of the curve y = f1(x) at (4. 7)...Ch. 3.10 - Prob. 60ECh. 3.10 - Explain why or why not Determine whether the...Ch. 3.10 - Prob. 62ECh. 3.10 - Graphing f and f a. Graph f with a graphing...Ch. 3.10 - Prob. 64ECh. 3.10 - Prob. 65ECh. 3.10 - Prob. 66ECh. 3.10 - Derivatives of inverse functions Consider the...Ch. 3.10 - Prob. 68ECh. 3.10 - Prob. 69ECh. 3.10 - Prob. 70ECh. 3.10 - Derivatives of inverse functions Consider the...Ch. 3.10 - Derivatives of inverse functions Consider the...Ch. 3.10 - Derivatives of inverse functions Consider the...Ch. 3.10 - Derivatives of inverse functions Consider the...Ch. 3.10 - Derivatives of inverse functions Consider the...Ch. 3.10 - Derivatives of inverse functions Consider the...Ch. 3.10 - Prob. 77ECh. 3.10 - Prob. 78ECh. 3.10 - Towing a boat A boat is towed toward a dock by a...Ch. 3.10 - Tracking a dive A biologist standing at the bottom...Ch. 3.10 - Angle to a particle, part I A particle travels...Ch. 3.10 - Angle to a particle (part 2) The figure in...Ch. 3.10 - Derivative of the inverse sine Find the derivative...Ch. 3.10 - Derivative of the inverse cosine Find the...Ch. 3.10 - Prob. 85ECh. 3.10 - Prob. 86ECh. 3.10 - Identity proofs Prove the following identities and...Ch. 3.10 - Identity proofs Prove the following identities and...Ch. 3.10 - Identity proofs Prove the following identities and...Ch. 3.10 - Prob. 90ECh. 3.11 - In Example 1, what is the rate of change of the...Ch. 3.11 - Assuming the same pane speeds as In Example 2, how...Ch. 3.11 - In Example 3, what is the rate of change of the...Ch. 3.11 - In Example 4, notice that as the balloon rises (as...Ch. 3.11 - Give an example in which one dimension of a...Ch. 3.11 - Charles law states that for a fixed mass of gas...Ch. 3.11 - If two opposite sides of a rectangle increase in...Ch. 3.11 - The temperature F in degrees Fahrenheit is related...Ch. 3.11 - A rectangular swimming pool 10 ft wide by 20 ft...Ch. 3.11 - At all times, the length of a rectangle is twice...Ch. 3.11 - The volume V of a sphere of radius r changes over...Ch. 3.11 - At all times, the length of the long leg of a...Ch. 3.11 - Assume x, y, and z are functions of t with z=x+y3....Ch. 3.11 - Prob. 10ECh. 3.11 - Expanding square The sides of a square increase in...Ch. 3.11 - Shrinking square The sides of a square decrease in...Ch. 3.11 - Expanding isosceles triangle The legs of an...Ch. 3.11 - Shrinking isosceles triangle The hypotenuse of an...Ch. 3.11 - Expanding circle The area of a circle increases at...Ch. 3.11 - Prob. 16ECh. 3.11 - Shrinking circle A circle has an initial radius of...Ch. 3.11 - Prob. 18ECh. 3.11 - Balloons A spherical balloon is inflated and its...Ch. 3.11 - Expanding rectangle A rectangle initially has...Ch. 3.11 - Melting snowball A spherical snowball melts at a...Ch. 3.11 - Divergent paths Two beats leave a pert at the same...Ch. 3.11 - Time-lagged flights An airliner passes over an...Ch. 3.11 - Flying a kite Once Kates kite reaches a height of...Ch. 3.11 - Rope on a boat A rope passing through a capstan on...Ch. 3.11 - Bug on a parabola A bug is moving along the right...Ch. 3.11 - Another balloon story A hot-air balloon is 150 ft...Ch. 3.11 - Baseball runners Runners stand at first and second...Ch. 3.11 - Another fishing story An angler hooks a trout and...Ch. 3.11 - Parabolic motion An arrow is shot into the air and...Ch. 3.11 - Draining a water heater A water heater that has...Ch. 3.11 - Drinking a soda At what rate is soda being sucked...Ch. 3.11 - Piston compression A piston is seated at the top...Ch. 3.11 - Filling two pools Two cylindrical swimming pools...Ch. 3.11 - Growing sandpile Sand falls from an overhead bin...Ch. 3.11 - Draining a tank An inverted conical water tank...Ch. 3.11 - Prob. 37ECh. 3.11 - Two tanks A conical tank with an upper radius of 4...Ch. 3.11 - Filling a hemispherical tank A hemispherical tank...Ch. 3.11 - Surface area of hemispherical tank Per the...Ch. 3.11 - Ladder against the wall A 13-foot ladder is...Ch. 3.11 - Ladder against the wall again A 12-foot ladder is...Ch. 3.11 - Moving shadow A 5-foot-tall woman walks at 8 ft/s...Ch. 3.11 - Another moving shadow A landscape light at ground...Ch. 3.11 - Watching an elevator An observer is 20 m above the...Ch. 3.11 - Observing a launch An observer stands 300 ft from...Ch. 3.11 - Viewing angle The bottom of a large theater screen...Ch. 3.11 - Altitude of a jet A jet ascends at a 10 angle from...Ch. 3.11 - Rate of dive of a submarine A surface ship is...Ch. 3.11 - A lighthouse problem A lighthouse stands 500 m off...Ch. 3.11 - Filming a race A camera is set up at the starting...Ch. 3.11 - Fishing reel An angler hooks a trout and begins...Ch. 3.11 - Wind energy The kinetic energy E (in joules) of a...Ch. 3.11 - Fishing reel An angler hooks a trout and begins...Ch. 3.11 - Clock hands The hands of the clock in the tower of...Ch. 3.11 - Divergent paths Two boats leave a port at the same...Ch. 3.11 - Filling a pool A swimming pool is 50 m long and 20...Ch. 3.11 - Disappearing triangle An equilateral triangle...Ch. 3.11 - Oblique tracking A port and a radar station are 2...Ch. 3.11 - Oblique tracking A ship leaves port traveling...Ch. 3.11 - Prob. 61ECh. 3.11 - Watching a Ferris wheel An observer stands 20 m...Ch. 3.11 - Draining a trough A trough in the shape of a half...Ch. 3.11 - Searchlightwide beam A revolving searchlight,...Ch. 3 - Explain why or why not Determine whether the...Ch. 3 - Evaluate the derivative of each of the following...Ch. 3 - Evaluate the derivative of each of the following...Ch. 3 - Evaluate the derivative of each of the following...Ch. 3 - Use differentiation to verify each equation....Ch. 3 - Use differentiation to verify each equation....Ch. 3 - Use differentiation to verify each equation....Ch. 3 - Use differentiation to verify each equation....Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 10.y=4x4lnxx4Ch. 3 - Evaluate and simplify y'. 11.y=2xCh. 3 - Evaluate and simplify y'. 12.y=2x2Ch. 3 - Evaluate and simplify y'. 13.y=e2Ch. 3 - Evaluate and simplify y'. 14.y=(2x3)x3/2Ch. 3 - Evaluate and simplify y'. 15.y=(1+x4)3/2Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 19.y=ex(x2+2x+2)Ch. 3 - Evaluate and simplify y'. 20.y=lnxlnx+a, where a...Ch. 3 - Evaluate and simplify y'. 21.y=sec2wsec2w+1Ch. 3 - Evaluate and simplify y'. 22.y=(sinxcosx+1)1/3Ch. 3 - Evaluate and simplify y'. 23.y=ln|sec3x|Ch. 3 - Evaluate and simplify y'. 24.y=ln|csc7x+cot7x|Ch. 3 - Evaluate and simplify y'. 25.y=(5t2+10)100Ch. 3 - Evaluate and simplify y'. 26.y=esinx+2x+1Ch. 3 - Evaluate and simplify y'. 27.y=ln(sinx3)Ch. 3 - Evaluate and simplify y'. 28.y=etanx(tanx1)Ch. 3 - Evaluate and simplify y'. 29.y=tan1t21Ch. 3 - Evaluate and simplify y'. 30.y=xx+1Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 33.y=lnww5Ch. 3 - Evaluate and simplify y'. 34.y=seas, where a is a...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 38.y=(vv+1)4/3Ch. 3 - Evaluate and simplify y'. 39.y=sincos2x+1Ch. 3 - Evaluate and simplify y'. 40.y=esin(cosx)Ch. 3 - Evaluate and simplify y'. 41.y=lnet+1Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 43.y=x2+2xtan1(cotx)Ch. 3 - Evaluate and simplify y'. 44.y=1x4+x2sin1x2Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 46.y=e6xsinxCh. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 48.y=10sinx+sin10xCh. 3 - Evaluate and simplify y'. 49.y=(x2+1)lnxCh. 3 - Evaluate and simplify y'. 50.y=xcos2xCh. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluate and simplify y'. 53.y=6xcot13x+ln(9x2+1)Ch. 3 - Evaluate and simplify y'. 54.y=2x2cos1x+sin1xCh. 3 - Evaluate and simplify y'. 55.x=cos(xy)Ch. 3 - Evaluate and simplify y'. 56.xy4+x4y=1Ch. 3 - Implicit differentiation Calculate y(x) for the...Ch. 3 - Implicit differentiation Calculate y(x) for the...Ch. 3 - Implicit differentiation Calculate y(x) for the...Ch. 3 - Evaluate and simplify y'....Ch. 3 - Evaluate and simplify y'....Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Evaluating derivatives Evaluate and simplify the...Ch. 3 - Higher-order derivatives Find and simplify y....Ch. 3 - Higher-order derivatives Find and simplify y....Ch. 3 - Higher-order derivatives Find and simplify y....Ch. 3 - Higher-order derivatives Find and simplify y....Ch. 3 - Higher-order derivatives Find and simplify y....Ch. 3 - Higher-order derivatives Find and simplify y....Ch. 3 - Tangent lines Find an equation of the line tangent...Ch. 3 - Tangent lines Find an equation of the line tangent...Ch. 3 - Tangent lines Find an equation of the line tangent...Ch. 3 - Tangent lines Find an equation of the line tangent...Ch. 3 - Tangent lines Find an equation of the line tangent...Ch. 3 - Derivative formulas Evaluate the following...Ch. 3 - Derivative formulas Evaluate the fallowing...Ch. 3 - Derivative formulas Evaluate the following...Ch. 3 - Derivative formulas Evaluate the following...Ch. 3 - Matching functions and derivatives Match the...Ch. 3 - Sketching a derivative graph Sketch a graph of f...Ch. 3 - Sketching a derivative graph Sketch a graph of g...Ch. 3 - Use the given graphs of f and g to find each...Ch. 3 - Finding derivatives from a table Find the values...Ch. 3 - Derivative of the inverse at a point Consider the...Ch. 3 - Derivative of the inverse at a point Consider the...Ch. 3 - Derivative of the inverse Find the derivative of...Ch. 3 - Derivative of the inverse Find the derivative of...Ch. 3 - Derivatives from a graph If possible, evaluate the...Ch. 3 - Derivatives from a graph If possible, evaluate the...Ch. 3 - The line tangent to y=f(x) at x = 3 is y=4x10 and...Ch. 3 - The line tangent to y=f(x) at x = 3 is y=4x10 and...Ch. 3 - Horizontal motion The position of an object moving...Ch. 3 - Projectile on Mars Suppose an object is fired...Ch. 3 - Beak length The length of the culmen (the upper...Ch. 3 - Prob. 97RECh. 3 - Antibiotic decay The half-life of an antibiotic in...Ch. 3 - Population of the United States The population of...Ch. 3 - Growth rate of bacteria Suppose the following...Ch. 3 - Velocity of a skydiver Assume the graph represents...Ch. 3 - A function and its inverse function The function...Ch. 3 - Prob. 103RECh. 3 - Limits The following limits represent the...Ch. 3 - Limits The following limits represent the...Ch. 3 - Velocity of a probe A small probe is launched...Ch. 3 - Prob. 107RECh. 3 - Marginal and average cost Suppose a company...Ch. 3 - Population growth Suppose p(t) = 1.7t3 + 72t2 +...Ch. 3 - Position of a piston The distance between the head...Ch. 3 - Boat rates Two boats leave a dock at the same...Ch. 3 - Rate of inflation of a balloon A spherical balloon...Ch. 3 - Rate of descent of a hot-air balloon A rope is...Ch. 3 - Filling a tank Water flows into a conical tank at...Ch. 3 - Angle of elevation A jet flies horizontally 500 ft...Ch. 3 - Viewing angle A man whose eye level is 6 ft above...Ch. 3 - Shadow length A street light is fastened to the...Ch. 3 - Quadratic functions a. Show that if (a, f(a)) is...Ch. 3 - Derivative of the inverse in two ways Let...Ch. 3 - A parabola property Let f(x) = x2. a. Show that...
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