Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
3rd Edition
ISBN: 9780134995991
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Textbook Question
Chapter 3.6, Problem 57E
A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and φ(t), respectively, where 0 ≤ t ≤ 4 and t is measured in minutes (see figure). These angles are measured in radians, where θ = φ = 0 represent the starting position and θ = φ = 2π represent the finish position. The angular velocities of the runners are θ′(t) and φ′(t).
- a. Compare in words the angular velocity of the two runners and the progress of the race.
- b. Which runner has the greater average angular velocity?
- c. Who wins the race?
- d. Jean’s position is given by θ(t) = πt2/8. What is her angular velocity at t = 2 and at what time is her angular velocity the greatest?
- e. Juan’s position is given by φ(t) = πt(8 − t)/8. What is his angular velocity at t = 2 and at what time is his angular velocity the greatest?
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Chapter 3 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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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