Thomas' Calculus (14th Edition)
14th Edition
ISBN: 9780134438986
Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher: PEARSON
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Chapter 3, Problem 11AAE
(a)
To determine
Calculate the time taken for the paper clip to reach the maximum height and the initial velocity of the paper clip when it leaves the hand.
(b)
To determine
Calculate the time taken for the paper clip to reach the maximum height and the maximum height reached.
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1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.
Chapter 3 Solutions
Thomas' Calculus (14th Edition)
Ch. 3.1 - In Exercises 1–4, use the grid and a straight edge...Ch. 3.1 - In Exercises 1-4, use the grid and a straight edge...Ch. 3.1 - In Exercises 1−4, use the grid and a straight edge...Ch. 3.1 - In Exercises 1−4, use the grid and a straight edge...Ch. 3.1 - In Exercises 510, find an equation for the tangent...Ch. 3.1 - In Exercises 510, find an equation for the tangent...Ch. 3.1 - In Exercises 510, find an equation for the tangent...Ch. 3.1 - In Exercises 510, find an equation for the tangent...Ch. 3.1 - In Exercises 510, find an equation for the tangent...Ch. 3.1 - In Exercises 510, find an equation for the tangent...
Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 11–18, find the slope of the...Ch. 3.1 - In Exercises 19–22, find the slope of the curve at...Ch. 3.1 - Prob. 20ECh. 3.1 - In Exercises 19–22, find the slope of the curve at...Ch. 3.1 - In Exercises 19–22, find the slope of the curve at...Ch. 3.1 - Interpreting Derivative Values
Growth of yeast...Ch. 3.1 - Effectiveness of a drug On a scale from 0 to 1,...Ch. 3.1 - At what points do the graphs of the functions in...Ch. 3.1 - At what points do the graphs of the functions in...Ch. 3.1 - Find equations of all lines having slope −1...Ch. 3.1 - Find an equation of the straight line having slope...Ch. 3.1 - Object dropped from a tower An object is dropped...Ch. 3.1 - Speed of a rocket At t sec after liftoff, the...Ch. 3.1 - Circle’s changing area What is the rate of change...Ch. 3.1 - Prob. 32ECh. 3.1 - Show that the line y = mx + b is its own tangent...Ch. 3.1 - Find the slope of the tangent line to the curve y...Ch. 3.1 - Does the graph of
have a tangent line at the...Ch. 3.1 - Does the graph of
have a tangent line at the...Ch. 3.1 - We say that a continuous curve y = f(x) has a...Ch. 3.1 - We say that a continuous curve y = f(x) has a...Ch. 3.1 - Graph the curves in Exercises 39–48.
Where do the...Ch. 3.1 - Graph the curves in Exercises 39–48.
Where do the...Ch. 3.1 - Prob. 41ECh. 3.1 - Graph the curves in Exercises 39–48.
Where do the...Ch. 3.1 - Graph the curves in Exercises 39–48.
Where do the...Ch. 3.1 - Graph the curves in Exercises 39–48.
Where do the...Ch. 3.1 - Prob. 45ECh. 3.1 - Prob. 46ECh. 3.1 - Graph the curves in Exercises 39–48.
Where do the...Ch. 3.1 - Graph the curves in Exercises 39–48.
Where do the...Ch. 3.2 - Using the definition, calculate the derivatives of...Ch. 3.2 - Using the definition, calculate the derivatives of...Ch. 3.2 - Using the definition, calculate the derivatives of...Ch. 3.2 - Using the definition, calculate the derivatives of...Ch. 3.2 - Using the definition, calculate the derivatives of...Ch. 3.2 - Using the definition, calculate the derivatives of...Ch. 3.2 - In Exercises 7–12, find the indicated...Ch. 3.2 - Prob. 8ECh. 3.2 - In Exercises 7–12, find the indicated...Ch. 3.2 - In Exercises 7–12, find the indicated...Ch. 3.2 - In Exercises 7–12, find the indicated...Ch. 3.2 - Prob. 12ECh. 3.2 - In Exercises 13–16, differentiate the functions...Ch. 3.2 - Prob. 14ECh. 3.2 - In Exercises 13−16, differentiate the functions...Ch. 3.2 - In Exercises 13–16, differentiate the functions...Ch. 3.2 - In Exercises 17–18, differentiate the functions....Ch. 3.2 - In Exercises 17–18, differentiate the functions....Ch. 3.2 - In Exercises 19–22, find the values of the...Ch. 3.2 - Prob. 20ECh. 3.2 - In Exercises 19–22, find the values of the...Ch. 3.2 - In Exercises 19-22, find the values of the...Ch. 3.2 - Use the formula
to find the derivative of the...Ch. 3.2 - Use the formula
to find the derivative of the...Ch. 3.2 - Prob. 25ECh. 3.2 - Prob. 26ECh. 3.2 - Match the functions graphed in Exercises 27–30...Ch. 3.2 - Match the functions graphed in Exercises 27–30...Ch. 3.2 - Match the functions graphed in Exercises 27–30...Ch. 3.2 - Match the functions graphed in Exercises 27–30...Ch. 3.2 - Consider the function f graphed here. The domain...Ch. 3.2 - Prob. 32ECh. 3.2 - Growth in the economy The graph in the...Ch. 3.2 - Prob. 34ECh. 3.2 - Temperature The given graph shows the outside...Ch. 3.2 - Average single-family home prices P (in thousands...Ch. 3.2 - Prob. 37ECh. 3.2 - Prob. 38ECh. 3.2 - Compute the right-hand and left-hand derivatives...Ch. 3.2 - Compute the right-hand and left-hand derivatives...Ch. 3.2 - In Exercises 41–44, determine whether the...Ch. 3.2 - Prob. 42ECh. 3.2 - In Exercises 41–44, determine whether the...Ch. 3.2 - Prob. 44ECh. 3.2 - Prob. 45ECh. 3.2 - Each figure in Exercises 45-50 shows the graph of...Ch. 3.2 - Each figure in Exercises 45-50 shows the graph of...Ch. 3.2 - Each figure in Exercises 45–50 shows the graph of...Ch. 3.2 - Prob. 49ECh. 3.2 - Prob. 50ECh. 3.2 - In Exercises 51–54,
Find the derivative f'(x) of...Ch. 3.2 - Prob. 52ECh. 3.2 - Prob. 53ECh. 3.2 - Prob. 54ECh. 3.2 - Tangent line to a parabola Does the parabola y =...Ch. 3.2 - Tangent line to Does any tangent line to the...Ch. 3.2 - Derivative of – f Does knowing that a function...Ch. 3.2 - Prob. 58ECh. 3.2 - Prob. 59ECh. 3.2 - Prob. 60ECh. 3.2 - Prob. 61ECh. 3.2 - Prob. 62ECh. 3.2 - Prob. 63ECh. 3.2 - Prob. 64ECh. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Prob. 9ECh. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - Derivative Calculations
In Exercises 112, find the...Ch. 3.3 - In Exercises 1316, find y'(a) by applying the...Ch. 3.3 - Prob. 14ECh. 3.3 - Prob. 15ECh. 3.3 - In Exercises 1316, find y' (a) by applying the...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Prob. 23ECh. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of the functions in Exercises...Ch. 3.3 - Find the derivatives of all orders of the...Ch. 3.3 - Find the derivatives of all orders of the...Ch. 3.3 - Prob. 31ECh. 3.3 - Prob. 32ECh. 3.3 - Find the first and second derivatives of the...Ch. 3.3 - Find the first and second derivatives of the...Ch. 3.3 - Prob. 35ECh. 3.3 - Prob. 36ECh. 3.3 - Prob. 37ECh. 3.3 - Find the first and second derivatives of the...Ch. 3.3 - Suppose u and v are functions of x that are...Ch. 3.3 - Prob. 40ECh. 3.3 - Normal line to a curve Find an equation for the...Ch. 3.3 - Horizontal tangent lines Find equations for the...Ch. 3.3 - Find the tangent lines to Newton’s serpentine...Ch. 3.3 - Find the tangent line to the Witch of Agnesi...Ch. 3.3 - Quadratic tangent to identity function The curve y...Ch. 3.3 - Quadratics having a common tangent line The curves...Ch. 3.3 - Find all points (x, y) on the graph of f(x) = 3x2...Ch. 3.3 - Find all points (x, y) on the graph of with...Ch. 3.3 - Find all points (x, y) on the graph of y = x/(x −...Ch. 3.3 - Find all points (x, y) on the graph of f(x) = x2...Ch. 3.3 - Assume that functions f and g are differentiable...Ch. 3.3 - Assume that functions f and g are differentiable...Ch. 3.3 - Prob. 53ECh. 3.3 - Find an equation for the line that is tangent to...Ch. 3.3 - For Exercises 69 and 70, evaluate each limit by...Ch. 3.3 - For Exercises 69 and 70, evaluate each limit by...Ch. 3.3 - Find the value of a that makes the following...Ch. 3.3 - Find the values of a and b that make the following...Ch. 3.3 - Prob. 59ECh. 3.3 - The body’s reaction to medicine The reaction of...Ch. 3.3 - Prob. 61ECh. 3.3 - Prob. 62ECh. 3.3 - Generalizing the Product Rule The Derivative...Ch. 3.3 - Prob. 64ECh. 3.3 - Cylinder pressure If gas in a cylinder is...Ch. 3.3 - The best quantity to order One of the formulas for...Ch. 3.4 - Exercises 1–6 give the positions s = f(t) of a...Ch. 3.4 - Exercises 1–6 give the positions s = f(t) of a...Ch. 3.4 - Exercises 1–6 give the positions s = f(t) of a...Ch. 3.4 - Exercises 1–6 give the positions s = f(t) of a...Ch. 3.4 - Exercises 1–6 give the positions s = f(t) of a...Ch. 3.4 - Exercises 1–6 give the positions s = f(t) of a...Ch. 3.4 - Particle motion At time t, the position of a body...Ch. 3.4 - Particle motion At time t ≥ 0, the velocity of a...Ch. 3.4 - Free fall on Mars and Jupiter The equations for...Ch. 3.4 - Lunar projectile motion A rock thrown vertically...Ch. 3.4 - Prob. 11ECh. 3.4 - Speeding bullet A 45-caliber bullet shot straight...Ch. 3.4 - Prob. 13ECh. 3.4 - Galileo's free-fall formula Galileo developed a...Ch. 3.4 - The accompanying figure shows the velocity v =...Ch. 3.4 - Prob. 16ECh. 3.4 - Prob. 17ECh. 3.4 - The accompanying figure shows the velocity v =...Ch. 3.4 - Two falling balls The multiflash photograph in the...Ch. 3.4 - Prob. 20ECh. 3.4 - The graphs in the accompanying figure show the...Ch. 3.4 - The graphs in the accompanying figure show the...Ch. 3.4 - Prob. 23ECh. 3.4 - Marginal revenue Suppose that the revenue from...Ch. 3.4 - Prob. 25ECh. 3.4 - Body surface area A typical male’s body surface...Ch. 3.4 - Draining a tank It takes 12 hours to drain a...Ch. 3.4 - Draining a tank The number of gallons of water in...Ch. 3.4 - Prob. 29ECh. 3.4 - Inflating a balloon The volume V = (4/3)πr3 of a...Ch. 3.4 - Airplane takeoff Suppose that the distance an...Ch. 3.4 - Volcanic lava fountains Although the November 1959...Ch. 3.4 - Exercises 33–36 give the position function of an...Ch. 3.4 - Prob. 34ECh. 3.4 - Prob. 35ECh. 3.4 - Exercises 33–36 give the position function of an...Ch. 3.5 - In Exercises 1–18, find dy/dx.
1. y = −10x + 3 cos...Ch. 3.5 - In Exercises 1–18, find dy/dx.
2.
Ch. 3.5 - Prob. 3ECh. 3.5 - In Exercises 1–18, find dy/dx.
4.
Ch. 3.5 - In Exercises 1–18, find dy/dx.
5.
Ch. 3.5 - In Exercises 1–18, find dy/dx.
6.
Ch. 3.5 - In Exercises 1–18, find dy/dx.
7. f(x) = sin x tan...Ch. 3.5 - In Exercises 1–18, find dy/dx.
8.
Ch. 3.5 - Derivatives
In Exercises 1–18, find dy/dx.
9.
Ch. 3.5 - In Exercises 1–18, find dy/dx.
10. y = (sin x +...Ch. 3.5 - In Exercises 1–18, find dy/dx.
11.
Ch. 3.5 - In Exercises 1–18, find dy/dx.
12.
Ch. 3.5 - In Exercises 1–18, find dy/dx.
13.
Ch. 3.5 - In Exercises 1–18, find dy/dx.
14.
Ch. 3.5 - In Exercises 1-18, find dy/dx.
Ch. 3.5 - In Exercises 1-18, find dy/dx.
Ch. 3.5 - In Exercises 1-18, find dy/dx.
Ch. 3.5 - Prob. 18ECh. 3.5 - In Exercises 19-22, find ds/dt.
Ch. 3.5 - In Exercises 19–22, find ds/dt.
20. s = t2 − sec t...Ch. 3.5 - In Exercises 19-22, find ds/dt.
Ch. 3.5 - In Exercises 19-22, find ds/dt.
Ch. 3.5 - In Exercises 23-26, find dr/dθ.
Ch. 3.5 - In Exercises 23-26, find dr/dθ.
Ch. 3.5 - Prob. 25ECh. 3.5 - Prob. 26ECh. 3.5 - In Exercises 27–32, find dp/dq.
27.
Ch. 3.5 - Prob. 28ECh. 3.5 - Prob. 29ECh. 3.5 - In Exercises 27-32, find dp/dq.
30.
Ch. 3.5 - In Exercises 27–32, find dp/dq.
31.
Ch. 3.5 - In Exercises 27–32. Find dp/dq.
32.
Ch. 3.5 - Prob. 33ECh. 3.5 - Find y(4) = d4y/dx4 if
y = −2 sin x.
y = 9 cos x.
Ch. 3.5 - In Exercises 35–38, graph the curves over the...Ch. 3.5 - Prob. 36ECh. 3.5 - Prob. 37ECh. 3.5 - In Exercises 35–38, graph the curves over the...Ch. 3.5 - Do the graphs of the functions in Exercises 39–44...Ch. 3.5 - Do the graphs of the functions in Exercises 39–44...Ch. 3.5 - Prob. 41ECh. 3.5 - Prob. 42ECh. 3.5 - Prob. 43ECh. 3.5 - Do the graphs of the functions in Exercises 39–44...Ch. 3.5 - Find all points on the curve y = tan x, −π/2 < x <...Ch. 3.5 - Find all points on the curve y = cot x, 0 < x < π,...Ch. 3.5 - In Exercises 47 and 48, find an equation for (a)...Ch. 3.5 - Prob. 48ECh. 3.5 - Find the limits in Exercises 49–56.
49.
Ch. 3.5 - Find the limits in Exercises 49–56.
50.
Ch. 3.5 - Find the limits in Exercises 49–56.
51.
Ch. 3.5 - Find the limits in Exercises 49–56.
52.
Ch. 3.5 - Trigonometric Limits
Find the limits in Exercises...Ch. 3.5 - Prob. 54ECh. 3.5 - Prob. 55ECh. 3.5 - Find the limits in Exercises 49–56.
56.
Ch. 3.5 - The equations in Exercises 57 and 58 give the...Ch. 3.5 - The equations in Exercises 57 and 58 give the...Ch. 3.5 - Is there a value of c that will make
continuous...Ch. 3.5 - Is there a value of b that will make
continuous...Ch. 3.5 - By computing the first few derivatives and looking...Ch. 3.5 - Derive the formula for the derivative with respect...Ch. 3.5 - A weight is attached to a spring and reaches its...Ch. 3.5 - Prob. 64ECh. 3.5 - Prob. 65ECh. 3.5 - Prob. 66ECh. 3.5 - Prob. 67ECh. 3.5 - A caution about centered difference...Ch. 3.5 - Prob. 69ECh. 3.5 - Prob. 70ECh. 3.5 - Prob. 71ECh. 3.5 - Prob. 72ECh. 3.6 - Derivative Calculations
In Exercises 1–8, given y...Ch. 3.6 - Derivative Calculations
In Exercises 1–8, given y...Ch. 3.6 - Derivative Calculation
In Exercises 1–8, given y =...Ch. 3.6 - Prob. 4ECh. 3.6 - Prob. 5ECh. 3.6 - Derivation Calculations
In Exercises 1–8, given y...Ch. 3.6 - Prob. 7ECh. 3.6 - Prob. 8ECh. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - In Exercises 9–22, write the function in the form...Ch. 3.6 - Prob. 18ECh. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Prob. 21ECh. 3.6 - Prob. 22ECh. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Prob. 25ECh. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Prob. 27ECh. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Prob. 31ECh. 3.6 - Prob. 32ECh. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - Prob. 36ECh. 3.6 - Prob. 37ECh. 3.6 - Prob. 38ECh. 3.6 - Prob. 39ECh. 3.6 - Find the derivatives of the functions in Exercises...Ch. 3.6 - In Exercises 51–70, find dy/dt.
51. y = sin2(πt −...Ch. 3.6 - In Exercises 51–70, find dy/dt.
52. y = sec2 πt
Ch. 3.6 - Prob. 43ECh. 3.6 - In Exercises 51–70, find dy/dt.
54. y = (1 + cot...Ch. 3.6 - Prob. 45ECh. 3.6 - In Exercises 51–70, find dy/dt.
56. y = (t−3/4 sin...Ch. 3.6 - In Exercises 51–70, find dy/dt.
59.
Ch. 3.6 - In Exercises 51–70, find dy/dt.
60.
Ch. 3.6 - In Exercises 51–70, find dy/dt.
61. y = sin (cos...Ch. 3.6 - In Exercises 51–70, find dy/dt.
62.
Ch. 3.6 - In Exercises 51–70, find dy/dt.
63.
Ch. 3.6 - Prob. 52ECh. 3.6 - Prob. 53ECh. 3.6 - Prob. 54ECh. 3.6 - Prob. 55ECh. 3.6 - In Exercises 51–70, find dy/dt.
68. y = cos4(sec2...Ch. 3.6 - In Exercises 51–70, find dy/dt.
69. y = 3t(2t2 −...Ch. 3.6 - In Exercises 51–70, find dy/dt.
70.
Ch. 3.6 - Second Derivatives
Find y″ in Exercises 71–78.
71....Ch. 3.6 - Prob. 60ECh. 3.6 - Second Derivatives
Find y″ in Exercises 71–78.
73....Ch. 3.6 - Second Derivatives
Find y″ in Exercises 71–78.
74....Ch. 3.6 - Prob. 63ECh. 3.6 - Second Derivatives
Find y″ in Exercises 71–78.
76....Ch. 3.6 - For each of the following functions, solve both...Ch. 3.6 - Prob. 66ECh. 3.6 - Finding Derivative values
In Exercises 81–86, find...Ch. 3.6 - Prob. 68ECh. 3.6 - Prob. 69ECh. 3.6 - Prob. 70ECh. 3.6 - Finding Derivative values
In Exercises 81–86, find...Ch. 3.6 - Finding Derivative values
In Exercises 81–86, find...Ch. 3.6 - Assume that f′(3) = −1, g′(2) = 5, g(2) = 3, and y...Ch. 3.6 - Prob. 74ECh. 3.6 - Suppose that functions f and g and their...Ch. 3.6 - Suppose that the functions f and g and their...Ch. 3.6 - Prob. 77ECh. 3.6 - Prob. 78ECh. 3.6 - Prob. 79ECh. 3.6 - What happens if you can write a function as a...Ch. 3.6 - Find the tangent line to at x= 0.
Ch. 3.6 - Find the tangent line to at x = 2.
Ch. 3.6 - Prob. 83ECh. 3.6 - Prob. 84ECh. 3.6 - Suppose that a piston is moving straight up and...Ch. 3.6 - Prob. 86ECh. 3.6 - Prob. 87ECh. 3.6 - Suppose that the velocity of a falling body is ...Ch. 3.6 - Prob. 89ECh. 3.6 - A particle moves along the x-axis with velocity...Ch. 3.6 - Prob. 91ECh. 3.6 - Prob. 92ECh. 3.6 - Prob. 93ECh. 3.6 - Prob. 94ECh. 3.6 - Prob. 95ECh. 3.6 - Prob. 96ECh. 3.6 - Consider the function
Show that f is continuous...Ch. 3.6 - Prob. 98ECh. 3.6 - Verify each of the following statements.
If f is...Ch. 3.6 - Prob. 100ECh. 3.6 - (Continuation of Exercise 100.) In Exercise 100,...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Use implicit differentiation to find dy/dx in...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Differentiating implicitly
Use implicit...Ch. 3.7 - Prob. 10ECh. 3.7 - Differentiating implicitly
Use implicit...Ch. 3.7 - Use implicit differentiation to find dy/dx in...Ch. 3.7 - Differentiating Implicitly
Use implicit...Ch. 3.7 - Use implicit differentiation to find dy/dx in...Ch. 3.7 - Find dr/dθ in Exercises 17-20.
Ch. 3.7 - Prob. 16ECh. 3.7 - Find dr / d in Exercises 17–20.
Ch. 3.7 - Find dr/dθ in Exercises 15–18.
18. cos r + cot θ =...Ch. 3.7 - In Exercises 21−28, use implicit differentiation...Ch. 3.7 - In Exercises 21−28, use implicit differentiation...Ch. 3.7 - Second Derivatives
In Exercises 19–26, use...Ch. 3.7 - In Exercises 21–28, use implicit differentiation...Ch. 3.7 - In Exercises 21–28, use implicit differentiation...Ch. 3.7 - In Exercises 21–28, use implicit differentiation...Ch. 3.7 - In Exercises 21–28, use implicit differentiation...Ch. 3.7 - In Exercises 21–28, use implicit differentiation...Ch. 3.7 - Prob. 27ECh. 3.7 - If , find the value of at the point (0, –1).
Ch. 3.7 - In Exercises 31 and 32, find the slope of the...Ch. 3.7 - In Exercises 31 and 32, find the slope of the...Ch. 3.7 - In Exercises 33−42, verify that the given point is...Ch. 3.7 - In Exercises 33−42, verify that the given point is...Ch. 3.7 - In Exercises 33−42, verify that the given point is...Ch. 3.7 - In Exercises 33−42, verify that the given point is...Ch. 3.7 - In Exercises 33−42, verify that the given point is...Ch. 3.7 - Prob. 36ECh. 3.7 - In Exercises 33−42, verify that the given point is...Ch. 3.7 - Exercises 33–42, verify that the given point is on...Ch. 3.7 - In Exercises 33−42, verify that the given point is...Ch. 3.7 - Prob. 40ECh. 3.7 - Prob. 41ECh. 3.7 - Normal lines parallel to a line Find the normal...Ch. 3.7 - Prob. 43ECh. 3.7 - Prob. 44ECh. 3.7 - The devil’s curve (Gabriel Gramer, 1750) Find the...Ch. 3.7 - The folium of Descartes (See Figure 3.27)
Find the...Ch. 3.7 - Prob. 47ECh. 3.7 - Prob. 48ECh. 3.7 - Normal lines to a parabola Show that if it is...Ch. 3.7 - Prob. 50ECh. 3.7 - Prob. 51ECh. 3.7 - The graph of is called a semicubical parabola and...Ch. 3.7 - Prob. 53ECh. 3.7 - Prob. 54ECh. 3.8 - Area Suppose that the radius r and area A = πr2 of...Ch. 3.8 - Surface area Suppose that the radius r and surface...Ch. 3.8 - Prob. 3ECh. 3.8 - Prob. 4ECh. 3.8 - Prob. 5ECh. 3.8 - Prob. 6ECh. 3.8 - Prob. 7ECh. 3.8 - Prob. 8ECh. 3.8 - Prob. 9ECh. 3.8 - Prob. 10ECh. 3.8 - If the original 24 m edge length x of a cube...Ch. 3.8 - Prob. 12ECh. 3.8 - Volume The radius r and height h of a right...Ch. 3.8 - Prob. 14ECh. 3.8 - Prob. 15ECh. 3.8 - Prob. 16ECh. 3.8 - Distance Let x and y be functions of t, and let ...Ch. 3.8 - Diagonals If x, y, and z are lengths of the edges...Ch. 3.8 - Area The area A of a triangle with sides of...Ch. 3.8 - Heating a plate When a circular plate of metal is...Ch. 3.8 - Changing dimensions in a rectangle The length l of...Ch. 3.8 - Prob. 22ECh. 3.8 - A sliding ladder A 13-ft ladder is leaning against...Ch. 3.8 - Commercial air traffic Two commercial airplanes...Ch. 3.8 - Flying a kite A girl flies a kite at a height of...Ch. 3.8 - Prob. 26ECh. 3.8 - Prob. 27ECh. 3.8 - A draining conical reservoir Water is flowing at...Ch. 3.8 - A draining hemispherical reservoir Water is...Ch. 3.8 - A growing raindrop Suppose that a drop of mist is...Ch. 3.8 - The radius of an inflating balloon A spherical...Ch. 3.8 - Hauling in a dinghy A dinghy is pulled toward a...Ch. 3.8 - A balloon and a bicycle A balloon is rising...Ch. 3.8 - Making coffee Coffee is draining from a conical...Ch. 3.8 - Prob. 35ECh. 3.8 - Prob. 36ECh. 3.8 - Prob. 37ECh. 3.8 - Videotaping a moving car You are videotaping a...Ch. 3.8 - A moving shadow A light shines from the top of a...Ch. 3.8 - Prob. 40ECh. 3.8 - Prob. 41ECh. 3.8 - Highway patrol A highway patrol plane flies 3 mi...Ch. 3.8 - Prob. 43ECh. 3.8 - Prob. 44ECh. 3.8 - Prob. 45ECh. 3.8 - Prob. 46ECh. 3.8 - A lighthouse beam A lighthouse sits 1 km offshore,...Ch. 3.9 - find the linearization L(x) of f(x) at x = a.
Ch. 3.9 - find the linearization L(x) of f(x) at x = a.
Ch. 3.9 - Prob. 3ECh. 3.9 - Prob. 4ECh. 3.9 - find the linearization L(x) of f(x) at x = a.
Ch. 3.9 - Common linear approximations at x = 0 Find the...Ch. 3.9 - find a linearization at a suitably chosen integer...Ch. 3.9 - Prob. 8ECh. 3.9 - find a linearization at a suitably chosen integer...Ch. 3.9 - find a linearization at a suitably chosen integer...Ch. 3.9 - find a linearization at a suitably chosen integer...Ch. 3.9 - Prob. 12ECh. 3.9 - Show that the linearization of f(x) = (1 + x)k at...Ch. 3.9 - Use the linear approximation to find an...Ch. 3.9 - Use the approximation (1 + x)k ≈
1 + kx to...Ch. 3.9 - Find the linearization of at x = 0. How is it...Ch. 3.9 - find dy.
Ch. 3.9 - find dy.
Ch. 3.9 - Prob. 19ECh. 3.9 - Prob. 20ECh. 3.9 - Prob. 21ECh. 3.9 - Prob. 22ECh. 3.9 - Prob. 23ECh. 3.9 - Prob. 24ECh. 3.9 - Prob. 25ECh. 3.9 - find dy.
Ch. 3.9 - Prob. 27ECh. 3.9 - find dy.
Ch. 3.9 - In Exercises 39-14, each function f(x) changes...Ch. 3.9 - Prob. 30ECh. 3.9 - Prob. 31ECh. 3.9 - Prob. 32ECh. 3.9 - Prob. 33ECh. 3.9 - Prob. 34ECh. 3.9 - In Exercises 45-50, write a differential formula...Ch. 3.9 - In Exercises 45-50, write a differential formula...Ch. 3.9 - Prob. 37ECh. 3.9 - Prob. 38ECh. 3.9 - In Exercises 45-50, write a differential formula...Ch. 3.9 - Prob. 40ECh. 3.9 - The radius of a circle is increased from 2.00 to...Ch. 3.9 - Prob. 42ECh. 3.9 - Prob. 43ECh. 3.9 - Prob. 44ECh. 3.9 - Prob. 45ECh. 3.9 - The edge x of a cube is measured with an error of...Ch. 3.9 - Prob. 47ECh. 3.9 - About how accurately must the interior diameter of...Ch. 3.9 - Prob. 49ECh. 3.9 - Prob. 50ECh. 3.9 - The effect of flight maneuvers on the heart The...Ch. 3.9 - Drug concentration The concentration C in...Ch. 3.9 - Prob. 53ECh. 3.9 - Measuring acceleration of gravity When the length...Ch. 3.9 - Quadratic approximations
Let Q(x) = b0 + b1 (x −...Ch. 3.9 - Prob. 56ECh. 3 - Prob. 1GYRCh. 3 - What role does the derivative play in defining...Ch. 3 - How can you sometimes graph the derivative of a...Ch. 3 - Prob. 4GYRCh. 3 - Prob. 5GYRCh. 3 - Prob. 6GYRCh. 3 - Prob. 7GYRCh. 3 - Prob. 8GYRCh. 3 - Prob. 9GYRCh. 3 - Prob. 10GYRCh. 3 - Prob. 11GYRCh. 3 - Prob. 12GYRCh. 3 - Prob. 13GYRCh. 3 - Prob. 14GYRCh. 3 - Prob. 15GYRCh. 3 - Prob. 16GYRCh. 3 - Prob. 17GYRCh. 3 - Prob. 18GYRCh. 3 - What is the rule for calculating the derivative of...Ch. 3 - Prob. 20GYRCh. 3 - Prob. 21GYRCh. 3 - Prob. 22GYRCh. 3 - Prob. 23GYRCh. 3 - Prob. 24GYRCh. 3 - Prob. 25GYRCh. 3 - Prob. 1PECh. 3 - Prob. 2PECh. 3 - Find the derivatives of the functions in Exercises...Ch. 3 - Prob. 4PECh. 3 - Prob. 5PECh. 3 - Prob. 6PECh. 3 - Prob. 7PECh. 3 - Prob. 8PECh. 3 - Prob. 9PECh. 3 - Prob. 10PECh. 3 - Prob. 11PECh. 3 - Prob. 12PECh. 3 - Prob. 13PECh. 3 - Prob. 14PECh. 3 - Prob. 15PECh. 3 - Prob. 16PECh. 3 - Prob. 17PECh. 3 - Prob. 18PECh. 3 - Prob. 19PECh. 3 - Prob. 20PECh. 3 - Prob. 21PECh. 3 - Prob. 22PECh. 3 - Prob. 23PECh. 3 - Prob. 24PECh. 3 - Prob. 25PECh. 3 - Prob. 26PECh. 3 - Prob. 27PECh. 3 - Prob. 28PECh. 3 - Prob. 29PECh. 3 - Prob. 30PECh. 3 - Prob. 31PECh. 3 - Prob. 32PECh. 3 - Prob. 33PECh. 3 - Prob. 34PECh. 3 - Prob. 35PECh. 3 - Prob. 36PECh. 3 - Prob. 37PECh. 3 - Prob. 38PECh. 3 - Prob. 39PECh. 3 - Prob. 40PECh. 3 - Prob. 41PECh. 3 - Prob. 42PECh. 3 - Prob. 43PECh. 3 - Prob. 44PECh. 3 - Prob. 45PECh. 3 - Prob. 46PECh. 3 - Prob. 47PECh. 3 - Prob. 48PECh. 3 - Prob. 49PECh. 3 - Prob. 50PECh. 3 - Prob. 51PECh. 3 - Prob. 52PECh. 3 - Prob. 53PECh. 3 - Prob. 54PECh. 3 - Prob. 55PECh. 3 - Prob. 56PECh. 3 - Prob. 57PECh. 3 - Prob. 58PECh. 3 - Prob. 59PECh. 3 - Find the value of dr / dt at t =0 if r = (θ2 +...Ch. 3 - Prob. 61PECh. 3 - Prob. 62PECh. 3 - Prob. 63PECh. 3 - Prob. 64PECh. 3 - Prob. 65PECh. 3 - Prob. 66PECh. 3 - Prob. 67PECh. 3 - Prob. 68PECh. 3 - Prob. 69PECh. 3 - Prob. 70PECh. 3 - Prob. 71PECh. 3 - Prob. 72PECh. 3 - Prob. 73PECh. 3 - Prob. 74PECh. 3 - Prob. 75PECh. 3 - Prob. 76PECh. 3 - Prob. 77PECh. 3 - Prob. 78PECh. 3 - Prob. 79PECh. 3 - Prob. 80PECh. 3 - Prob. 81PECh. 3 - Prob. 82PECh. 3 - Prob. 83PECh. 3 - Prob. 84PECh. 3 - Prob. 85PECh. 3 - Prob. 86PECh. 3 - Prob. 87PECh. 3 - Prob. 88PECh. 3 - Prob. 89PECh. 3 - Prob. 90PECh. 3 - Prob. 91PECh. 3 - Prob. 92PECh. 3 - Prob. 93PECh. 3 - Prob. 94PECh. 3 - Prob. 95PECh. 3 - Find the limits in Exercises 125–132.
126.
Ch. 3 - Prob. 97PECh. 3 - Find the limits in Exercises 125–132.
128.
Ch. 3 - Prob. 99PECh. 3 - Prob. 100PECh. 3 - Prob. 101PECh. 3 - Find the limits in Exercises 125–132.
132.
Ch. 3 - Prob. 103PECh. 3 - Prob. 104PECh. 3 - Prob. 105PECh. 3 - Prob. 106PECh. 3 - Prob. 107PECh. 3 - Prob. 108PECh. 3 - Resistors connected in parallel If two resistors...Ch. 3 - Prob. 110PECh. 3 - Prob. 111PECh. 3 - Prob. 112PECh. 3 - Prob. 113PECh. 3 - Prob. 114PECh. 3 - Prob. 115PECh. 3 - Prob. 116PECh. 3 - Prob. 117PECh. 3 - Prob. 118PECh. 3 - Prob. 119PECh. 3 - Prob. 120PECh. 3 - Prob. 121PECh. 3 - Prob. 122PECh. 3 - Prob. 123PECh. 3 - Prob. 124PECh. 3 - An equation like sin2θ + cos2 θ = 1 is called an...Ch. 3 - Prob. 2AAECh. 3 - Find values for the constants a, b, and c that...Ch. 3 - Prob. 4AAECh. 3 - Prob. 5AAECh. 3 - Prob. 6AAECh. 3 - Prob. 7AAECh. 3 - Prob. 8AAECh. 3 - Prob. 9AAECh. 3 - Prob. 10AAECh. 3 - Prob. 11AAECh. 3 - Prob. 12AAECh. 3 - Prob. 13AAECh. 3 - Prob. 14AAECh. 3 - Prob. 15AAECh. 3 - Prob. 16AAECh. 3 - Prob. 17AAECh. 3 - Prob. 18AAECh. 3 - Prob. 19AAECh. 3 - Prob. 20AAECh. 3 - Prob. 21AAECh. 3 - Prob. 22AAECh. 3 - Prob. 23AAECh. 3 - Prob. 24AAECh. 3 - Prob. 25AAECh. 3 - Prob. 26AAECh. 3 - Prob. 27AAECh. 3 - Leibniz’s rule for higher-order derivatives of...Ch. 3 - The period of a clock pendulum The period T of a...Ch. 3 - The melting ice cube Assume that an ice cube...
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