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 2PE
To determine
Calculate the derivative of the function
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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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