Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
14th Edition
ISBN: 9780134438986
Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher: PEARSON
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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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Chapter 7.2, Problem 68E
To determine

Calculate the derivative of y by using logarithmic differentiation.

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Chapter 7 Solutions

Thomas' Calculus (14th Edition)

Ch. 7.1 - Which of the functions graphed in Exercise are...Ch. 7.1 - Which of the functions graphed in Exercise are...Ch. 7.1 - Which of the functions graphed in Exercise are...Ch. 7.1 - Which of the functions graphed in Exercises 1–6...Ch. 7.1 - Which of the functions graphed in Exercise are...Ch. 7.1 - Which of the functions graphed in Exercise are...Ch. 7.1 - Determine from its graph whether the function is...Ch. 7.1 - Determine from its graph whether the function is...Ch. 7.1 - Determine from its graph whether the function is...Ch. 7.1 - Determine from its graph whether the function is...
Ch. 7.1 - Each of Exercise shows the graph of a function y =...Ch. 7.1 - Prob. 12ECh. 7.1 - Each of Exercise shows the graph of a function y =...Ch. 7.1 - Prob. 14ECh. 7.1 - Prob. 15ECh. 7.1 - Prob. 16ECh. 7.1 - Graph the function , 0 ≤ x ≤ 1. What symmetry does...Ch. 7.1 - Prob. 18ECh. 7.1 - Formulas for Inverse Functions Each of Exercises...Ch. 7.1 - Formulas for Inverse Functions Each of Exercises...Ch. 7.1 - Formulas for Inverse Functions Each of Exercises...Ch. 7.1 - Prob. 22ECh. 7.1 - Formulas for Inverse Functions Each of Exercises...Ch. 7.1 - Prob. 24ECh. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Prob. 30ECh. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - Each of Exercises 25–36 gives a formula for a...Ch. 7.1 - In Exercises 1–4: Find f–1(x). Graph f and f–1...Ch. 7.1 - In Exercises 1–4: Find f–1(x). Graph f and f–1...Ch. 7.1 - Prob. 37ECh. 7.1 - Prob. 38ECh. 7.1 - Show that f(x) = x3 and are inverses of one...Ch. 7.1 - Prob. 40ECh. 7.1 - Let f(x) = x3 – 3x2 – 1, x ≥ 2. Find the value of...Ch. 7.1 - Let f(x) = x2 – 4x – 5, x > 2. Find the value of...Ch. 7.1 - Prob. 43ECh. 7.1 - Prob. 44ECh. 7.1 - Find the inverse of the function f(x) = mx, where...Ch. 7.1 - Show that the graph of the inverse of f(x) = mx +...Ch. 7.1 - Prob. 47ECh. 7.1 - Find the inverse of f(x) = − x + 1. Graph the line...Ch. 7.1 - Show that increasing functions and decreasing...Ch. 7.1 - Prob. 50ECh. 7.1 - Use the results of Exercise 85 to show that the...Ch. 7.1 - Prob. 52ECh. 7.1 - Use the results of Exercise 85 to show that the...Ch. 7.1 - Use the results of Exercise 85 to show that the...Ch. 7.1 - If f(x) is one-to-one, can anything be said about...Ch. 7.1 - Prob. 56ECh. 7.1 - Suppose that the range of g lies in the domain of...Ch. 7.1 - If a composition f ◦ g is one-to-one, must g be...Ch. 7.1 - Assume that ƒ and g are differentiable functions...Ch. 7.1 - Prob. 60ECh. 7.2 - Express the following logarithms in terms of ln 2...Ch. 7.2 - Express the following logarithms in terms of ln 5...Ch. 7.2 - Use the properties of logarithms to write the...Ch. 7.2 - Prob. 4ECh. 7.2 - In Exercises 5 and 6, solve for t. 5. Ch. 7.2 - In Exercises 5 and 6, solve for t. 6. ln(t − 2) =...Ch. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - Derivatives of Logarithms In Exercises 11–40, find...Ch. 7.2 - Prob. 10ECh. 7.2 - Derivatives of Logarithms In Exercises 11–40, find...Ch. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - Derivatives of Logarithms In Exercises 11-40, find...Ch. 7.2 - Derivatives of Logarithms In Exercises 11-40, find...Ch. 7.2 - Prob. 17ECh. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - Prob. 19ECh. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - Prob. 22ECh. 7.2 - Prob. 23ECh. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - Prob. 28ECh. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - Prob. 30ECh. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - Prob. 32ECh. 7.2 - Prob. 33ECh. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - In Exercises 11-40, find the derivative of y with...Ch. 7.2 - Prob. 36ECh. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - In Exercises 7–38, find the derivative of y with...Ch. 7.2 - Evaluate the integrals in Exercises 1–46. 1. Ch. 7.2 - Evaluate the integrals in Exercises 1–46. 2. Ch. 7.2 - Evaluate the integrals in Exercises 1–46. 3. Ch. 7.2 - Evaluate the integrals in Exercises 1–46. 4. Ch. 7.2 - Evaluate the integrals in Exercises 39–56. Ch. 7.2 - Evaluate the integrals in Exercises 39–56. 44. Ch. 7.2 - Evaluate the integrals in Exercises 39–56. 45. Ch. 7.2 - Evaluate the integrals in Exercises 39–56. 46. Ch. 7.2 - Evaluate the integrals in Exercises 39–56. 47. Ch. 7.2 - Evaluate the integrals in Exercises 39–56. 48. Ch. 7.2 - Prob. 49ECh. 7.2 - Prob. 50ECh. 7.2 - Prob. 51ECh. 7.2 - Evaluate the integrals in Exercises 39–56. 52. Ch. 7.2 - Prob. 53ECh. 7.2 - Prob. 54ECh. 7.2 - Evaluate the integrals in Exercises 1–46. 7. Ch. 7.2 - Prob. 56ECh. 7.2 - In Exercises 41-54, use logarithmic...Ch. 7.2 - Prob. 58ECh. 7.2 - Prob. 59ECh. 7.2 - Prob. 60ECh. 7.2 - Prob. 61ECh. 7.2 - Prob. 62ECh. 7.2 - In Exercises 41-54, use logarithmic...Ch. 7.2 - Prob. 64ECh. 7.2 - Prob. 65ECh. 7.2 - In Exercises 41-54, use logarithmic...Ch. 7.2 - In Exercises 41-54, use logarithmic...Ch. 7.2 - In Exercises 41-54, use logarithmic...Ch. 7.2 - In Exercises 41-54, use logarithmic...Ch. 7.2 - In Exercises 41-54, use logarithmic...Ch. 7.2 - 71. Locate and identify the absolute extreme...Ch. 7.2 - 72. a. Prove that ƒ(x) = x − ln x is increasing...Ch. 7.2 - Prob. 73ECh. 7.2 - Prob. 74ECh. 7.2 - Prob. 75ECh. 7.2 - Prob. 76ECh. 7.2 - 77. The region in the first quadrant bounded by...Ch. 7.2 - Prob. 78ECh. 7.2 - Prob. 79ECh. 7.2 - Prob. 80ECh. 7.2 - Find the lengths of the following curves. y =...Ch. 7.2 - Prob. 82ECh. 7.2 - Prob. 83ECh. 7.2 - Prob. 84ECh. 7.2 - 85. Use a derivative to show that ƒ(x) = ln (x3 −...Ch. 7.2 - 86. Use a derivative to show that is one-to-one. Ch. 7.2 - Solve the initial value problems in Exercises 87...Ch. 7.2 - Solve the initial value problems in Exercises 87...Ch. 7.2 - Prob. 89ECh. 7.2 - 90. Use the same-derivative argument, as was done...Ch. 7.2 - Prob. 91ECh. 7.2 - 92. Does the graph of , x > 0, have an infection...Ch. 7.3 - In Exercises 1–4, solve for t. 1. e−0.3t =...Ch. 7.3 - In Exercises 1–4, solve for t. 2. e−0.01t =...Ch. 7.3 - In Exercises 1–4, solve for t. 3. Ch. 7.3 - Prob. 4ECh. 7.3 - e2t − 3et = 0 Ch. 7.3 - e−2t + 6 = 5e−t Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - Prob. 8ECh. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - Prob. 13ECh. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - Prob. 15ECh. 7.3 - In Exercises 55-62, find the derivative of y with...Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - Prob. 19ECh. 7.3 - In Exercises 55-62, find the derivative of y with...Ch. 7.3 - Prob. 21ECh. 7.3 - In Exercises 55-62, find the derivative of y with...Ch. 7.3 - In Exercises 55-62, find the derivative of y with...Ch. 7.3 - In Exercises 55-62, find the derivative of y with...Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - In Exercises 7–26, find the derivative of y with...Ch. 7.3 - In Exercises 63-66, find dy/dx. 63. ln y = ey sin...Ch. 7.3 - Prob. 28ECh. 7.3 - In Exercises 27–32, find . 29. Ch. 7.3 - Prob. 30ECh. 7.3 - In Exercises 27–32, find dy/dx. 31. 3 + sin y = y...Ch. 7.3 - Prob. 32ECh. 7.3 - Evaluate the integrals in Exercises 33–54. 33. Ch. 7.3 - Evaluate the integrals in Exercises 33–54. 34. Ch. 7.3 - Prob. 35ECh. 7.3 - Prob. 36ECh. 7.3 - Prob. 37ECh. 7.3 - Prob. 38ECh. 7.3 - Prob. 39ECh. 7.3 - Evaluate the integrals in Exercises 33–54. 40. Ch. 7.3 - Evaluate the integrals in Exercises 1–46. 15. Ch. 7.3 - Evaluate the integrals in Exercises 1–46. 16. Ch. 7.3 - Evaluate the integrals in Exercises 1–46. 17. Ch. 7.3 - Prob. 44ECh. 7.3 - Evaluate the integrals in Exercises 1–46. 19. Ch. 7.3 - Prob. 46ECh. 7.3 - Prob. 47ECh. 7.3 - Evaluate the integrals in Exercises 33–54. 48. Ch. 7.3 - Prob. 49ECh. 7.3 - Prob. 50ECh. 7.3 - Evaluate the integrals in Exercises 1–46. 23. Ch. 7.3 - Evaluate the integrals in Exercises 1–46. 24. Ch. 7.3 - Prob. 53ECh. 7.3 - Evaluate the integrals in Exercises 1–46. 26. Ch. 7.3 - Solve the initial value problems in Exercises...Ch. 7.3 - Solve the initial value problems in Exercises...Ch. 7.3 - Solve the initial value problems in Exercises...Ch. 7.3 - Prob. 58ECh. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - Prob. 60ECh. 7.3 - Prob. 61ECh. 7.3 - In Exercises 67–88, find the derivative of y with...Ch. 7.3 - Prob. 63ECh. 7.3 - Prob. 64ECh. 7.3 - Prob. 65ECh. 7.3 - In Exercises 59–86, find the derivative of y with...Ch. 7.3 - Prob. 67ECh. 7.3 - In Exercises 59–86, find the derivative of y with...Ch. 7.3 - Prob. 69ECh. 7.3 - In Exercises 59–86, find the derivative of y with...Ch. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - Prob. 73ECh. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - Prob. 75ECh. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - Prob. 77ECh. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - Prob. 79ECh. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - In Exercises 67-88, find the derivative of y with...Ch. 7.3 - Prob. 82ECh. 7.3 - Prob. 83ECh. 7.3 - Prob. 84ECh. 7.3 - Prob. 85ECh. 7.3 - Prob. 86ECh. 7.3 - Evaluate the integrals in Exercises 87–96. 87. Ch. 7.3 - Evaluate the integrals in Exercises 87–96. 88. Ch. 7.3 - Prob. 89ECh. 7.3 - Prob. 90ECh. 7.3 - Evaluate the integrals in Exercises 87–96. 91. Ch. 7.3 - Evaluate the integrals in Exercises 87–96. 92. Ch. 7.3 - Prob. 93ECh. 7.3 - Prob. 94ECh. 7.3 - Prob. 95ECh. 7.3 - Evaluate the integrals in Exercises 87–96. 96. Ch. 7.3 - Evaluate the integrals in Exercises 97–110. 97. Ch. 7.3 - Prob. 98ECh. 7.3 - Prob. 99ECh. 7.3 - Prob. 100ECh. 7.3 - Prob. 101ECh. 7.3 - Prob. 102ECh. 7.3 - Prob. 103ECh. 7.3 - Evaluate the integrals in Exercises 97–110. 104. Ch. 7.3 - Prob. 105ECh. 7.3 - Prob. 106ECh. 7.3 - Prob. 107ECh. 7.3 - Prob. 108ECh. 7.3 - Prob. 109ECh. 7.3 - Evaluate the integrals in Exercises 97–110. 110. Ch. 7.3 - Evaluate the integrals in Exercises...Ch. 7.3 - Prob. 112ECh. 7.3 - Prob. 113ECh. 7.3 - Prob. 114ECh. 7.3 - In Exercises 89-100, use logarithmic...Ch. 7.3 - Prob. 116ECh. 7.3 - Prob. 117ECh. 7.3 - Prob. 118ECh. 7.3 - Prob. 119ECh. 7.3 - Prob. 120ECh. 7.3 - Prob. 121ECh. 7.3 - Prob. 122ECh. 7.3 - In Exercises 89−100, use logarithmic...Ch. 7.3 - In Exercises 89−100, use logarithmic...Ch. 7.3 - In Exercises 89−100, use logarithmic...Ch. 7.3 - In Exercises 89−100, use logarithmic...Ch. 7.3 - For Exercises 127 and 128 find a function ƒ...Ch. 7.3 - Prob. 128ECh. 7.3 - Find the absolute maximum and minimum values of...Ch. 7.3 - Where does the periodic function ƒ(x) = 2esin(x/2)...Ch. 7.3 - Prob. 131ECh. 7.3 - Prob. 132ECh. 7.3 - Prob. 133ECh. 7.3 - Prob. 134ECh. 7.3 - Prob. 135ECh. 7.3 - Prob. 136ECh. 7.3 - Prob. 137ECh. 7.3 - Prob. 138ECh. 7.3 - Prob. 139ECh. 7.3 - Prob. 140ECh. 7.3 - Prob. 141ECh. 7.3 - Prob. 142ECh. 7.3 - Show that ∫ ln x dx = x ln x − x + C. Find the...Ch. 7.3 - Prob. 144ECh. 7.3 - Prob. 145ECh. 7.3 - The linearization of ex at x = 0 Derive the linear...Ch. 7.3 - Prob. 147ECh. 7.3 - Prob. 148ECh. 7.3 - Prob. 149ECh. 7.3 - Prob. 150ECh. 7.3 - Prob. 151ECh. 7.3 - Prob. 152ECh. 7.3 - Prob. 153ECh. 7.3 - Prob. 154ECh. 7.3 - Prob. 155ECh. 7.3 - Prob. 156ECh. 7.4 - In Exercises 1–4, show that each function y = f(x)...Ch. 7.4 - Prob. 2ECh. 7.4 - In Exercises 1–4, show that each function y = f(x)...Ch. 7.4 - Prob. 4ECh. 7.4 - In Exercises 5–8, show that each function is a...Ch. 7.4 - In Exercises 5–8, show that each function is a...Ch. 7.4 - In Exercises 5–8, show that each function is a...Ch. 7.4 - Prob. 8ECh. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Prob. 13ECh. 7.4 - Prob. 14ECh. 7.4 - Prob. 15ECh. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Prob. 18ECh. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Prob. 20ECh. 7.4 - Solve the differential equation in Exercises...Ch. 7.4 - Prob. 22ECh. 7.4 - Human evolution continues The analysis of tooth...Ch. 7.4 - Prob. 24ECh. 7.4 - First-order chemical reactions In some chemical...Ch. 7.4 - The inversion of sugar The processing of raw sugar...Ch. 7.4 - Working underwater The intensity L(x) of light x...Ch. 7.4 - Prob. 28ECh. 7.4 - Prob. 29ECh. 7.4 - Growth of bacteria A colony of bacteria is grown...Ch. 7.4 - Prob. 31ECh. 7.4 - Drug concentration An antibiotic is administered...Ch. 7.4 - Endangered species Biologists consider a species...Ch. 7.4 - The U.S. population The U.S. Census Bureau keeps a...Ch. 7.4 - Oil depletion Suppose the amount of oil pumped...Ch. 7.4 - Prob. 36ECh. 7.4 - Plutonium-239 The half-life of the plutonium...Ch. 7.4 - Prob. 38ECh. 7.4 - The mean life of a radioactive nucleus Physicists...Ch. 7.4 - Californium-252 What costs $60 million per gram...Ch. 7.4 - Cooling soup Suppose that a cup of soup cooled...Ch. 7.4 - Prob. 42ECh. 7.4 - Prob. 43ECh. 7.4 - Prob. 44ECh. 7.4 - The age of Crater Lake The charcoal from a tree...Ch. 7.4 - Prob. 46ECh. 7.4 - Carbon-14 The oldest known frozen human mummy,...Ch. 7.4 - Art forgery A painting attributed to Vermeer...Ch. 7.4 - Prob. 49ECh. 7.4 - Prob. 50ECh. 7.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 7.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 7.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 7.5 - Prob. 4ECh. 7.5 - Prob. 5ECh. 7.5 - Prob. 6ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 9ECh. 7.5 - Prob. 10ECh. 7.5 - Prob. 11ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 13ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 17ECh. 7.5 - Prob. 18ECh. 7.5 - Prob. 19ECh. 7.5 - Prob. 20ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 25ECh. 7.5 - Prob. 26ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 29ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 31ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 36ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 39ECh. 7.5 - Use 1’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 41ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 43ECh. 7.5 - Prob. 44ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 46ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Prob. 48ECh. 7.5 - Prob. 49ECh. 7.5 - Use l’Hôpital’s rule to find the limits in...Ch. 7.5 - Find the limits in Exercises 51–66. 51. Ch. 7.5 - Prob. 52ECh. 7.5 - Find the limits in Exercises 51–66. 53. Ch. 7.5 - Find the limits in Exercises 51–66. 54. Ch. 7.5 - Find the limits in Exercises 51–66. 55. Ch. 7.5 - Prob. 56ECh. 7.5 - Prob. 57ECh. 7.5 - Find the limits in Exercises 51–66. 58. Ch. 7.5 - Find the limits in Exercises 51–66. 59. Ch. 7.5 - Find the limits in Exercises 51–66. 60. Ch. 7.5 - Prob. 61ECh. 7.5 - Find the limits in Exercises 51–66. 62. Ch. 7.5 - Find the limits in Exercises 51–66. 63. Ch. 7.5 - Find the limits in Exercises 51–66. 64. Ch. 7.5 - Prob. 65ECh. 7.5 - Prob. 66ECh. 7.5 - L’Hôpital’s Rule does not help with the limits in...Ch. 7.5 - Prob. 68ECh. 7.5 - Prob. 69ECh. 7.5 - Prob. 70ECh. 7.5 - Prob. 71ECh. 7.5 - Prob. 72ECh. 7.5 - Prob. 73ECh. 7.5 - Prob. 74ECh. 7.5 - Prob. 75ECh. 7.5 - Prob. 76ECh. 7.5 - Prob. 77ECh. 7.5 - Prob. 78ECh. 7.5 - Prob. 79ECh. 7.5 - Prob. 80ECh. 7.5 - Prob. 81ECh. 7.5 - Prob. 82ECh. 7.5 - Prob. 83ECh. 7.5 - Prob. 84ECh. 7.5 - Prob. 85ECh. 7.5 - Prob. 86ECh. 7.5 - Prob. 87ECh. 7.5 - Prob. 88ECh. 7.5 - The continuous extension of (sin x)x to [0,...Ch. 7.5 - Prob. 90ECh. 7.6 - Use reference triangles in an appropriate...Ch. 7.6 - Use reference triangles in an appropriate...Ch. 7.6 - Prob. 3ECh. 7.6 - Use reference triangles in an appropriate...Ch. 7.6 - Prob. 5ECh. 7.6 - Prob. 6ECh. 7.6 - Use reference triangles in an appropriate...Ch. 7.6 - Use reference triangles in an appropriate...Ch. 7.6 - Prob. 9ECh. 7.6 - Find the values in Exercises 9–12. 10. Ch. 7.6 - Prob. 11ECh. 7.6 - Find the values in Exercises 9–12. 12. Ch. 7.6 - Prob. 13ECh. 7.6 - Find the limits in Exercises 13–20. (If in doubt,...Ch. 7.6 - Find the limits in Exercises 13–20. (If in doubt,...Ch. 7.6 - Find the limits in Exercises 13–20. (If in doubt,...Ch. 7.6 - Find the limits in Exercises 13–20. (If in doubt,...Ch. 7.6 - Find the limits in Exercises 13–20. (If in doubt,...Ch. 7.6 - Prob. 19ECh. 7.6 - Find the limits in Exercises 13–20. (If in doubt,...Ch. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 23ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 25ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 27ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 29ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 31ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 33ECh. 7.6 - Prob. 34ECh. 7.6 - Prob. 35ECh. 7.6 - Prob. 36ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 38ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 40ECh. 7.6 - Prob. 41ECh. 7.6 - In Exercises 21–42, find the derivative of y with...Ch. 7.6 - Prob. 43ECh. 7.6 - For problems 43-46 use implicit differentiation to...Ch. 7.6 - For problems 43-46 use implicit differentiation to...Ch. 7.6 - For problems 43-46 use implicit differentiation to...Ch. 7.6 - Evaluate the integrals in Exercises 47–70. 47. Ch. 7.6 - Evaluate the integrals in Exercises 47–70. 48. Ch. 7.6 - Prob. 49ECh. 7.6 - Evaluate the integrals in Exercises 47–70. 50. Ch. 7.6 - Prob. 53ECh. 7.6 - Prob. 55ECh. 7.6 - Evaluate the integrals in Exercises 47–70. 56. Ch. 7.6 - Prob. 59ECh. 7.6 - Evaluate the integrals in Exercises 47–70. 60. Ch. 7.6 - Prob. 61ECh. 7.6 - Evaluate the integrals in Exercises 47–70. 62. Ch. 7.6 - Prob. 63ECh. 7.6 - Evaluate the integrals in Exercises 47–70. 64. Ch. 7.6 - Evaluate the integrals in Exercises 47–70. 65. Ch. 7.6 - Prob. 66ECh. 7.6 - Prob. 67ECh. 7.6 - Prob. 69ECh. 7.6 - Prob. 70ECh. 7.6 - Prob. 71ECh. 7.6 - Prob. 72ECh. 7.6 - Prob. 73ECh. 7.6 - Prob. 74ECh. 7.6 - Prob. 75ECh. 7.6 - Prob. 76ECh. 7.6 - Prob. 77ECh. 7.6 - Evaluate the integrals in Exercises 71–84. 78. Ch. 7.6 - Evaluate the integrals in Exercises 71–84. 79. Ch. 7.6 - Evaluate the integrals in Exercises 71–84. 80. Ch. 7.6 - Prob. 81ECh. 7.6 - Prob. 82ECh. 7.6 - Prob. 83ECh. 7.6 - Prob. 84ECh. 7.6 - Prob. 85ECh. 7.6 - Evaluate the integrals in Exercises 71–84. 86. Ch. 7.6 - Prob. 87ECh. 7.6 - Evaluate the integrals in Exercises 71–84. 88. Ch. 7.6 - Prob. 89ECh. 7.6 - Evaluate the integrals in Exercises 71–84. 90. Ch. 7.6 - Prob. 91ECh. 7.6 - Evaluate the integrals in Exercises 71–84. 92. Ch. 7.6 - Evaluate the integrals in Exercises 71–84. 93. Ch. 7.6 - Evaluate the integrals in Exercises 71–84. 94. Ch. 7.6 - Prob. 95ECh. 7.6 - Evaluate the integrals in Exercises 71–84. 96. Ch. 7.6 - Prob. 97ECh. 7.6 - Prob. 98ECh. 7.6 - Prob. 99ECh. 7.6 - Prob. 100ECh. 7.6 - Prob. 101ECh. 7.6 - Prob. 102ECh. 7.6 - Prob. 103ECh. 7.6 - Prob. 104ECh. 7.6 - Prob. 105ECh. 7.6 - Prob. 106ECh. 7.6 - Prob. 107ECh. 7.6 - Prob. 108ECh. 7.6 - Solve the initial value problems in Exercises...Ch. 7.6 - Prob. 110ECh. 7.6 - Prob. 111ECh. 7.6 - Prob. 112ECh. 7.6 - Prob. 113ECh. 7.6 - Prob. 114ECh. 7.6 - Prob. 115ECh. 7.6 - Prob. 116ECh. 7.6 - Prob. 117ECh. 7.6 - Prob. 119ECh. 7.6 - Prob. 120ECh. 7.6 - Use the identity to derive the formula for the...Ch. 7.6 - Prob. 122ECh. 7.6 - Prob. 123ECh. 7.6 - Prob. 124ECh. 7.6 - Prob. 125ECh. 7.6 - Prob. 126ECh. 7.6 - Find the volume of the solid of revolution shown...Ch. 7.6 - Prob. 128ECh. 7.6 - Prob. 129ECh. 7.6 - Prob. 130ECh. 7.6 - Prob. 131ECh. 7.6 - Prob. 132ECh. 7.6 - Prob. 133ECh. 7.6 - Prob. 134ECh. 7.6 - Prob. 135ECh. 7.6 - Prob. 136ECh. 7.6 - Prob. 137ECh. 7.6 - Prob. 138ECh. 7.6 - Prob. 139ECh. 7.6 - Prob. 140ECh. 7.7 - Prob. 1ECh. 7.7 - Prob. 2ECh. 7.7 - Prob. 3ECh. 7.7 - Prob. 4ECh. 7.7 - Prob. 5ECh. 7.7 - Prob. 6ECh. 7.7 - Prob. 7ECh. 7.7 - Prob. 8ECh. 7.7 - Prob. 9ECh. 7.7 - Prob. 10ECh. 7.7 - Prove the identities sinh (x + y) = sinh x cosh y...Ch. 7.7 - Prob. 12ECh. 7.7 - Prob. 13ECh. 7.7 - Prob. 14ECh. 7.7 - Prob. 15ECh. 7.7 - Prob. 16ECh. 7.7 - Prob. 17ECh. 7.7 - Prob. 18ECh. 7.7 - In Exercises 13–24, find the derivative of y with...Ch. 7.7 - In Exercises 13–24, find the derivative of y with...Ch. 7.7 - Prob. 21ECh. 7.7 - In Exercises 13–24, find the derivative of y with...Ch. 7.7 - Prob. 23ECh. 7.7 - In Exercises 13–24, find the derivative of y with...Ch. 7.7 - Prob. 25ECh. 7.7 - Prob. 26ECh. 7.7 - Prob. 27ECh. 7.7 - Prob. 28ECh. 7.7 - Prob. 29ECh. 7.7 - Prob. 30ECh. 7.7 - Prob. 31ECh. 7.7 - Prob. 32ECh. 7.7 - Prob. 33ECh. 7.7 - Prob. 34ECh. 7.7 - Prob. 35ECh. 7.7 - Prob. 36ECh. 7.7 - Prob. 37ECh. 7.7 - Prob. 38ECh. 7.7 - Prob. 39ECh. 7.7 - Prob. 40ECh. 7.7 - Prob. 41ECh. 7.7 - Evaluate the integrals in Exercises 41–60. 42. Ch. 7.7 - Prob. 43ECh. 7.7 - Evaluate the integrals in Exercises 41–60. 44. Ch. 7.7 - Prob. 45ECh. 7.7 - Evaluate the integrals in Exercises 41–60. 46. Ch. 7.7 - Prob. 47ECh. 7.7 - Prob. 48ECh. 7.7 - Prob. 49ECh. 7.7 - Prob. 50ECh. 7.7 - Prob. 51ECh. 7.7 - Evaluate the integrals in Exercises 41-60. 52. Ch. 7.7 - Evaluate the integrals in Exercises 41–60. 53. Ch. 7.7 - Evaluate the integrals in Exercises 41–60. 54. Ch. 7.7 - Prob. 55ECh. 7.7 - Prob. 56ECh. 7.7 - Prob. 57ECh. 7.7 - Evaluate the integrals in Exercises 41–60. 58. Ch. 7.7 - Prob. 59ECh. 7.7 - Prob. 60ECh. 7.7 - Prob. 61ECh. 7.7 - Prob. 62ECh. 7.7 - Prob. 63ECh. 7.7 - Prob. 64ECh. 7.7 - Prob. 65ECh. 7.7 - Prob. 66ECh. 7.7 - Prob. 67ECh. 7.7 - Prob. 68ECh. 7.7 - Prob. 69ECh. 7.7 - Prob. 70ECh. 7.7 - Prob. 71ECh. 7.7 - Prob. 72ECh. 7.7 - Prob. 73ECh. 7.7 - Evaluate the integrals in Exercises 67–74 in terms...Ch. 7.7 - Prob. 75ECh. 7.7 - Prob. 76ECh. 7.7 - Prob. 77ECh. 7.7 - Prob. 78ECh. 7.7 - Prob. 79ECh. 7.7 - Prob. 80ECh. 7.7 - Prob. 81ECh. 7.7 - Prob. 82ECh. 7.7 - Prob. 83ECh. 7.7 - Prob. 84ECh. 7.7 - Prob. 85ECh. 7.7 - Prob. 86ECh. 7.8 - Which of the following functions grow faster than...Ch. 7.8 - Which of the following functions grow faster than...Ch. 7.8 - Prob. 3ECh. 7.8 - Prob. 4ECh. 7.8 - Prob. 5ECh. 7.8 - Prob. 6ECh. 7.8 - Order the following functions from slowest-growing...Ch. 7.8 - Prob. 8ECh. 7.8 - True or false? As x → ∞, x − o(x) x − o(x + 5) x =...Ch. 7.8 - Prob. 10ECh. 7.8 - Prob. 11ECh. 7.8 - Prob. 12ECh. 7.8 - Prob. 13ECh. 7.8 - Prob. 14ECh. 7.8 - Prob. 15ECh. 7.8 - Prob. 16ECh. 7.8 - Prob. 17ECh. 7.8 - Prob. 18ECh. 7.8 - Prob. 19ECh. 7.8 - Prob. 20ECh. 7.8 - Prob. 21ECh. 7.8 - The function ln x grows slower than any...Ch. 7.8 - Prob. 23ECh. 7.8 - Prob. 24ECh. 7.8 - Prob. 25ECh. 7.8 - Prob. 26ECh. 7 - Prob. 1GYRCh. 7 - How are the domains, ranges, and graphs of...Ch. 7 - Prob. 3GYRCh. 7 - Under what circumstances can you be sure that the...Ch. 7 - Prob. 5GYRCh. 7 - Prob. 6GYRCh. 7 - Prob. 7GYRCh. 7 - Prob. 8GYRCh. 7 - Prob. 9GYRCh. 7 - Prob. 10GYRCh. 7 - Prob. 11GYRCh. 7 - Prob. 12GYRCh. 7 - Prob. 13GYRCh. 7 - Prob. 14GYRCh. 7 - Prob. 15GYRCh. 7 - Prob. 16GYRCh. 7 - Prob. 17GYRCh. 7 - Prob. 18GYRCh. 7 - What are the derivatives of the six basic...Ch. 7 - Prob. 20GYRCh. 7 - Prob. 21GYRCh. 7 - Prob. 22GYRCh. 7 - Prob. 23GYRCh. 7 - Prob. 24GYRCh. 7 - Prob. 25GYRCh. 7 - In Exercises 1–24, find the derivative of y with...Ch. 7 - In Exercises 1–24, find the derivative of y with...Ch. 7 - In Exercises 1–24, find the derivative of y with...Ch. 7 - Prob. 4PECh. 7 - In Exercises 1–24, find the derivative of y with...Ch. 7 - Prob. 6PECh. 7 - Prob. 7PECh. 7 - Prob. 8PECh. 7 - Prob. 9PECh. 7 - In Exercises 1–24, find the derivative of y with...Ch. 7 - Prob. 11PECh. 7 - In Exercises 1–24, find the derivative of y with...Ch. 7 - Prob. 13PECh. 7 - Prob. 14PECh. 7 - Prob. 15PECh. 7 - Prob. 16PECh. 7 - Prob. 17PECh. 7 - In Exercises 1–24, find the derivative of y with...Ch. 7 - Prob. 19PECh. 7 - Prob. 20PECh. 7 - Prob. 21PECh. 7 - Prob. 22PECh. 7 - Prob. 23PECh. 7 - Prob. 24PECh. 7 - Prob. 25PECh. 7 - Prob. 26PECh. 7 - Prob. 27PECh. 7 - Prob. 28PECh. 7 - Prob. 29PECh. 7 - Prob. 30PECh. 7 - Prob. 31PECh. 7 - Prob. 32PECh. 7 - Evaluate the integrals in Exercises 31–78. 33. Ch. 7 - Prob. 34PECh. 7 - Prob. 35PECh. 7 - Prob. 36PECh. 7 - Prob. 37PECh. 7 - Prob. 38PECh. 7 - Prob. 39PECh. 7 - Prob. 40PECh. 7 - Prob. 41PECh. 7 - Prob. 42PECh. 7 - Prob. 43PECh. 7 - Prob. 44PECh. 7 - Prob. 45PECh. 7 - Prob. 46PECh. 7 - Prob. 47PECh. 7 - Prob. 48PECh. 7 - Prob. 49PECh. 7 - Prob. 50PECh. 7 - Prob. 51PECh. 7 - Prob. 52PECh. 7 - Prob. 53PECh. 7 - Prob. 54PECh. 7 - Prob. 55PECh. 7 - Prob. 56PECh. 7 - Prob. 57PECh. 7 - Prob. 58PECh. 7 - Prob. 59PECh. 7 - Prob. 60PECh. 7 - Prob. 61PECh. 7 - Prob. 62PECh. 7 - Prob. 63PECh. 7 - Prob. 64PECh. 7 - Prob. 65PECh. 7 - Prob. 66PECh. 7 - Prob. 67PECh. 7 - Prob. 68PECh. 7 - Prob. 69PECh. 7 - Prob. 70PECh. 7 - Prob. 71PECh. 7 - Prob. 72PECh. 7 - Prob. 73PECh. 7 - Prob. 74PECh. 7 - Prob. 75PECh. 7 - Prob. 76PECh. 7 - Prob. 77PECh. 7 - Prob. 78PECh. 7 - Prob. 79PECh. 7 - Prob. 80PECh. 7 - Prob. 81PECh. 7 - Prob. 82PECh. 7 - Prob. 83PECh. 7 - Prob. 84PECh. 7 - Prob. 85PECh. 7 - Prob. 86PECh. 7 - Prob. 87PECh. 7 - Prob. 88PECh. 7 - Prob. 89PECh. 7 - Prob. 90PECh. 7 - Prob. 91PECh. 7 - Prob. 92PECh. 7 - Prob. 93PECh. 7 - Use l’Hôpital’s Rule to find the limits in...Ch. 7 - Prob. 95PECh. 7 - Prob. 96PECh. 7 - Prob. 97PECh. 7 - Prob. 98PECh. 7 - Prob. 99PECh. 7 - Prob. 100PECh. 7 - Prob. 101PECh. 7 - Prob. 102PECh. 7 - Prob. 103PECh. 7 - Prob. 104PECh. 7 - Prob. 105PECh. 7 - Prob. 106PECh. 7 - Prob. 107PECh. 7 - Prob. 108PECh. 7 - Prob. 109PECh. 7 - Prob. 110PECh. 7 - Prob. 111PECh. 7 - Prob. 112PECh. 7 - Prob. 113PECh. 7 - Prob. 114PECh. 7 - Prob. 115PECh. 7 - Prob. 116PECh. 7 - Prob. 117PECh. 7 - Prob. 118PECh. 7 - Prob. 119PECh. 7 - Prob. 120PECh. 7 - Prob. 121PECh. 7 - Prob. 122PECh. 7 - Prob. 123PECh. 7 - Prob. 124PECh. 7 - Prob. 125PECh. 7 - Prob. 126PECh. 7 - Prob. 127PECh. 7 - Prob. 128PECh. 7 - Prob. 129PECh. 7 - Prob. 130PECh. 7 - Prob. 131PECh. 7 - Prob. 132PECh. 7 - Prob. 133PECh. 7 - Prob. 134PECh. 7 - Prob. 135PECh. 7 - Prob. 136PECh. 7 - Prob. 1AAECh. 7 - Prob. 2AAECh. 7 - Prob. 3AAECh. 7 - Prob. 4AAECh. 7 - Find the limits in Exercises 1–6. 5. Ch. 7 - Prob. 6AAECh. 7 - Prob. 7AAECh. 7 - Prob. 8AAECh. 7 - Prob. 9AAECh. 7 - Prob. 10AAECh. 7 - Prob. 11AAECh. 7 - Prob. 12AAECh. 7 - Prob. 13AAECh. 7 - Prob. 14AAECh. 7 - Prob. 15AAECh. 7 - Prob. 16AAECh. 7 - Prob. 17AAECh. 7 - Prob. 18AAECh. 7 - Prob. 19AAECh. 7 - Prob. 20AAECh. 7 - Prob. 21AAECh. 7 - Prob. 22AAECh. 7 - Consider point (a, b) on the graph of y = ln x and...
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY