Thomas' Calculus: Early Transcendentals (14th Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
14th Edition
ISBN: 9780134439020
Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher: PEARSON
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Concept explainers

Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 4, Problem 38PE
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Find the first derivative of the function f(x) and g(x).

Find a conclusion about the graph of the functions f(x) and g(x).

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Chapter 4 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 2ECh. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 6ECh. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - Prob. 10E
Ch. 4.1 - In Exercises 11–14, match the table with a...Ch. 4.1 - In Exercises 11–14, match the table with a...Ch. 4.1 - In Exercises 11–14, match the table with a...Ch. 4.1 - In Exercises 11–14, match the table with a...Ch. 4.1 - In Exercises 15–20, sketch the graph of each...Ch. 4.1 - Prob. 16ECh. 4.1 - In Exercises 15-20, sketch the graph of each...Ch. 4.1 - Prob. 18ECh. 4.1 - In Exercises 15–20, sketch the graph of each...Ch. 4.1 - In Exercises 15–20, sketch the graph of each...Ch. 4.1 - Prob. 21ECh. 4.1 - In Exercises 21–36, find the absolute maximum and...Ch. 4.1 - Prob. 23ECh. 4.1 - Prob. 24ECh. 4.1 - Prob. 25ECh. 4.1 - Prob. 26ECh. 4.1 - Prob. 27ECh. 4.1 - Prob. 28ECh. 4.1 - In Exercises 21–36, find the absolute maximum and...Ch. 4.1 - Prob. 30ECh. 4.1 - In Exercises 21–36, find the absolute maximum and...Ch. 4.1 - In Exercises 21–36, find the absolute maximum and...Ch. 4.1 - In Exercises 21–36, find the absolute maximum and...Ch. 4.1 - Prob. 34ECh. 4.1 - Prob. 35ECh. 4.1 - Prob. 36ECh. 4.1 - In Exercises 21–40, find the absolute maximum and...Ch. 4.1 - In Exercises 21–40, find the absolute maximum and...Ch. 4.1 - In Exercises 21–40, find the absolute maximum and...Ch. 4.1 - In Exercises 21–40, find the absolute maximum and...Ch. 4.1 - In Exercises 41–44, find the function’s absolute...Ch. 4.1 - In Exercises 41–44, find the function’s absolute...Ch. 4.1 - In Exercises 41–44, find the function’s absolute...Ch. 4.1 - Prob. 44ECh. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - Prob. 46ECh. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - Prob. 48ECh. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - Prob. 52ECh. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - In Exercises 45–56, determine all critical points...Ch. 4.1 - Prob. 56ECh. 4.1 - Prob. 57ECh. 4.1 - Prob. 58ECh. 4.1 - Prob. 59ECh. 4.1 - Prob. 60ECh. 4.1 - Prob. 61ECh. 4.1 - In Exercises 57–64, find the critical points and...Ch. 4.1 - Prob. 63ECh. 4.1 - Prob. 64ECh. 4.1 - Prob. 65ECh. 4.1 - Prob. 66ECh. 4.1 - Prob. 67ECh. 4.1 - Prob. 68ECh. 4.1 - In Exercises 67–70, show that the function has...Ch. 4.1 - Prob. 70ECh. 4.1 - Prob. 71ECh. 4.1 - Prob. 72ECh. 4.1 - Prob. 73ECh. 4.1 - Prob. 74ECh. 4.1 - Prob. 75ECh. 4.1 - Prob. 76ECh. 4.1 - Prob. 77ECh. 4.1 - Prob. 78ECh. 4.1 - Prob. 79ECh. 4.1 - Prob. 80ECh. 4.1 - Prob. 81ECh. 4.1 - Prob. 82ECh. 4.2 - Checking the Mean Value Theorem Find the value or...Ch. 4.2 - Checking the Mean Value Theorem Find the value or...Ch. 4.2 - Checking the Mean Value Theorem Find the value or...Ch. 4.2 - Checking the Mean Value Theorem Find the value or...Ch. 4.2 - Checking the Mean Value Theorem Find the value or...Ch. 4.2 - Checking the Mean Value Theorem Find the value or...Ch. 4.2 - Checking the Mean Value Theorem Find the value or...Ch. 4.2 - Find the value or values of c that satisfy the...Ch. 4.2 - Which of the functions in Exercises 9-14 satisfy...Ch. 4.2 - Prob. 10ECh. 4.2 - Prob. 11ECh. 4.2 - Prob. 12ECh. 4.2 - Which of the functions in Exercises 9–14 satisfy...Ch. 4.2 - Which of the functions in Exercises 9–14 satisfy...Ch. 4.2 - The function is zero at x = 0 and x = 1 and...Ch. 4.2 - For what values of a, m, and b does the...Ch. 4.2 - Plot the zeros of each polynomial on a line...Ch. 4.2 - Suppose that f″ is continuous on [a, b] and that f...Ch. 4.2 - Show that if f″ > 0 throughout an interval [a, b],...Ch. 4.2 - Show that a cubic polynomial can have at most...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Show that the functions in Exercises 21–28 have...Ch. 4.2 - Suppose that f(–1) = 3 and that f′(x) = 0 for all...Ch. 4.2 - Suppose that f(0) = 5 and that f'(x) = 2 for all...Ch. 4.2 - Suppose that f'(x) = 2x for all x. Find f(2)...Ch. 4.2 - Prob. 32ECh. 4.2 - In Exercises 33–38, find all possible functions...Ch. 4.2 - In Exercises 33–38, find all possible functions...Ch. 4.2 - Prob. 35ECh. 4.2 - Prob. 36ECh. 4.2 - Prob. 37ECh. 4.2 - In Exercises 33–38, find all possible functions...Ch. 4.2 - In Exercises 39–42, find the function with the...Ch. 4.2 - In Exercises 39–42, find the function with the...Ch. 4.2 - In Exercises 39–42, find the function with the...Ch. 4.2 - In Exercises 39–42, find the function with the...Ch. 4.2 - Exercises 43–46 give the velocity v = ds/dt and...Ch. 4.2 - Prob. 44ECh. 4.2 - Exercises 43–46 give the velocity v = ds/dt and...Ch. 4.2 - Exercises 43–46 give the velocity v = ds/dt and...Ch. 4.2 - Prob. 47ECh. 4.2 - Prob. 48ECh. 4.2 - Prob. 49ECh. 4.2 - Exercises 47–50 give the acceleration , initial...Ch. 4.2 - Temperature change It took 14 sec for a mercury...Ch. 4.2 - A trucker handed in a ticket at a toll booth...Ch. 4.2 - Prob. 53ECh. 4.2 - A marathoner ran the 26.2-mi New York City...Ch. 4.2 - Prob. 55ECh. 4.2 - Free fall on the moon On our moon, the...Ch. 4.2 - The geometric mean of a and b The geometric mean...Ch. 4.2 - The arithmetic mean of a and b The arithmetic mean...Ch. 4.2 - Prob. 59ECh. 4.2 - Prob. 60ECh. 4.2 - Prob. 61ECh. 4.2 - Prob. 62ECh. 4.2 - Prob. 63ECh. 4.2 - Prob. 64ECh. 4.2 - Prob. 65ECh. 4.2 - Show that for any numbers a and b, the sine...Ch. 4.2 - Prob. 67ECh. 4.2 - Prob. 68ECh. 4.2 - Prob. 69ECh. 4.2 - Prob. 70ECh. 4.2 - Prob. 71ECh. 4.2 - Prob. 72ECh. 4.2 - Let f be differentiable at every value of x and...Ch. 4.2 - Let f(x) = px2 + qx + r be a quadratic function...Ch. 4.2 - Prob. 75ECh. 4.2 - Prob. 76ECh. 4.2 - Prob. 77ECh. 4.2 - Prob. 78ECh. 4.3 - Answer the following questions about the functions...Ch. 4.3 - Answer the following questions about the functions...Ch. 4.3 - Answer the following questions about the functions...Ch. 4.3 - Prob. 4ECh. 4.3 - Answer the following questions about the functions...Ch. 4.3 - Answer the following questions about the functions...Ch. 4.3 - Answer the following questions about the functions...Ch. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.3 - Answer the following questions about the functions...Ch. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Prob. 13ECh. 4.3 - Answer the following questions about the functions...Ch. 4.3 - In Exercises 15–18: Find the open intervals on...Ch. 4.3 - In Exercises 15–18: Find the open intervals on...Ch. 4.3 - Prob. 17ECh. 4.3 - In Exercises 15–18: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - Prob. 23ECh. 4.3 - Prob. 24ECh. 4.3 - Prob. 25ECh. 4.3 - Prob. 26ECh. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - Find the open intervals on which the function is...Ch. 4.3 - Prob. 32ECh. 4.3 - Prob. 33ECh. 4.3 - Prob. 34ECh. 4.3 - Find the open intervals on which the function is...Ch. 4.3 - Prob. 36ECh. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - Prob. 39ECh. 4.3 - Find the open intervals on which the function is...Ch. 4.3 - Prob. 41ECh. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - Prob. 44ECh. 4.3 - Prob. 45ECh. 4.3 - In Exercises 19–46: Find the open intervals on...Ch. 4.3 - In Exercises 47–58: Identify the function’s local...Ch. 4.3 - Prob. 48ECh. 4.3 - In Exercises 47–58: Identify the function’s local...Ch. 4.3 - Prob. 50ECh. 4.3 - Prob. 51ECh. 4.3 - Prob. 52ECh. 4.3 - Prob. 53ECh. 4.3 - Prob. 54ECh. 4.3 - Prob. 55ECh. 4.3 - Prob. 56ECh. 4.3 - Prob. 57ECh. 4.3 - In Exercises 47–58: Identify the function’s local...Ch. 4.3 - Prob. 59ECh. 4.3 - Prob. 60ECh. 4.3 - In Exercises 59–66: Find the local extrema of each...Ch. 4.3 - Prob. 62ECh. 4.3 - In Exercises 59–66: Find the local extrema of each...Ch. 4.3 - Prob. 64ECh. 4.3 - Prob. 65ECh. 4.3 - In Exercises 59–66: Find the local extrema of each...Ch. 4.3 - Prob. 67ECh. 4.3 - Prob. 68ECh. 4.3 - Prob. 69ECh. 4.3 - Prob. 70ECh. 4.3 - Prob. 71ECh. 4.3 - Prob. 72ECh. 4.3 - Prob. 73ECh. 4.3 - Prob. 74ECh. 4.3 - Prob. 75ECh. 4.3 - Prob. 76ECh. 4.3 - Determine the values of constants a and b so that...Ch. 4.3 - Determine the values of constants a, b, c, and d...Ch. 4.3 - Locate and identify the absolute extreme values...Ch. 4.3 - Prove that f(x) = x − ln x is increasing for x >...Ch. 4.3 - Find the absolute maximum and the absolute minimum...Ch. 4.3 - Where does the periodic function f(x) = 2esin...Ch. 4.3 - Find the absolute maximum value of f(x) = x2 ln...Ch. 4.3 - Prob. 84ECh. 4.3 - Prob. 85ECh. 4.3 - Prob. 86ECh. 4.3 - Prob. 87ECh. 4.3 - Prob. 88ECh. 4.3 - Prob. 89ECh. 4.3 - Prob. 90ECh. 4.4 - Identify the inflection points and local maxima...Ch. 4.4 - Identify the inflection points and local maxima...Ch. 4.4 - Identify the inflection points and local maxima...Ch. 4.4 - Identify the inflection points and local maxima...Ch. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Identify the inflection points and local maxima...Ch. 4.4 - Identify the inflection points and local maxima...Ch. 4.4 - In Exercises 9-58, identify the coordinates of any...Ch. 4.4 - In Exercises 9-58, identify the coordinates of any...Ch. 4.4 - Prob. 11ECh. 4.4 - In Exercises 9-58, identify the coordinates of any...Ch. 4.4 - In Exercises 9-58, identify the coordinates of any...Ch. 4.4 - In Exercises 9-58, identify the coordinates of any...Ch. 4.4 - Prob. 15ECh. 4.4 - Prob. 16ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 20ECh. 4.4 - Prob. 21ECh. 4.4 - Prob. 22ECh. 4.4 - Prob. 23ECh. 4.4 - Prob. 24ECh. 4.4 - In Exercises 9–58, identify the coordinates of...Ch. 4.4 - Prob. 26ECh. 4.4 - Prob. 27ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 30ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 36ECh. 4.4 - Prob. 37ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 39ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - In Exercises 9–58, identify the coordinates of...Ch. 4.4 - Prob. 42ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 44ECh. 4.4 - Prob. 45ECh. 4.4 - Prob. 46ECh. 4.4 - Prob. 47ECh. 4.4 - Prob. 48ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 50ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 52ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 56ECh. 4.4 - Prob. 57ECh. 4.4 - In Exercises 9–58, identify the coordinates of any...Ch. 4.4 - Prob. 59ECh. 4.4 - Prob. 60ECh. 4.4 - Prob. 61ECh. 4.4 - Prob. 62ECh. 4.4 - Prob. 63ECh. 4.4 - Each of Exercises 71–92 gives the first derivative...Ch. 4.4 - Each of Exercises 71–92 gives the first derivative...Ch. 4.4 - Prob. 66ECh. 4.4 - Prob. 67ECh. 4.4 - Prob. 68ECh. 4.4 - Prob. 69ECh. 4.4 - Prob. 70ECh. 4.4 - Prob. 71ECh. 4.4 - Each of Exercises 71–92 gives the first derivative...Ch. 4.4 - Prob. 73ECh. 4.4 - Each of Exercises 71–92 gives the first derivative...Ch. 4.4 - Prob. 75ECh. 4.4 - Prob. 76ECh. 4.4 - Prob. 77ECh. 4.4 - Prob. 78ECh. 4.4 - Prob. 79ECh. 4.4 - Prob. 80ECh. 4.4 - Prob. 81ECh. 4.4 - Each of Exercises 93–96 shows the graphs of the...Ch. 4.4 - Prob. 83ECh. 4.4 - Prob. 84ECh. 4.4 - Graph the rational functions in Exercises 85–102...Ch. 4.4 - Graph the rational functions in Exercises 85–102...Ch. 4.4 - Prob. 87ECh. 4.4 - Graph the rational functions in Exercises 85–102...Ch. 4.4 - Graph the rational functions in Exercises 85–102...Ch. 4.4 - Prob. 90ECh. 4.4 - Prob. 91ECh. 4.4 - Prob. 92ECh. 4.4 - Graph the rational functions in Exercises 85–102...Ch. 4.4 - Graph the rational functions in Exercises 85–102...Ch. 4.4 - Prob. 95ECh. 4.4 - Prob. 96ECh. 4.4 - Prob. 97ECh. 4.4 - Prob. 98ECh. 4.4 - Prob. 99ECh. 4.4 - Prob. 100ECh. 4.4 - Prob. 101ECh. 4.4 - Graph the rational functions in Exercises 85–102...Ch. 4.4 - Prob. 103ECh. 4.4 - Prob. 104ECh. 4.4 - Sketch the graph of a twice-differentiable...Ch. 4.4 - Prob. 106ECh. 4.4 - Prob. 107ECh. 4.4 - Prob. 108ECh. 4.4 - In Exercises 109 and 110, the graph of f′ is...Ch. 4.4 - In Exercises 109 and 110, the graph of f′ is...Ch. 4.4 - In Exercises 111 and 112, the graph of f′ is...Ch. 4.4 - Prob. 112ECh. 4.4 - Prob. 113ECh. 4.4 - Prob. 114ECh. 4.4 - Prob. 115ECh. 4.4 - Prob. 116ECh. 4.4 - Prob. 117ECh. 4.4 - Prob. 118ECh. 4.4 - Prob. 119ECh. 4.4 - Can anything be said about the graph of a function...Ch. 4.4 - Prob. 121ECh. 4.4 - Prob. 122ECh. 4.4 - Prob. 123ECh. 4.4 - Prob. 124ECh. 4.4 - Prob. 125ECh. 4.4 - Prob. 126ECh. 4.4 - Prob. 127ECh. 4.4 - Prob. 128ECh. 4.4 - Prob. 133ECh. 4.4 - Prob. 134ECh. 4.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 4.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 4.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 4.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 4.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 4.5 - In Exercises 1–6, use l’Hôpital’s Rule to evaluate...Ch. 4.5 - Prob. 7ECh. 4.5 - Prob. 8ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 10ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 18ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 24ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 26ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 30ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 36ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 38ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use 1’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 42ECh. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Use l’Hôpital’s rule to find the limits in...Ch. 4.5 - Prob. 50ECh. 4.5 - Find the limits in Exercises 51–66. 51. Ch. 4.5 - Prob. 52ECh. 4.5 - Find the limits in Exercises 51–66. 53. Ch. 4.5 - Prob. 54ECh. 4.5 - Find the limits in Exercises 51–66. 55. Ch. 4.5 - Find the limits in Exercises 51–66. 56. Ch. 4.5 - Prob. 57ECh. 4.5 - Find the limits in Exercises 51–66. 58. Ch. 4.5 - Find the limits in Exercises 51–66. 59. Ch. 4.5 - Find the limits in Exercises 51–66. 60. Ch. 4.5 - Find the limits in Exercises 51–66. 61. Ch. 4.5 - Prob. 62ECh. 4.5 - Find the limits in Exercises 51–66. 63. Ch. 4.5 - Find the limits in Exercises 51–66. 64. Ch. 4.5 - Prob. 65ECh. 4.5 - Find the limits in Exercises 51–66. 66. Ch. 4.5 - L’Hôpital’s Rule does not help with the limits in...Ch. 4.5 - Prob. 68ECh. 4.5 - Prob. 69ECh. 4.5 - L’Hôpital’s Rule does not help with the limits in...Ch. 4.5 - L’Hôpital’s Rule does not help with the limits in...Ch. 4.5 - L’Hôpital’s Rule does not help with the limits in...Ch. 4.5 - L’Hôpital’s Rule does not help with the limits in...Ch. 4.5 - Prob. 74ECh. 4.5 - Prob. 75ECh. 4.5 - Prob. 76ECh. 4.5 - Prob. 77ECh. 4.5 - Prob. 78ECh. 4.5 - Prob. 79ECh. 4.5 - Prob. 80ECh. 4.5 - Prob. 81ECh. 4.5 - Find . Ch. 4.5 - Prob. 83ECh. 4.5 - This exercise explores the difference between the...Ch. 4.5 - Show that Ch. 4.5 - Prob. 86ECh. 4.5 - Prob. 87ECh. 4.5 - Find f′(0) for Ch. 4.5 - Prob. 89ECh. 4.5 - Prob. 90ECh. 4.6 - Minimizing perimeter What is the smallest...Ch. 4.6 - Prob. 2ECh. 4.6 - The figure shows a rectangle inscribed in an...Ch. 4.6 - A rectangle has its base on the x-axis and its...Ch. 4.6 - You are planning to make an open rectangular box...Ch. 4.6 - You are planning to close off a corner of the...Ch. 4.6 - The best fencing plan A rectangular plot of...Ch. 4.6 - The shortest fence A 216 m2 rectangular pea patch...Ch. 4.6 - Designing a tank Your iron works has contracted to...Ch. 4.6 - Catching rainwater A 1125 ft3 open-top rectangular...Ch. 4.6 - Designing a poster You are designing a rectangular...Ch. 4.6 - Find the volume of the largest right circular cone...Ch. 4.6 - Two sides of a triangle have lengths a and b, and...Ch. 4.6 - Designing a can What are the dimensions of the...Ch. 4.6 - Designing a can You are designing a 1000 cm3 right...Ch. 4.6 - Designing a box with a lid A piece of cardboard...Ch. 4.6 - Prob. 17ECh. 4.6 - Prob. 18ECh. 4.6 - Find the dimensions of a right circular cylinder...Ch. 4.6 - The U.S. Postal Service will accept a box for...Ch. 4.6 - Prob. 21ECh. 4.6 - A window is in the form of a rectangle surmounted...Ch. 4.6 - A silo (base not included) is to be constructed in...Ch. 4.6 - The trough in the figure is to be made to the...Ch. 4.6 - Prob. 25ECh. 4.6 - Constructing cylinders Compare the answer to the...Ch. 4.6 - Constructing cones A right triangle whose...Ch. 4.6 - Prob. 28ECh. 4.6 - Find a positive number for which the sum of it and...Ch. 4.6 - Find a positive number for which the sum of its...Ch. 4.6 - Prob. 31ECh. 4.6 - Answer Exercise 31 if one piece is bent into a...Ch. 4.6 - Determine the dimensions of the rectangle of...Ch. 4.6 - Determine the dimensions of the rectangle of...Ch. 4.6 - Prob. 35ECh. 4.6 - Prob. 36ECh. 4.6 - Prob. 37ECh. 4.6 - Prob. 38ECh. 4.6 - Consider the accompanying graphs of y = 2x + 3 and...Ch. 4.6 - Find the point on the graph of y = 20x3 + 60x −...Ch. 4.6 - Prob. 41ECh. 4.6 - A cone is formed from a circular piece of material...Ch. 4.6 - Prob. 43ECh. 4.6 - Quickest route Jane is 2 mi offshore in a boat and...Ch. 4.6 - Shortest beam The 8-ft wall shown here stands 27...Ch. 4.6 - Motion on a line The positions of two particles on...Ch. 4.6 - Prob. 47ECh. 4.6 - Prob. 48ECh. 4.6 - Strength of a beam The strength S of a rectangular...Ch. 4.6 - Stiffness of a beam The stiffness S of a...Ch. 4.6 - Prob. 51ECh. 4.6 - Prob. 52ECh. 4.6 - Distance between two ships At noon, ship A was 12...Ch. 4.6 - Fermat’s principle in optics Light from a source A...Ch. 4.6 - Tin pest When metallic tin is kept below 13.2°C,...Ch. 4.6 - Airplane landing path An airplane is flying at...Ch. 4.6 - It costs you c dollars each to manufacture and...Ch. 4.6 - You operate a tour service that offers the...Ch. 4.6 - Wilson lot size formula One of the formulas for...Ch. 4.6 - Prob. 60ECh. 4.6 - Show that if r(x) = 6x and c(x) = x3 − 6x2 + 15x...Ch. 4.6 - Production level Suppose that c(x) = x3 − 20x2 +...Ch. 4.6 - You are to construct an open rectangular box with...Ch. 4.6 - Prob. 64ECh. 4.6 - Prob. 65ECh. 4.6 - How we cough When we cough, the trachea (windpipe)...Ch. 4.6 - Prob. 67ECh. 4.6 - Prob. 68ECh. 4.6 - Prob. 69ECh. 4.6 - You have been asked to determine whether the...Ch. 4.6 - Prob. 71ECh. 4.6 - Prob. 72ECh. 4.6 - Prob. 73ECh. 4.6 - Prob. 74ECh. 4.7 - Use Newton’s method to estimate the solutions of...Ch. 4.7 - Use Newton’s method to estimate the one real...Ch. 4.7 - Prob. 3ECh. 4.7 - Prob. 4ECh. 4.7 - Prob. 5ECh. 4.7 - Prob. 6ECh. 4.7 - Use Newton’s method to find an approximate...Ch. 4.7 - Prob. 8ECh. 4.7 - Use Newton’s method to find an approximate...Ch. 4.7 - Prob. 10ECh. 4.7 - Prob. 11ECh. 4.7 - Prob. 12ECh. 4.7 - Prob. 13ECh. 4.7 - Prob. 14ECh. 4.7 - Prob. 15ECh. 4.7 - Prob. 16ECh. 4.7 - Prob. 17ECh. 4.7 - Prob. 18ECh. 4.7 - Prob. 19ECh. 4.7 - Prob. 20ECh. 4.7 - Prob. 21ECh. 4.7 - Prob. 22ECh. 4.7 - Intersection of curves At what value(s) of x does...Ch. 4.7 - Prob. 24ECh. 4.7 - The graphs of y = x2(x + 1) and y = 1/x (x > 0)...Ch. 4.7 - Prob. 26ECh. 4.7 - Intersection of curves At what value(s) of x does ...Ch. 4.7 - Prob. 28ECh. 4.7 - Prob. 29ECh. 4.7 - Prob. 30ECh. 4.7 - Prob. 31ECh. 4.7 - Prob. 32ECh. 4.7 - Prob. 33ECh. 4.7 - Prob. 34ECh. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - Prob. 2ECh. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - Prob. 8ECh. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - Prob. 12ECh. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1 -24, find an antiderivative for...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - Prob. 16ECh. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1 -24, find an antiderivative for...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - Prob. 22ECh. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 1–24, find an antiderivative for each...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - Prob. 30ECh. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - Prob. 34ECh. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - Prob. 36ECh. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - Prob. 40ECh. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - Prob. 48ECh. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - Prob. 50ECh. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - Prob. 54ECh. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - Prob. 58ECh. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - Prob. 60ECh. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - Prob. 66ECh. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - Prob. 68ECh. 4.8 - In Exercises 25-70, find the most general...Ch. 4.8 - In Exercises 25–70, find the most general...Ch. 4.8 - Prob. 71ECh. 4.8 - Prob. 72ECh. 4.8 - Prob. 73ECh. 4.8 - Verify the formulas in Exercises 71–82 by...Ch. 4.8 - Prob. 75ECh. 4.8 - Prob. 76ECh. 4.8 - Prob. 77ECh. 4.8 - Prob. 78ECh. 4.8 - Verify the formulas in Exercises 71–82 by...Ch. 4.8 - Prob. 80ECh. 4.8 - Verify the formulas in Exercises 71–82 by...Ch. 4.8 - Prob. 82ECh. 4.8 - Prob. 83ECh. 4.8 - Prob. 84ECh. 4.8 - Right, or wrong? Say which for each formula and...Ch. 4.8 - Prob. 86ECh. 4.8 - Prob. 87ECh. 4.8 - Prob. 88ECh. 4.8 - Prob. 89ECh. 4.8 - Prob. 90ECh. 4.8 - Prob. 91ECh. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Prob. 100ECh. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Prob. 103ECh. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Prob. 106ECh. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Prob. 108ECh. 4.8 - Prob. 109ECh. 4.8 - Prob. 110ECh. 4.8 - Prob. 111ECh. 4.8 - Solve the initial value problems in Exercises...Ch. 4.8 - Find the curve y = f(x) in the xy-plane that...Ch. 4.8 - Find a curve y = f(x) with the following...Ch. 4.8 - Prob. 115ECh. 4.8 - Prob. 116ECh. 4.8 - Prob. 117ECh. 4.8 - Prob. 118ECh. 4.8 - Prob. 119ECh. 4.8 - Prob. 120ECh. 4.8 - Prob. 121ECh. 4.8 - Prob. 122ECh. 4.8 - Prob. 123ECh. 4.8 - Liftoff from Earth A rocket lifts off the surface...Ch. 4.8 - Stopping a car in time You are driving along a...Ch. 4.8 - Prob. 126ECh. 4.8 - Motion along a coordinate line A particle moves on...Ch. 4.8 - The hammer and the feather When Apollo 15...Ch. 4.8 - Prob. 129ECh. 4.8 - Prob. 130ECh. 4.8 - Prob. 131ECh. 4 - Prob. 1GYRCh. 4 - Prob. 2GYRCh. 4 - Prob. 3GYRCh. 4 - Prob. 4GYRCh. 4 - What are the hypotheses and conclusion of the Mean...Ch. 4 - Prob. 6GYRCh. 4 - Prob. 7GYRCh. 4 - Prob. 8GYRCh. 4 - Prob. 9GYRCh. 4 - Prob. 10GYRCh. 4 - Prob. 11GYRCh. 4 - Prob. 12GYRCh. 4 - Prob. 13GYRCh. 4 - Prob. 14GYRCh. 4 - Prob. 15GYRCh. 4 - Prob. 16GYRCh. 4 - Prob. 17GYRCh. 4 - Prob. 18GYRCh. 4 - Prob. 19GYRCh. 4 - Prob. 20GYRCh. 4 - Prob. 21GYRCh. 4 - Prob. 22GYRCh. 4 - Prob. 23GYRCh. 4 - Prob. 24GYRCh. 4 - Prob. 25GYRCh. 4 - Prob. 1PECh. 4 - Prob. 2PECh. 4 - In Exercises 1–16, find the extreme values...Ch. 4 - Prob. 4PECh. 4 - Prob. 5PECh. 4 - Prob. 6PECh. 4 - In Exercises 1–16, find the extreme values...Ch. 4 - Prob. 8PECh. 4 - Prob. 9PECh. 4 - Prob. 10PECh. 4 - In Exercises 1–16, find the extreme values...Ch. 4 - Prob. 12PECh. 4 - In Exercises 1–16, find the extreme values...Ch. 4 - Prob. 14PECh. 4 - Prob. 15PECh. 4 - Prob. 16PECh. 4 - Does f(x) = x3 + 2x + tan x have any local maximum...Ch. 4 - Prob. 18PECh. 4 - Prob. 19PECh. 4 - Prob. 20PECh. 4 - Prob. 21PECh. 4 - Prob. 22PECh. 4 - In Exercises 23 and 24, find the absolute maximum...Ch. 4 - Prob. 24PECh. 4 - Prob. 25PECh. 4 - Prob. 26PECh. 4 - Prob. 27PECh. 4 - Prob. 28PECh. 4 - Prob. 29PECh. 4 - Prob. 30PECh. 4 - Prob. 31PECh. 4 - Prob. 32PECh. 4 - Prob. 33PECh. 4 - Prob. 34PECh. 4 - Prob. 35PECh. 4 - Prob. 36PECh. 4 - Prob. 37PECh. 4 - Prob. 38PECh. 4 - Prob. 39PECh. 4 - Prob. 40PECh. 4 - Prob. 41PECh. 4 - Prob. 42PECh. 4 - Prob. 43PECh. 4 - Prob. 44PECh. 4 - Prob. 45PECh. 4 - Prob. 46PECh. 4 - Prob. 47PECh. 4 - Prob. 48PECh. 4 - Prob. 49PECh. 4 - Prob. 50PECh. 4 - Graph the curves in Exercises 43–58. 51. Ch. 4 - Prob. 52PECh. 4 - Graph the curves in Exercises 43–58. 53. Ch. 4 - Prob. 54PECh. 4 - Graph the curves in Exercises 43–58. 55. y = ln...Ch. 4 - Prob. 56PECh. 4 - Prob. 57PECh. 4 - Prob. 58PECh. 4 - Prob. 59PECh. 4 - Prob. 60PECh. 4 - Prob. 61PECh. 4 - Prob. 62PECh. 4 - Prob. 63PECh. 4 - Prob. 64PECh. 4 - Prob. 65PECh. 4 - Prob. 66PECh. 4 - Prob. 67PECh. 4 - Prob. 68PECh. 4 - Sketch the graphs of the rational functions in...Ch. 4 - Prob. 70PECh. 4 - Prob. 71PECh. 4 - Prob. 72PECh. 4 - Sketch the graphs of the rational functions in...Ch. 4 - Prob. 74PECh. 4 - Sketch the graphs of the rational functions in...Ch. 4 - Prob. 76PECh. 4 - Prob. 77PECh. 4 - Prob. 78PECh. 4 - Prob. 79PECh. 4 - Prob. 80PECh. 4 - Prob. 81PECh. 4 - Prob. 82PECh. 4 - Prob. 83PECh. 4 - Prob. 84PECh. 4 - Prob. 85PECh. 4 - Prob. 86PECh. 4 - Prob. 87PECh. 4 - Prob. 88PECh. 4 - Prob. 89PECh. 4 - Prob. 90PECh. 4 - Prob. 91PECh. 4 - Prob. 92PECh. 4 - Prob. 93PECh. 4 - Prob. 94PECh. 4 - Prob. 95PECh. 4 - Prob. 96PECh. 4 - Prob. 97PECh. 4 - Prob. 98PECh. 4 - Prob. 99PECh. 4 - Prob. 100PECh. 4 - Prob. 101PECh. 4 - Prob. 102PECh. 4 - The sum of two nonnegative numbers is 36. Find the...Ch. 4 - Prob. 104PECh. 4 - Prob. 105PECh. 4 - A customer has asked you to design an open-top...Ch. 4 - Find the height and radius of the largest right...Ch. 4 - The figure here shows two right circular cones,...Ch. 4 - Prob. 109PECh. 4 - Prob. 110PECh. 4 - Prob. 111PECh. 4 - Prob. 112PECh. 4 - Prob. 113PECh. 4 - Prob. 114PECh. 4 - Prob. 115PECh. 4 - Prob. 116PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 118PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 120PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 122PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 124PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 126PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 128PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 130PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 132PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 134PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 136PECh. 4 - Find the indefinite integrals (most general...Ch. 4 - Prob. 138PECh. 4 - Solve the initial value problems in Exercises...Ch. 4 - Prob. 140PECh. 4 - Solve the initial value problems in Exercise...Ch. 4 - Prob. 142PECh. 4 - Prob. 143PECh. 4 - Prob. 144PECh. 4 - Prob. 145PECh. 4 - Prob. 146PECh. 4 - Prob. 147PECh. 4 - Prob. 148PECh. 4 - Prob. 149PECh. 4 - Prob. 150PECh. 4 - Prob. 151PECh. 4 - Prob. 152PECh. 4 - Prob. 153PECh. 4 - Prob. 154PECh. 4 - Prob. 1AAECh. 4 - Prob. 2AAECh. 4 - Prob. 3AAECh. 4 - Prob. 4AAECh. 4 - Prob. 5AAECh. 4 - Prob. 6AAECh. 4 - Bounding a function Suppose that f is continuous...Ch. 4 - Prob. 8AAECh. 4 - Prob. 9AAECh. 4 - Prob. 10AAECh. 4 - Prob. 11AAECh. 4 - Prob. 12AAECh. 4 - Largest inscribed triangle Points A and B lie at...Ch. 4 - Prob. 14AAECh. 4 - Prob. 15AAECh. 4 - Prob. 16AAECh. 4 - Prob. 17AAECh. 4 - Prob. 18AAECh. 4 - Prob. 19AAECh. 4 - Prob. 20AAECh. 4 - Prob. 21AAECh. 4 - Prob. 22AAECh. 4 - Prob. 23AAECh. 4 - Prob. 24AAECh. 4 - Prob. 25AAECh. 4 - Prob. 26AAECh. 4 - Prob. 27AAECh. 4 - Prob. 28AAECh. 4 - Prob. 29AAECh. 4 - Prob. 30AAECh. 4 - Can there be a curve satisfying the following...Ch. 4 - Find the equation for the curve in the xy-plane...Ch. 4 - A particle moves along the x-axis. Its...Ch. 4 - A particle moves with acceleration . Assuming that...Ch. 4 - Given f(x) = ax2 + 2bx + c with a > 0. By...Ch. 4 - Prob. 36AAECh. 4 - Prob. 37AAECh. 4 - Prob. 38AAECh. 4 - Prob. 39AAE
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Finding Local Maxima and Minima by Differentiation; Author: Professor Dave Explains;https://www.youtube.com/watch?v=pvLj1s7SOtk;License: Standard YouTube License, CC-BY