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Problem 4.3.40
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x² - 2 ln x
Problem 4.1.23
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x² - 4x + 2
Problem 4.1.35
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 1 / x + ln x
Problem 4.5.30
Shortest ladder A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.
Problem 4.1.39
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x √(x-a)
Problem 4.1.43
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x² - 10 on [-2, 3]
Problem 4.1.11
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>
Problem 4.7.47
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_v→3 (v-1-√(v²-5)) / (v-3)
Problem 4.1.17
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Problem 4.8.27
{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.
y = sin x and y = x/2
Problem 4.9.27
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (5s + 3)² ds
Problem 4.3.92
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = x³ - 13x² - 9x
Problem 4.7.55
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 csc 6x sin 7x
Problem 4.1.79
{Use of Tech} Absolute maxima and minima
a. Find the critical points of f on the given interval.
b. Determine the absolute extreme values of f on the given interval.
c. Use a graphing utility to confirm your conclusions.
f(x) = 2ᶻ sin x on [-2,6]
Problem 4.3.78
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = 6x² - x³
Problem 4.7.36
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - x - 1) / 5x²
Problem 4.7.40
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (sin x - x) / 7x³
Problem 4.5.61
Metal rain gutters A rain gutter is made from sheets of metal 9 in wide. The gutters have a 3-in base and two 3-in sides, folded up at an angle Θ (see figure). What angle Θ maximizes the cross-sectional area of the gutter? <IMAGE>
Problem 4.7.70
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (log₂ x - log₃ x)
Problem 4.5.36
Covering a marble Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius 0 < r < 4 is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?
Problem 4.9.63
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1))dw
Problem 4.1.53
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = (2x)ˣ on [0.1,1]
Problem 4.7.83
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (x + cos x)¹/ˣ
Problem 4.9.105
105–106. {Use of Tech} Races The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other.
A : v(t) = sin t; s(0) = 0 B. V(t) = cos t; S(0) = 0
Problem 4.7.28
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (x - 3 √x) / (x - √x)
Problem 4.6.70
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = ln (1 - x)
Problem 4.9.91
Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.
v(t) = 2t + 4; s(0) = 0
Problem 4.7.35
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π (cos x +1 ) / (x - π )²
Problem 4.3.37
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = √(9 - x²) + sin⁻¹ (x/3)
Problem 4.6.65
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = 2 - a cos x, a constant