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Problem 4.8.50
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = tan x/2 on (-π,π)
Problem 4.4.73
{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.
x²/₃ + y²/₃ = 1 (Astroid or hypocycloid with four cusps)
Problem 4.6.67
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) = 3x³ - 4x
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 4.7.29
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (3 sin 4x) / 5x
Problem 4.7.42
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (tan⁻¹ x - π/2)/(1/x)
Problem 4.1.63
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) = sec x on [-(π/4),π/4]
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 4.1.39
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x √(x-a)
Problem 4.1.17
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Problem 4.3.78
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 4.3.92
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 4.9.71
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = 8x³ + sin x; F(0) = 2
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 4.2.5
5–7. For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.
a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .
b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.
ƒ(x) = x² / 4 + 1 ; [ -2, 4] <IMAGE>
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 4.9.27
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (5s + 3)² ds
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 4.7.70
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 4.7.40
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 4.3.35
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x²/₃ (x²-4)
Problem 4.6.19
Linear approximation Find the linear approximation to the following functions at the given point a.
f(x) = 4x² + x; a = 1
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 4.9.3
Describe the set of antiderivatives of ƒ(x) = 1
Problem 4.2.1
Explain Rolle’s Theorem with a sketch.
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 4.7.35
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 4.6.65
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