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Problem 4.7.34
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (sin² 3x) / x²
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 4.3.21
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = (x - 1)²
Problem 4.3.94
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) = 2x⁻³ - x⁻²
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 4.7.70
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (log₂ x - log₃ x)
Problem 4.3.85
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) = x²e⁻ˣ
Problem 4.1.16
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Problem 4.3.83
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) = 2x³ - 3x² + 12
Problem 4.4.11
Sketch a continuous function f on some interval that has the properties described. Answers will vary.
The function f satisfies f'(-2) = 2, f'(0) = 0, f'(1) = -3 and f'(4) = 1.
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 4.6.11
Suppose f is differentiable on (-∞,∞) and f(5.01) - f(5) = 0.25.Use linear approximation to estimate the value of f'(5).
Problem 4.4.66
{Use of Tech} A pursuit curve A man stands 1 mi east of a crossroads. At noon, a dog starts walking north from the crossroads at 1 mi/hr (see figure). At the same instant, the man starts walking and at all times walks directly toward the dog at s > 1 mi/hr . The path in the xy-plane followed by the man as he pursues the dog is given by the function y = ƒ(x) = s/2 ((x(ˢ⁺¹)/ˢ) /(s+1) - (x(ˢ⁺¹)/ˢ / s-1)) + s/ s² - 1
Select various values of s > 1 and graph this pursuit curve. Comment on the changes in the curve as s increases. <IMAGE>
Problem 4.5.51
Viewing angles An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 3 ft above eye level and the upper edge of the screen is 10 ft above eye level (see figure). How far from the screen should you stand to maximize your viewing angle? <IMAGE>
Problem 4.3.87
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) = 2x² ln x - 11x²
Problem 4.7.33
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (1 - cos 3x) / 8x²
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 4.7.82
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_z→∞ (1 + 10/z²)ᶻ²
Problem 4.9.65
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((1 + √x)/x)dx
Problem 4.1.67
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) = x²/₃(4 -x²) on -2,2
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 4.9.73
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(v) = sec v tan v; F(0) = 2, -π/2 < v < π/2
Problem 4.8.49
{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) = cos x
Problem 4.9.35
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((4x⁴ - 6x²) / x ) dx
Problem 4.7.39
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)
Problem 4.9.39
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (csc² Θ + 2Θ² - 3Θ) dΘ
Problem 4.8.11
{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.
f(x) = e⁻ˣ - x; x₀ = ln 2
Problem 4.7.57
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π/2 - x) sec x
Problem 4.4.15
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x² - 6x