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Problem 4.7.27
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (1 - ln x) / (1 + ln x)
Problem 4.9.13
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
ƒ(x) = 2 sinx + 1
Problem 4.9.25
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4√x - (4 /√x)) dx
Problem 4.6.59
Approximating changes
Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone of fixed height h = 6m when its radius decreases from r = 10 m to r = 9.9 m (S = πr√(r² + h²).
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 4.5.65
Crankshaft A crank of radius r rotates with an angular frequency w It is connected to a piston by a connecting rod of length L (see figure). The acceleration of the piston varies with the position of the crank according to the function <IMAGE>
a (Θ) = w²r (cos Θ + (r cos2Θ) / L) .
For fixed w , L, and r find the values of Θ, with 0 ≤ Θ ≤ 2π , for which the acceleration of the piston is a maximum and minimum.
Problem 4.7.25
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ e (ln x - 1) / (x - 1)
Problem 4.7.72
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π - 2x) tan x
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 4.1.21
Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.
ƒ' (x) and ƒ'3 are undefined; ƒ'(2) = 0; has a local maximum at x= 1; ƒ has local minimum at x = 2; and ƒ has an absolute maximum at x= 3; and ƒ has an absolute minimum at x = 4 .
Problem 4.6.9
Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.
Problem 4.6.68
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) = sin⁻¹ x
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 4.6.69
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) = tan x
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 4.3.85
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 4.7.70
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (log₂ x - log₃ x)
Problem 4.9.103
A car starting at rest accelerates at 16 ft/s² for 5 seconds on a straight road. How far does it travel during this time?
Problem 4.2.42
Avalanche forecasting Avalanche forecasters measure the temperature gradient dT/dh, which is the rate at which the temperature in a snowpack T changes with respect to its depth h. A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters use the following rule of thumb: If dT/dh exceeds 10° C/m anywhere in the snowpack, conditions are favorable for weak-layer formation, and the risk of avalanche increases. Assume the temperature function is continuous and differentiable.
a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0), the temperature is -16° C. At a depth of 1.1 m, the temperature is -2° C. Using the Mean Value Theorem, what can he conclude about the temperature gradient? Is the formation of a weak layer likely?
Problem 4.7.53
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 x csc x
Problem 4.9.23
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x⁵ - 5x⁹) dx
Problem 4.1.29
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x³ -4a²x
Problem 4.1.57
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) = 2x³ - 15x² + 24x on [0,5]
Problem 4.3.42
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x⁴/4 - 8x³/3 + 15x²/2 + 8
Problem 4.9.33
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3/x⁴ + 2 - 3/x²)
Problem 4.7.41
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (e¹/ₓ - 1)/(1/x)
Problem 4.1.61
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) = x/(x²+9)⁵ on [-2,2]
Problem 4.5.15
Minimum sum Find positive numbers x and y satisfying the equation xy = 12 such that the sum 2x + y is as small as possible.
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