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Problem 4.9.29
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx
Problem 4.20.1
{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = cos x - x/7
Problem 4.3.34
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = cos² x on [-π,π]
Problem 4.6.66
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) = (x+4)/(4-x)
Problem 4.5.11
Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)
Problem 4.8.15
{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = sin x + x - 1; x₀ = 0.5
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 4.3.82
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) = eˣ(x - 2)²
Problem 4.3.32
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = √2 sin x- x on [0, 2π]
Problem 4.3.36
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x²√(9 - x²) on (-3,3)
Problem 4.9.97
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = -32; v(0) = 20, s(0) = 0
Problem 4.9.57
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/x√(36x² - 36))dx
Problem 4.6.21
Linear approximation Find the linear approximation to the following functions at the given point a.
g(t) = √(2t + 9); a = -4
Problem 4.3.102
Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.
f"(x) > 0 on (-∞,-2); f"(-2) = 0; f'(1) = 0; f"(2) = 0; f'(3) = 0; f"(x) > 0 on (4,∞)
Problem 4.7.46
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (ln(3x + 5eˣ)) / (ln(7x + 3e²ˣ)
Problem 4.9.53
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (6/√(4 - 4x²))dx
Problem 4.9.55
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4/x√(x² - 1))dx
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 4.5.33
Maximum-volume cone A cone is constructed by cutting a sector from a circular sheet of metal with radius 20. The cut sheet is then folded up and welded (see figure). Find the radius and height of the cone with maximum volume that can be formed in this way. <IMAGE>
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 4.1.29
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x³ -4a²x
Problem 4.3.39
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = -12x⁵ + 75x⁴ - 80x³
Problem 4.9.101
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = 2 + 3 sin t; v(0) = 1, s(0) = 10
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 4.2.42
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 4.5.68
Cylinder and cones (Putnam Exam 1938) Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?
Problem 4.R.2e
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
e. On what intervals (approximately) is f concave up?
Problem 4.R.56
{Use of Tech} Newton’s method Use Newton’s method to approximate the roots of ƒ(x) = e⁻²ˣ + 2eˣ - 6 to six digits.
Problem 4.R.107
104–107. Functions from derivatives Find the function f with the following properties.
h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3)