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Problem 4.7.43
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ -1 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)
Problem 4.6.63
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) = 1/x³
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 4.6.5
Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).
Problem 4.8.33
{Use of Tech} Tumor size In a study conducted at Dartmouth College, mice with a particular type of cancerous tumor were treated with the chemotherapy drug Cisplatin. If the volume of one of these tumors at the time of treatment is V₀, then the volume of the tumor t days after treatment is modeled by the function V(t) = V₀ (0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ). (Source: Undergraduate Mathematics for the Life Sciences, MAA Notes No. 81, 2013)
Plot a graph of y = 0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ, for 0 ≤ t ≤ 16, and describe the tumor size over time. Use Newton’s method to determine when the tumor decreases to half of its original size.
Problem 4.9.95
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) = 6t² + 4t - 10; s(0) = 0
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 4.9.63
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1))dw
Problem 4.3.44
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = tan⁻¹ (x/(x²+2))
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 4.9.89
{Use of Tech} Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.
f'(x) = 3x + sinx; f(0) = 3
Problem 4.6.37
Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
1/203
Problem 4.9.57
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/x√(36x² - 36))dx
Problem 4.9.31
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x + 1) (4 - 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 4.8.31
{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 = 4√x and y = x² + 1
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 4.6.21
Linear approximation Find the linear approximation to the following functions at the given point a.
g(t) = √(2t + 9); a = -4
Problem 4.9.53
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (6/√(4 - 4x²))dx
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 4.6.57
Approximating changes
Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).
Problem 4.9.75
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = (3y + 5)/y; F(1) = 3. y > 0
Problem 4.8.17
{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) = tan x - 2x; x₀ = 1.2
Problem 4.9.55
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4/x√(x² - 1))dx
Problem 4.2.42
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 4.5.33
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 4.9.91
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.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.83
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
x¹⸍² and x¹⸍³
Problem 4.R.2d
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
d. Give the approximate coordinates of the zero(s) of f.