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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.
Problem 4.R.97
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ dx / (1 - sin² x)
Problem 4.R.2c
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
c. Give the approximate coordinates of the inflection point(s) of f.
Problem 4.R.5c
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
c. Determine where f has local maxima and minima.
Problem 4.R.118a
{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.
a. What is the domain of f (in terms of a)?
Problem 4.R.49
Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).
Problem 4.R.118c
{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.
c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.
Problem 4.R.63
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²
Problem 4.R.95
90–103. Indefinite integrals Determine the following indefinite integrals.
∫(1 + 3 cosΘ) dΘ
Problem 4.R.16
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x sin⁻¹ x on [-1, 1]
Problem 4.R.81
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx
Problem 4.R.5a
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.
Problem 4.R.118b
{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.
b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).
Problem 4.R.71
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→0 csc x sin⁻¹ x
Problem 4.R.2f
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
f. On what intervals (approximately) is f concave down?
Problem 4.R.10
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = x³ ln x on (0, ∞)
Problem 4.R.43
Maximum area A line segment of length 10 joins the points (0, p) and (q, 0) to form a triangle in the first quadrant. Find the values of p and q that maximize the area of the triangle.
Problem 4.R.77
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_ x→0 ⁺ | ln x | ˣ
Problem 4.R.73
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ ln ((x +1) / (x-1))
Problem 4.R.89
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
2ˣ and 4ˣ⸍²
Problem 4.R.45
45–46. Linear approximation
a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?
ƒ(x) = x²⸍³ ; a =27; ƒ(29)
Problem 4.R.1c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.
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.79
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ (1 - (3/x))ˣ
Problem 4.R.29
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = ln( x² + 3) / (x -1)
Problem 4.R.2b
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
b. Give the approximate coordinates of the absolute maximum and minimum values of ƒ (if they exist).
Problem 4.R.101
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
Problem 4.R.67
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_y→0⁺ (ln¹⁰ y) / √y
Problem 4.R.41
Maximum printable area A rectangular page in a text (with width x and length y) has an area of 98 in² , top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?
Problem 4.R.116a
Cosine limits Let n be a positive integer. Evaluate the following limits.
lim_x→0 (1 - cos xⁿ) / x²ⁿ