Back
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)
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.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.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.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.81
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx
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.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.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.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.75
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→π / 2- (sin x) ^tan 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.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.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.39
Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>
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.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.79
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ (1 - (3/x))ˣ
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.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.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.101
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx