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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.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.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.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.77
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_ x→0 ⁺ | ln x | ˣ
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.81
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx
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.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.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.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.35
Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>
Problem 4.R.37
{Use of Tech } Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)
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.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.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.44
Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.
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.31
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) = 10x² / (x² + 3)
Problem 4.R.27
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) = 4cos (π (x-1)) on [0, 2]
Problem 4.R.34
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.
{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ
Problem 4.R.91
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (2x +1)² dx
Problem 4.R.93
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ ((1/x²) - (2/(x⁵⸍²))) dx
Problem 4.R.95
90–103. Indefinite integrals Determine the following indefinite integrals.
∫(1 + 3 cosΘ) dΘ
Problem 4.R.97
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ dx / (1 - sin² x)
Problem 4.R.99
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (12/x)dx
Problem 4.R.101
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
Problem 4.R.103
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (⁴√x³ + √x⁵) dx
Problem 4.R.105
104–107. Functions from derivatives Find the function f with the following properties.
ƒ'(t) = sin t + 2t; ƒ(0) = 5
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)