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Problem 69
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ log₂ x / log₃ x
Problem 70
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = ³√(x - 4)
Problem 72
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = x⁴eˣ + x
Problem 72b
Another pen problem A rancher is building a horse pen on the corner of her property using 1000 ft of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure). <IMAGE>
b. Suppose there is already a fence along the side of the property opposite the side of length y. Find the lengths of the sides that maximize the area of the pen, using 1000 ft of fencing.
Problem 74
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
h(t) = 2 + cos 2t on [0,π]
Problem 74b
The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). <IMAGE>
b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.
Problem 75
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
g(t) = 3t⁵ - 30t⁴ + 80t³ + 100
Problem 76
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = 2x⁴ + 8x³ + 12x² - x - 2
Problem 76b
Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?
Problem 81b
{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.
b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.
Problem 87
Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.
lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)
Problem 89
More limits Evaluate the following limits.
lim_x→1 (x ln x - x + 1) / (xln²x)
Problem 100
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x² ln x; x³
Problem 101
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x²⁰ ; 1.0001ˣ
Problem 106
{Use of Tech} Graph carefully Graph the function f(x) = 60x⁵ - 901x³ + 27x in the window [-4,4] x [-10,000, 10,000]. How many extreme values do you see? Locate all the extreme values by analyzing f'.
Problem 107a
Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>
a. On what interval(s) is f increasing? Decreasing?
Problem 107c
The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>
c. At what point(s) does f have an inflection point?
Problem 111
Consider the lim_x→∞ (√ ax + b) / √cx + d where a, b, c, and d are positive real numbers. Show that l’Hôpital’s Rule fails for this limit. Find the limit using another method.
Problem 116
Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?
Problem 119
Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .
Problem 121a
Exponential growth rates
a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?
Problem 121b
Exponential growth rates
b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.