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)

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→∞ (1 - (3/x))ˣ

{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)?

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.

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→π / 2- (sin x) ^tan x

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (x² / (x⁴ + x²)) dx

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_y→0⁺ (ln¹⁰ y) / √y

Cosine limits Let n be a positive integer. Evaluate the following limits.

lim_x→0 (1 - cos xⁿ) / x²ⁿ

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (2x +1)² dx

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

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.

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) = sin 2x + 3 on [-π , π]

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]

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>

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.

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 - 6x² on [-1, 5]

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (⁴√x³ + √x⁵) dx

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

a . Give the approximate coordinates of the local maxima and minima of ƒ

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

ln x and log₁₀ x

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) = 4x¹⸍² - x⁵⸍² on [0, 4]

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ ((1/x²) - (2/(x⁵⸍²))) dx

104–107. Functions from derivatives Find the function f with the following properties.

ƒ'(t) = sin t + 2t; ƒ(0) = 5

{Use of Tech } Min​​imizing 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.)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The function f(x) = √x has a local maximum on the interval [0,∞).

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < -1

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

a. Find the velocity of the object for all relevant times. 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>

a. Find the function T that gives the travel time as a function of x, where 0 ≤ x ≤ 50.

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure).​​ <IMAGE>

a. Find the time at which the object first passes the rest position, y = 0.