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 .

c. Find the time when the object reaches its highest point. What is the height? 

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.

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 .

d. Find the time when the object strikes the ground.

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} 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>

d. Find the time and the displacement when the object reaches its high point for the second time.

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

d. Identify the local extreme values and inflection points of ƒ .

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).​​ 

d. Find the number and location of the fixed points of g for a = 2, 3, and 4 on the interval 0 ≤ x ≤ 1. 

Rectangles in triangles Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures.

d. An arbitrary triangle with a given area A (The result applies to any triangle, but first consider triangles for which all the angles are less than or equal to 90° .)

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

f' < 0 and f" < 0, for 8 < x < 10

Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>

f. Sketch one possible graph of f.

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

g. Use your work in parts (a) through (f) to sketch a graph of ƒ .

Let ƒ(x) = (x - 3) (x + 3)²

g. Use your work in parts (a) through (f) to sketch a graph of ƒ.

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

Plot a possible graph of f.

For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.

a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .

b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.

ƒ(x) = x⁵/16 ; [-2, 2] <IMAGE>

Evaluate lim_x→2 (x³ - 3x² + 2) / (x-2) using l’Hôpital’s Rule and then check your work by evaluating the limit using an appropriate Chapter 2 method.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→2 (x² - 2x / (x² - 6x + 8) 

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

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

Use ƒ' and ƒ" to complete parts (a) and (b). 

a. Find the intervals on which f is increasing and the intervals on which it is decreasing.

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x⁹/9 + 3x⁵ - 16x

Use ƒ' and ƒ" to complete parts (a) and (b). 

a. Find the intervals on which f is increasing and the intervals on which it is decreasing.

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x√(x +9)

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 1 ln x / (4x - x² - 3)

Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (3x⁴ - x²) / (6x⁴ + 12) 

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ ∞ (4x³ - 2x² + 6) / (πx³ + 4)

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) = (x⁴/2) - 3x² + 4x + 1

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) = 3x/(x² + 3)

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = 3x/(x² - 1)

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x²/(x - 2)

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = (x² + 12)/(2x + 1)

Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.

c. Find the elasticity function on the domain of the demand function.

Use the guidelines of this section to make a complete graph of f.

f(x) = tan⁻¹ (x²/√3)

Evaluate and simplify y'.

y = (3t²−1 / 3t²+1)^−3

Use the guidelines of this section to make a complete graph of f.

f(x) = x + 2 cos x on [-2π,2π)