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Problem 4.9.95
Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.
v(t) = 6t² + 4t - 10; s(0) = 0
Problem 4.9.97
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = -32; v(0) = 20, s(0) = 0
Problem 4.9.101
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = 2 + 3 sin t; v(0) = 1, s(0) = 10
Problem 4.9.103
A car starting at rest accelerates at 16 ft/s² for 5 seconds on a straight road. How far does it travel during this time?
Problem 4.9.89
{Use of Tech} Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.
f'(x) = 3x + sinx; f(0) = 3
Problem 4.9.105
105–106. {Use of Tech} Races The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other.
A : v(t) = sin t; s(0) = 0 B. V(t) = cos t; S(0) = 0
Problem 4.9.71
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = 8x³ + sin x; F(0) = 2
Problem 4.9.73
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(v) = sec v tan v; F(0) = 2, -π/2 < v < π/2
Problem 4.9.75
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = (3y + 5)/y; F(1) = 3. y > 0
Problem 4.9.111a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.
Problem 5d
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
Plot a possible graph of f.
Problem 7
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>
Problem 10
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.
Problem 17
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→2 (x² - 2x / (x² - 6x + 8)
Problem 18
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)
Problem 19
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
Problem 20
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)
Problem 21
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 ln x / (4x - x² - 3)
Problem 22
Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.
Problem 23
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (3x⁴ - x²) / (6x⁴ + 12)
Problem 24
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (4x³ - 2x² + 6) / (πx³ + 4)
Problem 25
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
Problem 26
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)
Problem 29
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 3x/(x² - 1)
Problem 31
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x²/(x - 2)
Problem 33
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = (x2 + 12)/(2x + 1)
Problem 34
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.
Problem 35
Use the guidelines of this section to make a complete graph of f.
f(x) = tan⁻¹ (x²/√3)
Problem 36
Evaluate and simplify y'.
y = (3t²−1 / 3t²+1)^−3
Problem 37
Use the guidelines of this section to make a complete graph of f.
f(x) = x + 2 cos x on [-2π,2π)