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Problem 4.6.67
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = 3x³ - 4x
Problem 4.6.68
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = sin⁻¹ x
Problem 4.6.69
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = tan x
Problem 4.6.70
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = ln (1 - x)
Problem 4.6.5
Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).
Problem 4.6.6
Suppose f is differentiable on (-∞,∞) and the equation of the line tangent to the graph of f at x = 2 is y = 5x -3. Use the linear approximation to f at x = 2 to approximate f(2.01).
Problem 4.6.7
Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.
Problem 4.6.8
Use linear approximation to estimate f (5.1) given that f(5) = 10 and f'(5) = -2.
Problem 4.6.9
Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.
Problem 4.6.11
Suppose f is differentiable on (-∞,∞) and f(5.01) - f(5) = 0.25.Use linear approximation to estimate the value of f'(5).
Problem 4.6.12
Suppose f is differentiable on (-∞,∞), f(5.99) = 7, and f(6) = 7.002. Use linear approximation to estimate the value of f'(6).
Problem 4.6.19
Linear approximation Find the linear approximation to the following functions at the given point a.
f(x) = 4x² + x; a = 1
Problem 4.6.21
Linear approximation Find the linear approximation to the following functions at the given point a.
g(t) = √(2t + 9); a = -4
Problem 4.6.45
Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
1/³√510
Problem 4.6.13
Estimating speed Use the linear approximation given in Example 1 to answer the following questions.
If you travel one mile in 59 seconds, what is your approximate average speed? What is your exact speed?
Problem 4.7.82
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_z→∞ (1 + 10/z²)ᶻ²
Problem 4.7.19
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 (x² + 2x) / (x +3)
Problem 4.7.20
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - 1) / (2x + 5)
Problem 4.7.22
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - 1) / (x² + 3x)
Problem 4.7.25
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ e (ln x - 1) / (x - 1)
Problem 4.7.26
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)
Problem 4.7.27
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (1 - ln x) / (1 + ln x)
Problem 4.7.28
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (x - 3 √x) / (x - √x)
Problem 4.7.29
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (3 sin 4x) / 5x
Problem 4.7.30
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→2π (x sin x + x² - 4π²) / (x - 2π)
Problem 4.7.36
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - x - 1) / 5x²
Problem 4.7.37
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ π/2⁻ (tanx ) / (3 / (2x - π))
Problem 4.7.38
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)
Problem 4.7.39
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)
Problem 4.7.40
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (sin x - x) / 7x³