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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 4.4.26
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x³ - 147x + 286
Problem 4.7.43
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ -1 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)
Problem 4.9.61
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (eˣ⁺²) dx
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 4.5.25
Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?
Problem 4.7.32
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_z→0 (tan 4z) / (tan 7z)
Problem 4.8.25
{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = ln x - x² + 3x - 1
Problem 4.6.66
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) = (x+4)/(4-x)
Problem 4.3.83
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = 2x³ - 3x² + 12
Problem 4.2.1
Explain Rolle’s Theorem with a sketch.
Problem 4.1.55
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x+ cos⁻¹x on [-1,1]
Problem 4.1.65
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x³e⁻ˣ on [-1,5]
Problem 4.9.5
Give the antiderivatives of xᵖ . For what values of p does your answer apply?
Problem 4.1.43
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x² - 10 on [-2, 3]
Problem 4.1.37
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x² √(x + 5)
Problem 4.8.5
Let ƒ(x) = 2x³ - 6x² + 4x. Use Newton’s method to find x₁ given that x₀ = 1.4. Use the graph of f (see figure) and an appropriate tangent line to illustrate how x₁ is obtained from x₀ . <IMAGE>
Problem 4.4.15
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x² - 6x
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 4.7.47
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_v→3 (v-1-√(v²-5)) / (v-3)
Problem 4.5.42
Light transmission A window consists of a rectangular pane of clear glass surmounted by a semicircular pane of tinted glass. The clear glass transmits twice as much light per unit of surface area as the tinted glass. Of all such windows with a fixed perimeter P, what are the dimensions of the window that transmits the most light?
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 4.3.40
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x² - 2 ln x
Problem 4.5.11
Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)
Problem 4.7.46
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (ln(3x + 5eˣ)) / (ln(7x + 3e²ˣ)
Problem 4.1.41
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(t) = 1/5 t⁵ - a⁴t
Problem 4.8.23
{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = e⁻ˣ - ((x + 4)/5)
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 4.9.25
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4√x - (4 /√x)) dx
Problem 4.4.20
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x⁴ + 4x³