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Problem 4.9.35
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((4x⁴ - 6x²) / x ) dx
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 4.5.61
Metal rain gutters A rain gutter is made from sheets of metal 9 in wide. The gutters have a 3-in base and two 3-in sides, folded up at an angle Θ (see figure). What angle Θ maximizes the cross-sectional area of the gutter? <IMAGE>
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 4.1.23
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x² - 4x + 2
Problem 4.9.61
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (eˣ⁺²) dx
Problem 4.8.5
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 4.1.55
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 4.4.73
{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.
x²/₃ + y²/₃ = 1 (Astroid or hypocycloid with four cusps)
Problem 4.7.31
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_u→ π/4 (tan u - cot u) / (u - π/4)
Problem 4.9.9
If F(x) = x² - 3x + C and F (-1) = 4 , what is the value of C?
Problem 4.9.5
Give the antiderivatives of xᵖ . For what values of p does your answer apply?
Problem 4.3.35
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x²/₃ (x²-4)
Problem 4.8.50
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = tan x/2 on (-π,π)
Problem 4.8.29
{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.
y = 1/x and y = 4 - x²
Problem 4.4.15
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x² - 6x
Problem 4.9.93
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) = 2√t; s(0) = 1
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 4.6.68
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 4.1.25
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x³ / 3 - 9x
Problem 4.9.19
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
ƒ(x) = eˣ
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 4.7.28
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 4.9.57
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (1/x√(36x² - 36))dx
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 4.9.43
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (sec² x - 1) dx
Problem 4.9.25
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4√x - (4 /√x)) dx
Problem 4.4.11
Sketch a continuous function f on some interval that has the properties described. Answers will vary.
The function f satisfies f'(-2) = 2, f'(0) = 0, f'(1) = -3 and f'(4) = 1.