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Problem 4.1.45
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) = x³ - 3x² on [-1, 3]
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 4.7.55
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 csc 6x sin 7x
Problem 4.3.94
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) = 2x⁻³ - x⁻²
Problem 4.1.21
Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.
ƒ' (x) and ƒ'3 are undefined; ƒ'(2) = 0; has a local maximum at x= 1; ƒ has local minimum at x = 2; and ƒ has an absolute maximum at x= 3; and ƒ has an absolute minimum at x = 4 .
Problem 4.2.5
5–7. 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) = x² / 4 + 1 ; [ -2, 4] <IMAGE>
Problem 4.8.33
{Use of Tech} Tumor size In a study conducted at Dartmouth College, mice with a particular type of cancerous tumor were treated with the chemotherapy drug Cisplatin. If the volume of one of these tumors at the time of treatment is V₀, then the volume of the tumor t days after treatment is modeled by the function V(t) = V₀ (0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ). (Source: Undergraduate Mathematics for the Life Sciences, MAA Notes No. 81, 2013)
Plot a graph of y = 0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ, for 0 ≤ t ≤ 16, and describe the tumor size over time. Use Newton’s method to determine when the tumor decreases to half of its original size.
Problem 4.1.16
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Problem 4.9.27
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (5s + 3)² ds
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.
Problem 4.6.61
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) = 2x + 1
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 4.7.75
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0⁺ x²ˣ
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 4.3.38
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x ln x - 2x + 3 on (0,∞)
Problem 4.7.33
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (1 - cos 3x) / 8x²
Problem 4.9.63
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1))dw
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
Problem 4.1.53
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) = (2x)ˣ on [0.1,1]
Problem 4.9.65
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((1 + √x)/x)dx
Problem 4.3.104
Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.
f'(x) > 0, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f"(0) = 0
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 4.8.45
{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.
Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?
Problem 4.8.49
{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) = cos x
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 4.7.39
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 4.9.39
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (csc² Θ + 2Θ² - 3Θ) dΘ
Problem 4.8.11
{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.
f(x) = e⁻ˣ - x; x₀ = ln 2
Problem 4.7.57
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π/2 - x) sec x
Problem 4.9.91
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) = 2t + 4; s(0) = 0