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Problem 7.3.145a
145. The linearization of eˣ at x = 0
a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.
Problem 7.3.155a
155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
a. Find an equation for the line through the origin tangent to the graph of
y = ln(x).
Problem 7.7.37a
Verify the integration formulas in Exercises 37–40.
37. a. ∫sech(x)dx = tan⁻¹(sinh x) + C
Problem 7.1.72a
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Problem 7.6.3a
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
3. a. arcsin(-1/2)
Problem 7.8.3a
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + 4x
Problem 7.1.44a
In Exercises 41–44:
a. Find f⁻¹(x).
44. f(x) = 2x², x ≥ 0, a = 5
Problem 7.8.10.a
10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)
Problem 7.3.143a
143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.
Problem 7.4.1a
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
a. y = e^(-x)
Problem 7.1.37a
Find the inverse of the function f(x)=mx, where m is a constant different from zero.
Problem 7.8.9.a
9. True, or false? As x→∞,
a. x = o(x)
Problem 7.2.84a
84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.
Problem 7.6.5a
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. a. arccos(1/2)
Problem 7.8.2.a
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. 10x^4 + 30x + 1
Problem 7.1.40a
Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?
Problem 7.5.80a
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
a. f(x) = x, g(x) = x², (a, b) = (-2, 0)
Problem 7.8.4a
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + √x
Problem 7.5.80b
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
b. f(x) = x, g(x) = x², (a, b) arbitrary
Problem 7.1.39b
Find the inverse of f(x)=x+b (b constant). How is the graph of f^(-1) related to the graph of f?
Problem 7.7.73b
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)
Problem 7.1.67b
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Problem 7.6.4b
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)
Problem 7.5.91.b
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
Problem 7.6.8b
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)
Problem 7.6.2b
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
2. b. tan^(-1)(√3)
Problem 7.2.1b
1. Express the following logarithms in terms of ln 2 and ln 3.
b. ln(4/9)
Problem 7.7.69b
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
69. ∫(from 5/4 to 2)dx/(1-x²)
Problem 7.3.132b
132. Let f(x) = e^x / (1 + e^(2x)).
b. Find all inflection points for f.
Problem 7.1.72b
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
