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

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

67. ∫(from 0 to 2√3)dx/√(4+x²)

Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at

b. x=2

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.

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

Verify the integration formulas in Exercises 37–40.

37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C

75. b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

g(x) = x(ln x)²

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

6. b. arccsc(-2/√3)

2. Express the following logarithms in terms of ln 5 and ln 7.

b. ln 9.8

Find the volumes of the solids in Exercises 135 and 136.

135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are

b. vertical squares whose base edges run from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).

3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.

b. ln(3x² - 9x) + ln(1/3x)

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

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.

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

2. y' = y²

b. y = -1/(x+3)

71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]

In Exercises 1–4, solve for t.

2. b. e^(kt) = 10

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

3. c. sin^(-1)(-√3/2)

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. √(x^4 + x^3)

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

c. 1/√x

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

1. c. tan^(-1)(1/√3)

86. This exercise explores the difference between

lim(x→∞)(1 + 1/x²)^x

and

lim(x→∞)(1 + 1/x)^x = e

c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

c. ln(√x)

In Exercises 1–4, solve for t.

1. c. e^((ln 0.2)t) = 0.4

9. True, or false? As x→∞,

c. x = O(x+5)

4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.

c. 3ln ∛(t² - 1) - ln(t+1)

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:

c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).

10. True, or false? As x→∞,

c. 1/x - 1/x² = o(1/x)

1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

c. √x

In Exercises 41–44:

c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that

(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a

44. f(x) = 2x², x ≥ 0, a = 5