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)).

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)

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)).

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

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

c. x=3

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

7. c. arcsec(-2)

c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −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:

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

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

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

c. x²e^(-x)

88. Given that x>0, find the maximum value, if any, of

c. x^(1/x^n) (n a positive integer)

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?

c. √(1+x^4)

What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?

82. Use the definitions of the hyperbolic functions to find each of the following limits.

c. lim(x→∞) sinh x

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

5. c. arccos(√3/2)

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

1. 2y' + 3y = e^(-x)

c. y = e^(-x) + Ce^(-(3/2)x)

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

d. ln 1225

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:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

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.

d. Conclude that

xᵉ < eˣ for all positive x ≠ e.

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:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

1. Express the following logarithms in terms of ln 2 and ln 3.

d. ln ∛9

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:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

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

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:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

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

e. x^3 - x^2

112. True, or false? Give reasons for your answers.

e. sec^(-1)x = O(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:

e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

9. True, or false? As x→∞,

e. e^x = o(e^(2x))

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?

e. (3/2)^x

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?

e. x

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?

e. e^(-x)

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

e. x ln(x)

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?

e. x - 2ln(x)