4. Applications of Derivatives

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→2 (x² - 2x / (x² - 6x + 8) 

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ x² ln( cos 1/x)

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→π / 2- (sin x) ^tan x

In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

5. lim (x → 0) (1 - cos x) / x²

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

49. lim (x → 0) (x - sin x) / (x tan 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?

a. x-3

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

12. lim (x → ∞) (x - 8x²) / (12x² + 5x)

{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 = 4√x and y = x² + 1

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)

Dependence on Initial Point

8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations

c. x_0=2

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)

Finding Functions from Derivatives

In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.

g'(x) = 1 / x² + 2x, P(−1, 1)

Finding Functions from Derivatives

Suppose that f'(x) = 2x for all x. Find f(2) if

b. f(1) = 0

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (eˣ - 1) / (x² + 3x)

Derivatives in Differential Form

In Exercises 17–28, find dy.

2y³/² + xy − x = 0