4. Applications of Derivatives

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x² ln x; x³

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

lim_Θ→π/2 (tan Θ - secΘ)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x).

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ

Checking the Mean Value Theorem

Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.

g(x) = {x³, −2 ≤ x ≤ 0

x², 0 < x ≤ 2

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.

f(x) = cos⁻¹ x - x; x₀ = 0.75

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

lim_x→1⁻ (x/(x-1) - x/(ln x) 

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)

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

Differential Estimates of Change

In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.

The change in the surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.

c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)

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

lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

63. lim (x → ∞) ((x + 2)/(x - 1))^x

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

lim_x→∞ log₂ x / log₃ x

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

35. lim (x → 0⁺) ln(x² + 2x) / ln x