4. Applications of Derivatives

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.

f(x) = e⁻ˣ - x; x₀ = ln 2

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

1. lim (x → -2) (x + 2) / (x² - 4)

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s

b. area?

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

lim_x→π/2⁻ (π - 2x) tan x  

21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].

b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.

ƒ(x) = x + 1/x; [1,3]

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = sin⁻¹ x

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

lim_x→ 1 (x² + 2x) / (x +3)

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

3. lim (x → ∞) (5x² - 3x) / (7x² + 1)

{Use of Tech} Modified Newton’s method The function f has a root of multiplicity 2 at r if f(r) = f'(r) = 0 and f"(r) ≠ 0. In this case, a slight modification of Newton’s method, known as the modified (or accelerated) Newton’s method, is given by the formula xₙ + 1 = xₙ - (2f(xₙ)/(f'(xₙ), for n = 0, 1, 2, . . . . This modified form generally increases the rate of convergence.

b. Apply Newton’s method and the modified Newton’s method using x₀ = 0.1 to find the value of x₃ in each case. Compare the accuracy of these values of x₃.

Running pace Explain why if a runner completes a 6.2-mi (10-km) race in 32 min, then he must have been running at exactly 11 mi/hr at least twice in the race. Assume the runner’s speed at the finish line is zero.

Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

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

lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = tan x

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = e⁻ˣ - ((x + 4)/5)

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = cos 2x - x² + 2x