4. Applications of Derivatives

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

lim_y→0⁺ (ln¹⁰ y) / √y

Find the limits in Exercises 1–6.

3. lim(x→0⁺) (cox(√x))^(1/x)

Cosine limits Let n be a positive integer. Evaluate the following limits.

lim_x→0 (1 - cos xⁿ) / x²ⁿ

Quadratic approximations

[Technology Exercise] c. Graph f(x) = 1/(1 − x) and its quadratic approximation at x = 0. Then zoom in on the two graphs at the point (0,1). Comment on what you see.

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

b. Based on the Mean Value Theorem, what can you conclude about the instantaneous growth rate of bamboo measured in millimeters per second between 10:00 A.M. and 3:00 P.M.?

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

93. lim(x→0) (csc(x) - cot(x))

Approximating changes

Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).

Explain the Mean Value Theorem with a sketch.

{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.

Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?

11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.

ƒ(x) = 1 - x²⸍³ ; [-1, 1]

Tolerance

a. About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value?

Quadratic approximations

b. Find the quadratic approximation to f(x) = 1/(1 − x) at x = 0.

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = cos(x²)

Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

f(x) = x⁴, x₀ = 1, dx = 0.1

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

b. f(x) = x, g(x) = x², (a, b) arbitrary