4. Applications of Derivatives

Derivatives in Differential Form

In Exercises 17–28, find dy.

2y³/² + xy − x = 0

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sec(x² − 1)

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = 3 csc(1 − 2√x)

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|.

<IMAGE>

f(x) = x² + 2x, x₀ = 1, dx = 0.1

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?

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

a. circumference?

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 lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change

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 lateral surface area S = πr√(r² + h²) of a right circular cone when the radius changes from r₀ to r₀ + dr and the height does not change

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

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 volume V = x³ of a cube when the edge lengths change from x₀ to x₀ + dx

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 volume V = (4/3)πr³ of a sphere when the radius changes from r₀ to r₀ + dr

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.

f(x) =√(x − 1), [1, 3]

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

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = x²ᐟ³, [−1, 8]