4. Applications of Derivatives

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = x√(1 − x²)

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|.

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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