Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Differentials

Calculus

4. Applications of Derivatives

Differentials

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1:42 minutes

Problem 3.9.35

Textbook Question

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

Verified step by step guidance

Start by recalling the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \).

To find the differential \( dV \), differentiate the volume formula with respect to \( r \). This gives \( dV = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \cdot dr \).

Calculate the derivative: \( \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) = 4 \pi r^2 \).

Substitute the derivative back into the differential formula: \( dV = 4 \pi r^2 \cdot dr \).

This differential formula \( dV = 4 \pi r^2 \cdot dr \) estimates the change in volume when the radius changes from \( r_0 \) to \( r_0 + dr \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Differential Calculus

Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. In this context, it helps estimate how a small change in one variable, such as the radius of a sphere, affects another variable, like the volume. Understanding derivatives is crucial for applying differential formulas to approximate changes.

Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius. This formula is essential for understanding how the volume changes with respect to the radius. Knowing this relationship allows us to apply calculus concepts to find the differential change in volume when the radius changes slightly.

Differential Formula

A differential formula is used to approximate the change in a function's value due to a small change in its input. For a function V(r), the differential dV is given by dV = V'(r)dr, where V'(r) is the derivative of V with respect to r. This formula helps estimate the change in volume of a sphere when its radius changes by a small amount dr.

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Textbook Question

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

Textbook Question

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

Textbook Question

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

Textbook Question

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

Textbook Question

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]

Textbook Question

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

Textbook Question

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]

Textbook Question

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) = x⁴ᐟ⁵, [0, 1]

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Differential Estimates of ChangeIn 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

Key Concepts

Differential Calculus

Volume of a Sphere

Differential Formula

Watch next

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