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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2:43 minutes

Problem 31b

Textbook Question

Finding Functions from Derivatives

In Exercises 31–36, find all possible functions with the given derivative.

b. y′ = x²

Verified step by step guidance

To find the original function from its derivative, we need to perform integration. The given derivative is \( y' = x^2 \).

The process of finding the original function is called integration. We need to integrate \( x^2 \) with respect to \( x \).

The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Here, \( n = 2 \).

Apply the power rule for integration: \( \int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C \).

Thus, the original function \( y \) is \( y = \frac{x^3}{3} + C \), where \( C \) is any constant, representing the family of functions with the given derivative.

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Key Concepts

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

Antiderivatives

An antiderivative of a function is another function whose derivative is the original function. To find a function from its derivative, we perform the process of integration, which is essentially finding the antiderivative. For example, if y' = x², the antiderivative is y = (1/3)x³ + C, where C is the constant of integration.

Indefinite Integration

Indefinite integration is the process of finding the antiderivative of a function. It involves integrating the function without specific limits, resulting in a general form that includes a constant of integration, C. This constant accounts for all possible vertical shifts of the antiderivative, reflecting the family of functions that share the same derivative.

Constant of Integration

The constant of integration, denoted as C, arises when performing indefinite integration. It represents an arbitrary constant added to the antiderivative, accounting for the fact that multiple functions can have the same derivative. For example, if y' = x², the antiderivative is y = (1/3)x³ + C, where C can be any real number.