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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2:45 minutes

Problem 31a

Textbook Question

Finding Functions from Derivatives

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

a. y′ = x

Verified step by step guidance

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

Integrate the derivative \( y' = x \) with respect to \( x \). This means we need to find \( \int x \, dx \).

The integral of \( x \) with respect to \( x \) is \( \frac{x^2}{2} \). This is because the power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

After integrating, we have \( y = \frac{x^2}{2} + C \), where \( C \) is an arbitrary constant. This represents the family of functions whose derivative is \( x \).

The constant \( C \) can be any real number, which means there are infinitely many functions that satisfy the given derivative, each differing by a constant.

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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, you need to determine its antiderivative. For example, if y' = x, the antiderivative of x is (1/2)x^2 plus a constant C, representing all possible functions with the given derivative.

Integration

Integration is the process of finding the antiderivative of a function. It involves calculating the integral of the function, which can be indefinite or definite. In this context, finding the indefinite integral of y' = x will yield the general form of the function y = (1/2)x^2 + C, where C is an arbitrary constant.

Constant of Integration

The constant of integration, denoted as C, arises when computing indefinite integrals. It represents an infinite number of possible functions that differ by a constant. When finding functions from derivatives, this constant accounts for all vertical shifts of the antiderivative, ensuring the solution encompasses all possible functions with the given derivative.