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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2:39 minutes

Problem 34a

Textbook Question

Finding Functions from Derivatives

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

a. y' = 1 / 2√x

Verified step by step guidance

Step 1: Recognize that finding a function from its derivative involves integration. The given derivative is \( y' = \frac{1}{2\sqrt{x}} \).

Step 2: Rewrite the derivative in a form that is easier to integrate. Note that \( \frac{1}{2\sqrt{x}} \) can be expressed as \( \frac{1}{2}x^{-1/2} \).

Step 3: Integrate the expression \( \frac{1}{2}x^{-1/2} \) with respect to \( x \). Use the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \).

Step 4: Apply the power rule to \( \frac{1}{2}x^{-1/2} \). The integral becomes \( \frac{1}{2} \int x^{-1/2} \, dx = \frac{1}{2} \cdot \frac{x^{1/2}}{1/2} + C \). Simplify this expression.

Step 5: Simplify the result from the integration to find the general form of the function \( y(x) \). Don't forget to include the constant of integration \( C \), which represents all possible functions that have 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

Antiderivatives, or indefinite integrals, are functions that reverse the process of differentiation. To find a function from its derivative, you need to determine its antiderivative. This involves integrating the derivative function, which in this case is y' = 1 / 2√x, to find the original function y(x).

Integration Techniques

Integration techniques are methods used to find antiderivatives. For the derivative y' = 1 / 2√x, recognizing the form of the function is crucial. The expression 1 / 2√x can be rewritten as x^(-1/2), which is a power function. Applying the power rule for integration, you add 1 to the exponent and divide by the new exponent to find the antiderivative.

Constant of Integration

The constant of integration is an arbitrary constant added to the antiderivative of a function. When integrating a derivative to find the original function, the constant accounts for any vertical shifts in the graph of the function. Since differentiation eliminates constants, the antiderivative must include this constant to represent all possible original functions.