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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2:49 minutes

Problem 27

Textbook Question

Finding Functions from Derivatives

Suppose that f(−1) = 3 and that f'(x) = 0 for all x. Must f(x) = 3 for all x? Give reasons for your answer.

Verified step by step guidance

First, understand the meaning of the derivative f'(x) = 0 for all x. This implies that the function f(x) has a constant slope of 0, meaning it is a horizontal line.

Since the derivative represents the rate of change of the function, a derivative of 0 indicates that the function does not change as x changes. Therefore, f(x) must be a constant function.

Given that f(x) is a constant function, it must take the same value for all x.

We are provided with the information that f(−1) = 3. Since f(x) is constant, this value must be the same for all x.

Thus, we conclude that f(x) = 3 for all x, because a constant function with f(−1) = 3 implies that the constant value is 3 everywhere.

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Key Concepts

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

Constant Function

A constant function is a function that always returns the same value, no matter the input. In mathematical terms, if f'(x) = 0 for all x, it implies that the function f(x) does not change as x changes, indicating that f(x) is a constant function.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. If the derivative f'(x) is zero for all x, it means the function has no slope and is flat, suggesting that the function is constant across its domain.

Initial Condition

An initial condition provides specific information about a function at a particular point, which helps determine the constant of integration when solving differential equations. In this problem, f(−1) = 3 serves as an initial condition, confirming that the constant value of the function f(x) is 3.