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

7. Antiderivatives & Indefinite Integrals

Antiderivatives

Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

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Multiple Choice

If $F (x)$ and $G (x)$ are both antiderivatives of $f (x)$ , what must be true about $F (x)$ and $G (x)$ ?

F (x) + G (x) = f (x)

F (x) - G (x)

is a constant for all x

F (x) = G (x)

for all x

F (x) \times G (x)

is a constant for all x

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Verified step by step guidance

Step 1: Understand the concept of antiderivatives. An antiderivative of a function f(x) is a function whose derivative is f(x). For example, if F(x) is an antiderivative of f(x), then F'(x) = f(x).

Step 2: Recognize that if F(x) and G(x) are both antiderivatives of f(x), their derivatives are equal, i.e., F'(x) = G'(x) = f(x). This implies that the difference between F(x) and G(x) must be constant because the derivative of a constant is zero.

Step 3: Express the relationship mathematically. Since F'(x) = G'(x), we can write F(x) - G(x) = C, where C is a constant. This means that F(x) and G(x) differ only by a constant.

Step 4: Eliminate incorrect options. The statement 'F(x) + G(x) = f(x)' is incorrect because the sum of two antiderivatives does not equal the original function. Similarly, 'F(x) = G(x) for all x' is incorrect because F(x) and G(x) can differ by a constant.

Step 5: Conclude that the correct answer is 'F(x) - G(x) is a constant for all x,' as this is the fundamental property of antiderivatives of the same function.