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

8. Definite Integrals

Fundamental Theorem of Calculus

Calculus

8. Definite Integrals

Fundamental Theorem of Calculus

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Multiple Choice

Let $g (x) = \int_{0}^{x} f (t) d t$ , where $f$ is the function whose graph is shown. Which of the following statements is true about $g^{'} (x)$ ?

g^{'} (x) = f (x)

g^{'} (x) = \int_{0}^{x} f^{'} (t) d t

g^{'} (x) = f (0)

g^{'} (x) = x f (x)

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Verified step by step guidance

The problem involves the Fundamental Theorem of Calculus, which states that if g(x) = ∫₀^x f(t) dt, then g'(x) = f(x). This theorem connects differentiation and integration.

To understand why g'(x) = f(x), recall that the derivative of an integral with a variable upper limit is the integrand evaluated at that upper limit. In this case, the upper limit is x, so g'(x) = f(x).

The other options can be ruled out as follows: g'(x) = ∫₀^x f'(t) dt is incorrect because the derivative of g(x) is not another integral; g'(x) = f(0) is incorrect because it evaluates f at the lower limit, not the upper limit; and g'(x) = x f(x) is incorrect because it introduces an extra factor of x that is not part of the Fundamental Theorem of Calculus.

To summarize, the correct answer is g'(x) = f(x), which directly follows from the Fundamental Theorem of Calculus.

This problem highlights the importance of understanding the relationship between differentiation and integration, as well as the role of the Fundamental Theorem of Calculus in solving such problems.