6.4 Second Fundamental Theorem of Calculus

Question 1

What is the Second Fundamental Theorem of Calculus (FTC)?

Answer 1

if F(x) is the integral of a function f(t), then the derivative of F(x) is the original function f(x).

Question 2

What does the Second Fundamental Theorem of Calculus help us do? What is the relationship of differentiation and integration?

Answer 2

It helps us evaluate the derivative of an integral. If F(x)=∫ f(x)dx, then F′(x)=f(x), showing that differentiation and integration are inverse operations.

Question 3

What are the conditions for the Second Fundamental Theorem of Calculus?

Answer 3

The function f(t) must be continuous on the interval [a,b] for the theorem to apply.

Question 4

What is the derivative of the function
F(x)=∫(a on the bottom, x on the top) f(t)dt and why?

Answer 4

The derivative is simply f(x), according to the Second Fundamental Theorem of Calculus:F′(x)=f(x)

Question 5

What happens if the upper limit of integration is a function of x, say g(x), in the Second Fundamental Theorem of Calculus?

Answer 5

If the upper limit is a function g(x), then the chain rule applies:d/dx (∫a on the bottom, g(x) on the top) f(t)dt)=f(g(x))⋅g’ (x)

Question 6

How does the Second Fundamental Theorem of Calculus relate to the First Fundamental Theorem?

Answer 6

The Second FTC shows how to differentiate an integral, while the First FTC connects the integral of a function to its antiderivative, allowing us to evaluate definite integrals.

Question 7

Example: Differentiate F(x)=∫(0 on the bottom, x on the top) sin(t)dt.

Answer 7

F′(x)=sin(x)

Question 8

Differentiate F(x)=∫(2 on the bottom, x on the top) e^(t^2)dt.

Answer 8

By the Second Fundamental Theorem of Calculus: F′(x)=e^(x^2)