Question 1

Suppose $$f is a $$continuous function on $$[a, b]. $$According to the Fundamental Theorem of Calculus, Part 1, if $$g(x) = \int_a^x f(t)\,dt$$, what is $$g'(x)$$?

Accepted Answer

$$g'(x) = f(x)$$

Question 2

Let $$f(t)$$ represent the rate at which water flows into a tank (in liters per minute), and $$g(x) = \$$int_0$$^x f(t)\,dt is $$the total amount of water in the tank after $$x$$ minutes. What does $$g'(x)$$ represent in this context?

Accepted Answer

The instantaneous rate at which water is flowing into the tank at time $$x$$

Question 3

Evaluate $$\frac{d}{dx} \$$int_2$$^x \sqrt{1 + t^2$}\,dt.$$

Accepted Answer

$$\sqrt{1 + x^2$}$$

Question 4

If $$F'(x) = f(x)$$ and $$f is $$continuous on $$[a, b]$$, what does the Fundamental Theorem of Calculus, Part 2, state about $$\int_a^b f(x)\,dx$$?

Accepted Answer

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Question 5

Given $$g(x) = \$$int_0$$^x t\,dt$$, find $$g(x)$$ and $$g'(x).$$

Accepted Answer

$$g(x) = \frac{x^2$}{2}$$ and $$g'(x) = x$$

Question 6

Suppose $$f(t) is $$positive for $$t  3. If$$ $$g(x) = \$$int_0$$^x f(t)\,dt$$, what can you say about the behavior of $$g(x)$$ for $$x  3$$?

Accepted Answer

$$g(x)$$ increases for $$x  3$$

Question 7

A car's velocity (in m/s) at time $$t$$ seconds is given by $$v(t). $$What does $$\$$int_0$$^T v(t)\,dt$$ represent?

Accepted Answer

The total displacement of the car from $$t=0 to$$ $$t=T$$

Question 8

If $$g(x) = \$$int_1$$^x f(t)\,dt$$ and $$f is $$continuous, which of the following is true?

Accepted Answer

$$g'(x) = f(x)$$

Question 9

Let $$f(x) be $$continuous on $$[a, b]. $$Which statement best describes the relationship between differentiation and integration according to the Fundamental Theorem of Calculus?

Accepted Answer

Differentiation and integration are inverse processes

Question 10

Suppose $$g(x) = \$$int_0$$^x f(t)\,dt$$ and $$f is $$continuous. If $$f(2) = 5$$, what is $$g'(2)$$?

Accepted Answer

$$g'(2) = 5$$

Question 11

If $$F(x) is an $$antiderivative of $$f(x)$$, which of the following is always true for continuous $$f on$$ $$[a, b]$$?

Accepted Answer

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Question 12

A medication is administered at a rate $$r(t) mg/$$hour. What does $$\$$int_0$$^8 r(t)\,dt$$ represent in a medical context?

Accepted Answer

The total amount of medication administered in 8 hours

Question 13

If $$g(x) = \$$int_0$$^x \sqrt{1 + t^2$}\,dt$$, what is $$g'(x)$$?

Accepted Answer

$$\sqrt{1 + x^2$}$$

Question 14

Which of the following best describes the meaning of $$\frac{d}{dx} \int_a^x f(t)\,dt$$ for a continuous function $$f$$?

Accepted Answer

It gives back the original function $$f(x)$$

Question 15

Suppose $$f is $$continuous and $$F is $$any antiderivative of $$f. $$Which formula allows you to compute the definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

$$F(b) - F(a)$$

Question 16

If $$g(x) = \$$int_0$$^x f(t)\,dt$$ and $$f is $$continuous, which of the following statements is true?

Accepted Answer

$$g is $$differentiable and $$g'(x) = f(x)$$

Question 17

A population grows at a rate $$P'(t)$$ individuals per year. What does $$\$$int_0$$^{10} P'(t)\,dt$$ represent?

Accepted Answer

The net change in population over 10 years

The Fundamental Theorem of Calculus: Concepts, Examples, and Applications

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