Q: Suppose the birth rate in France $$t$$ years after 1970 is $$13 + t (in $$thousands per year). What is the total number of births (in thousands) from the end of 1970 to the end of 1990?

$$\displaystyle \$$int_0$$^{20} (13 + t)\,dt = 20 \times 13 + \frac{1}{2} \times 20^2$ = 260 + 200 = 460$$

Question 1

Which of the following is the correct antiderivative of $$5 \sec^2$(x)$$?

Accepted Answer

$$5 	an(x) + C$$

Question 2

Evaluate the definite integral $$\displaystyle \int_{-\pi}^{\pi} \sin(3x)\,dx.$$

Accepted Answer

$$0$$

Question 3

Suppose $$f'(t)$$ represents the instantaneous rate of change of $$f(t)$$ with respect to $$t. $$Which integral gives the net change in $$f$$ from $$t=a to$$ $$t=b$$?

Accepted Answer

$$\displaystyle \int_a^b f'(t)\,dt$$

Question 4

Given $$f(x) = \displaystyle \$$int_0$$^x \frac{dt}{\sqrt{1 + t^3$}}$$, what is $$f'(7)$$?

Accepted Answer

$$\displaystyle \frac{1}{\sqrt{1 + 343}}$$

Question 5

Which of the following best describes what the area under the velocity curve from $$t=0 to$$ $$t=80$$ seconds represents in the context of the car on the gravel road?

Accepted Answer

The total distance traveled by the car

Question 6

Which of the following is NOT a requirement for applying Part 2 of the Fundamental Theorem of Calculus to compute $$\int_a^b f(x)\,dx$$?

Accepted Answer

$$f(x)$$ must be differentiable on $$[a, b]$$

Question 7

Suppose the birth rate in France $$t$$ years after 1970 is $$13 + t (in $$thousands per year). What is the total number of births (in thousands) from the end of 1970 to the end of 1990?

Accepted Answer

$$\displaystyle \$$int_0$$^{20} (13 + t)\,dt = 20 	imes 13 + \frac{1}{2} 	imes 20^2$ = 260 + 200 = 460$$

Question 8

Which of the following is the correct statement of Part 1 of the Fundamental Theorem of Calculus?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$F(x) = \int_a^x f(t)\,dt$$, then $$F'(x) = f(x)$$

Question 9

Calculate $$\displaystyle \frac{d}{dx} \left( \int_{3}^{\cos x} \sin(t^2$)\,dt \right).$$

Accepted Answer

$$-\sin((\cos x)^2$) \sin x$$

Question 10

Which of the following definite integrals CANNOT be computed using Part 2 of the Fundamental Theorem of Calculus?

Accepted Answer

$$\displaystyle \$$int_0$$^{10} \frac{1}{x-5}\,dx$$

Question 11

You are estimating the length of a road by recording velocity at 10-second intervals and using a left-endpoint Riemann sum with $$n=4$$ subintervals over $$[0,80]. $$What is the width $$\Delta t of $$each rectangle?

Accepted Answer

$$20$$ seconds

Question 12

Which of the following best describes the physical meaning of the area of the first rectangle (for $$t in$$ $$[0,20]) in $$your Riemann sum for the car's velocity?

Accepted Answer

The approximate distance traveled by the car in the first 20 seconds

Question 13

Which of the following is the correct antiderivative of $$x^2$$ - \sqrt{x}$$?

Accepted Answer

$$\displaystyle \frac{1}{3}x^3$ - \frac{2}{3}x$$^{3/2}$$ + C$$

Question 14

If $$F(x) is $$any antiderivative of $$f(x)$$, which of the following expresses the value of $$\int_a^b f(x)\,dx$$ according to the Fundamental Theorem of Calculus?

Accepted Answer

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

Question 15

Which of the following best describes the meaning of the definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

The signed area between $$y = f(x)$$ and the $$x-$$axis from $$x = a to$$ $$x = b$$

Integral Concepts & Fundamental Theorem of Calculus: Guided Study

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