Question 1

Evaluate the definite integral $$\int_{-1}^{3} (x+4)\,dx.$$

Accepted Answer

$$13$$

Question 2

Which of the following is the correct value for $$\int_{1}^{4} x$$^{-1/2}$$\,dx$$?

Accepted Answer

$$3$$

Question 3

Evaluate $$\int_{1}^{2} (t^6$ - t^2$)\,dt.$$

Accepted Answer

$$12$$

Question 4

A certain object moves so that its velocity after time $$t (in $$seconds) is $$v(t) = t^2$ + 4t + 6. $$What is the total distance traveled in the first 4 seconds?

Accepted Answer

$$77.3 m$$

Question 5

A new firm finds that its rate of profit (in hundreds of dollars) after $$t$$ years is $$P'(t) = 2t^2$ + 4t + 8. $$What is the total profit earned in the first 4 years?

Accepted Answer

$$10,400$$

Question 6

Which statement best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates the derivative of a function to its integral.

Question 7

Evaluate $$\int_{1}^{5} e$$^{5x}$$\,dx.$$

Accepted Answer

$$\frac{1}{5}(e$$^{25}$$ - e$$^{5}$$)$$

Question 8

If $$F(x) is an $$antiderivative of $$f(x)$$, which expression gives the value of $$\int_{a}^{b} f(x)\,dx$$?

Accepted Answer

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

Question 9

A company’s revenue rate is $$R'(t) = 3t^2$ + 2t + 5 (in $$thousands of dollars per year). What is the total revenue earned from year 0 to year 3?

Accepted Answer

$$50$$ thousand dollars

Question 10

Evaluate $$\int_{0}^{2} (4x^3$)\,dx.$$

Accepted Answer

$$64$$

Question 11

Which of the following is the correct value for $$\int_{0}^{5} (1 + x^4$)\,dx$$?

Accepted Answer

$$320$$

Question 12

A medication is administered at a rate of $$M'(t) = 2t + 3 mg/$$hour. How much medication is given in the first 6 hours?

Accepted Answer

$$54 mg$$

Question 13

If $$f(x) = x^2$, what is $$\int_{2}^{5} f(x)\,dx$$?

Accepted Answer

$$117$$

Question 14

A car’s acceleration is $$a(t) = 6t. If $$the car starts from rest, what is its velocity after 3 seconds?

Accepted Answer

$$27 m/s$$

Question 15

Which of the following is NOT a requirement for the Fundamental Theorem of Calculus to apply?

Accepted Answer

The function must be differentiable everywhere.

Question 16

Evaluate $$\int_{0}^{3} (2x + 1)\,dx.$$

Accepted Answer

$$10.5$$

Question 17

A population grows at a rate $$P'(t) = 5e$$^{0.2t}$$ people per year. How many people are added in the first 5 years?

Accepted Answer

$$5(e^{1} - 1$$)$$

Question 18

If $$f(x) = 3x^2$, what is the average value of f(x$$)$$ on $$[0,2]$$?

Accepted Answer

$$6$$

Question 19

A tank is being filled at a rate $$F'(t) = 4t^2$ liters per minute. How much water is added in the first 3 minutes?

Accepted Answer

$$36$$ liters

Question 20

Which of the following best explains why definite integrals can be used to calculate total change over an interval?

Accepted Answer

Because the definite integral sums the instantaneous rates of change over the interval.

Business Calculus: Definite Integrals and Applications (Fundamental Theorem of Calculus)

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