Question 1

Which of the following definite integrals correctly represents the Riemann sum $$\sum_{i=1}^{n} \frac{2}{n} \left(1 + \frac{2i}{n}\$$right)^3$$ as n$$ 	o \infty$$?

Accepted Answer

$$\int_{1}^{3} x^3$ \, dx$$

Question 2

Suppose $$F(x) = \int_{0}^{x} \sin(t^2$) \, dt. $$What is $$F'(x)$$ according to the Fundamental Theorem of Calculus?

Accepted Answer

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

Question 3

Evaluate the indefinite integral $$\int 2x \cos(x^2$) \, dx$$ using $$u-$$substitution.

Accepted Answer

$$\sin(x^2$) + C$$

Question 4

Find the value of the definite integral $$\int_{0}^{2} (3x^2$ + 2x) \, dx.$$

Accepted Answer

$$16$$

Question 5

Evaluate the summation $$\sum_{k=1}^{4} (2k - 1).$$

Accepted Answer

$$12$$

Question 6

The average value of $$f(x) = x^2$ on the interval $$[0, 2]$$ is:

Accepted Answer

$$\frac{4}{3}$$

Question 7

Given the differential equation $$\frac{dy}{dx} = 3x^2$ and y(1) = 5$, find the particular solution.

Accepted Answer

$$y(x) = x^3$ + 4$$

Question 8

A car's velocity is given by $$v(t) = 4t$$ meters per second. How far does the car travel from $$t = 0 to$$ $$t = 5$$ seconds?

Accepted Answer

$$50$$ meters

Question 9

Use the Trapezoidal Rule with $$n=2 to $$estimate $$\int_{0}^{2} x^2$ \, dx.$$

Accepted Answer

$$4$$

Question 10

Which of the following is the correct Simpson's Rule formula for approximating $$\int_{a}^{b} f(x) \, dx$$ with $$n$$ subintervals?

Accepted Answer

$$\frac{\Delta x}{3} \left[ f(x_0$) + 4f(x_1$) + 2f(x_2$) + \cdots + 4f(x_{n-1}) + f(x_n) \right]$$

Question 11

Rewrite the Riemann sum $$\sum_{i=1}^{n} \frac{1}{n} \sqrt{1 + \frac{i}{n}} as a $$definite integral as $$n 	o \infty.$$

Accepted Answer

$$\int_{1}^{2} \sqrt{x} \, dx$$

Question 12

If $$F(x) = \int_{2}^{x} \ln(t) \, dt$$, what is $$F'(x)$$?

Accepted Answer

$$\ln(x)$$

Question 13

Evaluate $$\int_{0}^{\pi} \sin(x) \, dx.$$

Accepted Answer

$$2$$

Question 14

Find the average value of $$f(x) = 3x on$$ $$[0, 4].$$

Accepted Answer

$$6$$

Question 15

A medication is administered at a rate $$r(t) = 5e$$^{-0.2t}$$ mg/hour. How much medication is given in the first 5 hours?

Accepted Answer

$$\int_{0}^{5} 5e$$^{-0.2t}$$ \, dt = 5 \left[ -5e$$^{-0.2t}$$ \right]_{0}^{5} = 5(1 - e$$^{-1}$$) \approx 3.16 mg$$

Question 16

Given $$\frac{dv}{dt} = -9.8$$ and $$v(0) = 20$$, what is $$v(t)$$?

Accepted Answer

$$v(t) = 20 - 9.8t$$

Question 17

Use $$u-$$substitution to evaluate $$\int x e$$^{x^2}$$ \, dx.$$

Accepted Answer

$$\frac{1}{2} e$$^{x^2}$$ + C$$

Question 18

A ball is thrown upward with velocity $$v(t) = 30 - 9.8t. $$How high does it rise in the first 2 seconds?

Accepted Answer

$$\int_{0}^{2} (30 - 9.8t) \, dt = 30(2) - 9.8(2^2$)/2 = 60 - 19.6 = 40.4$$ meters

Question 19

Which of the following is NOT a valid application of the Fundamental Theorem of Calculus?

Accepted Answer

Using $$\int_{a}^{b} f(x) \, dx to $$find the derivative of $$f(x)$$

Calculus 1 Chapter 4: Integrals, Applications, and Numerical Methods – Study Guide

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