Question 1

Which of the following is the correct formula for integration by parts?

Accepted Answer

$$\int u \, dv = uv - \int v \, du$$

Question 2

A pharmaceutical company needs to determine the total amount of a drug in the bloodstream over time, given the rate of absorption $$R(t) = 5e$$^{-0.2t}$$ mg/hr. Which integral represents the total amount absorbed from t=0$ to t=10$ hours?

Accepted Answer

$$\int_{0}^{10} 5e$$^{-0.2t}$$ \, dt$$

Question 3

Evaluate the integral $$\int x e$$^{x}$$ \, dx$$ using integration by parts.

Accepted Answer

$$x e$$^{x}$$ - e$$^{x}$$ + C$$

Question 4

Which substitution would be most appropriate for evaluating $$\int \sin^3$(x) \cos(x) \, dx$$?

Accepted Answer

Let $$u = \sin(x)$$

Question 5

A water tank has a cross-sectional area that varies with height as $$A(y) = 2y. $$What is the volume of water in the tank from $$y=0 to$$ $$y=3$$ meters?

Accepted Answer

$$\int_{0}^{3} 2y \, dy$$

Question 6

Which of the following integrals can be solved using the trigonometric substitution $$x = a \sin(	heta)$$?

Accepted Answer

$$\int \sqrt{a^2$ - x^2$} \, dx$$

Question 7

A solid has a base in the $$xy-$$plane bounded by $$y = x^2$ and y = 4$, and cross-sections perpendicular to the x$-axis are squares. What is the correct integral to find the volume of the solid?

Accepted Answer

$$\int_{-2}^{2} (4 - x^2$)^2$$ \, dx$

Question 8

Which of the following is the correct formula for the volume of a solid of revolution generated by rotating $$y = f(x)$$ about the $$x-$$axis from $$x=a to$$ $$x=b$$?

Accepted Answer

$$V = \pi \int_{a}^{b} [f(x)]^2 \, dx$$

Question 9

A cable hangs between two poles and forms the curve $$y = \cosh(x). $$What is the length of the cable between $$x = 0$$ and $$x = 2$$?

Accepted Answer

$$\int_{0}^{2} \sqrt{1 + \$$sinh^2$$(x)} \, dx$$

Question 10

Which of the following series is a geometric series?

Accepted Answer

$$\sum_{n=0}^{\infty} 3 \left(\frac{1}{2}\right)^n$$

Question 11

A medical researcher models the decay of a drug in the bloodstream with the series $$\sum_{n=1}^{\infty} \frac{1}{2^n}. $$What is the sum of this series?

Accepted Answer

$$1$$

Question 12

Which test would be most appropriate to determine the convergence of the series $$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$?

Accepted Answer

p-test

Question 13

Evaluate the improper integral $$\int_{1}^{\infty} \frac{1}{x^2$} \, dx.$$

Accepted Answer

$$1$$

Question 14

Which of the following is the Maclaurin series for $$e^x$$?

Accepted Answer

$$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Question 15

A biologist models the population of bacteria with the function $$P(t) = 100e$$^{0.3t}$$. What is the rate of change of the population at t=2$ hours?

Accepted Answer

$$30e^{0.6}$$

Question 16

Which of the following is the correct formula for the surface area of a solid of revolution generated by rotating $$y = f(x)$$ about the $$x-$$axis from $$x=a to$$ $$x=b$$?

Accepted Answer

$$2\pi \int_{a}^{b} f(x) \sqrt{1 + [f'(x)]^2} \, dx$$

Question 17

Which of the following is the general solution to the differential equation $$\frac{dy}{dx} = ky$$?

Accepted Answer

$$y = Ce$$^{kx}$$

Question 18

A region is bounded by $$y = x^2$ and y = 4$. What is the area of the region between these curves from x = -2$ to x = 2$?

Accepted Answer

$$\int_{-2}^{2} (4 - x^2$) \, dx$$

Question 19

Which of the following is the correct formula for the $$nth $$term of a Taylor series for $$f(x)$$ centered at $$a$$?

Accepted Answer

$$\frac{f$$^{(n)}$$(a)}{n!}(x-a)^n$$

Calculus I Course Schedule and Assessment Overview

Guida di studio - Domande di pratica

Study guide - Flashcard