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 $n$th 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

Study Guide - Practice Questions

Study Guide - Flashcards