Q: Which of the following is the correct application of L'Hôpital's Rule to evaluate the limit $$\lim_{x \to 0} \frac{\sin x}{x}$$?

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

Q: Which of the following statements about the Ratio Test is correct?

If $$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \u003c 1$$, the series converges.

Question 1

Which of the following is the correct application of L'Hôpital's Rule to evaluate the limit $$\lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\sin x}{x} = 1$$

Question 2

A radioactive substance decays according to the differential equation $$\frac{dN}{dt} = -kN$$, where $$N is $$the amount of substance and $$k \u003e 0. If$$ $$N(0) = N_0$, what is the solution for N(t$$)$$?

Accepted Answer

$$N(t) = N_0$ e$$^{-kt}$$

Question 3

Which of the following integrals is considered an improper integral?

Accepted Answer

$$\int_{0}^{1} \frac{1}{x} dx$$

Question 4

Given the power series $$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$, what is its radius of convergence?

Accepted Answer

$$R = \infty$$

Question 5

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

Accepted Answer

Comparison Test

Question 6

A force $$F(x) = 3x (in $$Newtons) moves an object from $$x = 0 to$$ $$x = 4$$ meters. What is the total work done by the force?

Accepted Answer

$$W = \int_{0}^{4} 3x dx = 24$$ Joules

Question 7

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

Accepted Answer

$$1$$

Question 8

Which of the following is the Taylor polynomial of degree 2 for $$f(x) = e^x$$ centered at $$x = 0$$?

Accepted Answer

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

Question 9

Which of the following series converges conditionally but not absolutely?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$

Question 10

A tank is being filled with water at a rate proportional to the amount already in the tank. If $$V(t) is $$the volume at time $$t$$, which differential equation models this scenario?

Accepted Answer

$$\frac{dV}{dt} = kV$$

Question 11

Which of the following is an example of an indeterminate form that can be evaluated using L'Hôpital's Rule?

Accepted Answer

$$\frac{0}{0}$$

Question 12

Given the series $$\sum_{n=1}^{\infty} \frac{1}{n}$$, which test shows that the series diverges?

Accepted Answer

Divergence Test

Question 13

Find the center of mass of a thin rod of length $$L$$ with linear density $$\lambda(x) = 2x$$ for $$0 \leq x \leq L.$$

Accepted Answer

$$\frac{2L}{3}$$

Question 14

Which of the following is the correct form of the Taylor series for $$\sin x$$ centered at $$x = 0$$?

Accepted Answer

$$x - \frac{x^3$}{3!} + \frac{x^5$}{5!} - \frac{x^7$}{7!} + \cdots$$

Question 15

Which of the following statements about the Ratio Test is correct?

Accepted Answer

If $$\lim_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight| \u003c 1$$, the series converges.

Question 16

A population of bacteria grows according to $$\frac{dP}{dt} = rP. If $$the initial population is $$P_0$$, what is the population after time $$t$$?

Accepted Answer

$$P(t) = P_0$ e$$^{rt}$$

Question 17

Which of the following is the correct integral to find the area under the curve $$y = x^2$ from x = 1$ to x = 3$?

Accepted Answer

$$\int_{1}^{3} x^2$ dx$$

Question 18

Which of the following series converges absolutely?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2$}$$

Question 19

If $$\int_{0}^{1} f(x) dx$$ diverges, which of the following could be a possible reason?

Accepted Answer

The function $$f(x) is $$unbounded near $$x = 0.$$

MATH 151: Calculus I Syllabus and Course Overview

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