Quiz about Calculus II Unit 3 Review

Question 1

$$\lim_{x\rightarrow0}\ \frac{x}{e^x}$$

Accepted Answer

0

Answer

$$-\infty$$

Answer

1

Answer

$$\infty$$

Question 2

$$\lim_{x\rightarrow\infty}\ \frac{x}{e^x}$$

Accepted Answer

0

Answer

$$-\infty$$

Answer

1

Answer

$$\infty$$

Question 3

$$\lim_{x\rightarrow\infty}\ \frac{25x^5}{e^x}$$

Accepted Answer

0

Answer

$$-\infty$$

Answer

1

Answer

$$\infty$$

Question 4

With the following limit, when plugging in the value of "a" to the function, we get an indeterminate product. $$\lim_{x\rightarrow\infty}\ x\sin\left(\frac{1}{x}\right)$$

We can use L'Hopital's rule after rewriting this product as a quotient. Which of the following forms will be easiest to work with?

Accepted Answer

$$\lim_{x\rightarrow\infty}\ \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}}$$

Answer

$$\lim_{x\rightarrow\infty}\ \frac{x}{\csc\left(\frac{1}{x}\right)}$$

Answer

$$\lim_{x\rightarrow\infty}\ \frac{x}{\frac{1}{\sin\left(\frac{1}{x}\right)}}$$

Question 5

With the following limit, when plugging in the value of "a" to the function, we get an indeterminate product. $$\lim_{x\rightarrow-\infty}\ x^2e^x$$

We can use L'Hopital's rule after rewriting this product as a quotient. Which of the following forms will be easiest to work with?

Accepted Answer

$$\lim_{x\rightarrow-\infty}\ \frac{x^2}{e^{-x}}$$

Answer

$$\lim_{x\rightarrow-\infty}\ \frac{e^x}{\frac{1}{x^2}}$$

Answer

$$\lim_{x\rightarrow-\infty}\ \frac{e^x}{x^{-2}}$$

Answer

$$\lim_{x\rightarrow-\infty}\ \frac{x^{-2}}{e^x}$$

Question 6

When finding the error bounds for the following integral, what would the K value be if using Simpson's Rule?

$$\int_0^{\frac{\pi}{2}}45\cos\left(x\right)\ dx$$

Accepted Answer

$$45$$

Answer

$$\frac{45\pi}{2}$$

Answer

$$\frac{45}{2}\ or\ 22.5$$

Answer

$$\frac{\pi}{2}$$

Question 7

When finding the error bounds for the following integral, what would the K value be if using the Midpoint Rule?

$$\int_1^2e^{\frac{1}{x}}\ dx$$

Accepted Answer

$$3e$$

Answer

$$e$$

Answer

$$\frac{5\sqrt[]{e}}{16}$$

Answer

$$\sqrt[]{e}$$

Question 8

When finding the error bounds for the following integral, what would the K value be if using Simpson's Rule?

$$\int_4^5\ \frac{2}{x-3}\ dx$$

Accepted Answer

$$48$$

Answer

$$1$$

Answer

$$2$$

Answer

$$\frac{48}{5}$$

Question 9

Which of the following is an example of a correct Simpson's Rule summation?

a) $$\frac{1}{3}\left[f\left(1\right)+4f\left(2\right)+2f\left(3\right)+4f\left(4\right)+2f\left(5\right)+f\left(6\right)\right]$$

b) $$\frac{1}{6}\left[f\left(1\right)+4f\left(2\right)+2f\left(3\right)+4f\left(4\right)+2f\left(5\right)+4f\left(6\right)+f\left(7\right)\right]$$

c) $$\frac{1}{3}\left[f\left(1\right)+4f\left(2\right)+2f\left(3\right)+4f\left(4\right)+2f\left(5\right)+4f\left(6\right)+f\left(7\right)\right]$$

d) $$\frac{1}{6}\left[f\left(1\right)+4f\left(2\right)+2f\left(3\right)+4f\left(4\right)+2f\left(5\right)+f\left(6\right)\right]$$

Accepted Answer

c

Answer

a

Answer

b

Answer

d

Question 10

True or False: If we find $$\int_{-\infty}^0\ \frac{1}{x}\ dx$$ to be divergent, we can assume that the integral $$\int_{-\infty}^{\infty}\ \frac{1}{x}\ dx$$ is also divergent.

Accepted Answer

True

Answer

False

Question 11

Consider the improper integral $$\int_1^4\ \frac{1}{x^2-4}dx$$ . What would be the correct way to write this as a limit?

Accepted Answer

$$\lim_{t\rightarrow2^-}\ \int_1^t\ \frac{1}{x^2-4}dx\ +\ \lim_{t\rightarrow2^+}\ \int_t^4\ \frac{1}{x^2-4}dx$$

Answer

$$\lim_{t\rightarrow4}\ \int_t^4\ \frac{1}{x^2-4}dx$$

Answer

$$\lim_{t\rightarrow2}\ \int_1^4\ \frac{1}{x^2-4}dx$$

Question 12

For which of the following integrals can u-substitution be used? (can be more than one)

Accepted Answer

$$\int\ \frac{\ln\left(x\right)}{x}dx$$

Accepted Answer

$$\int xe^{x^2\ }dx$$

Answer

$$\int x\ln\left(x\right)dx$$

Answer

$$\int xe^{x\ }dx$$

Calculus II Unit 3 Review

15 questions

Place the following limits in ascending order.

For any integral with an even value of n, order the method of approximation from least to most accurate.

Create a free account and access millions of resources

Similar Resources on Wayground

Popular Resources on Wayground

Discover more resources for Mathematics