Wayground
Calculus II Unit 3 Review - Printable Mathematics Worksheets - Wayground

Calculus II Unit 3 Review

15 questions

University - University

Mathematics

Calculus II Unit 3 Review is a free printable Mathematics worksheet for Grade 13 students that reviews key skills involving limits, L'Hopital's Rule, numerical integration, error bounds, improper integrals, and u-substitution. Its 15 items include ordering tasks, multiple-choice questions, a drag-and-drop limit activity, and a multiple-select question, giving students varied ways to apply calculus procedures and interpret results. Students compare limits, choose useful quotient forms for indeterminate products, rank approximation methods, identify K values for Simpson's and Midpoint Rules, recognize a correct Simpson's Rule summation, express an improper integral as a limit, evaluate limits, and select integrals suitable for u-substitution. The worksheet works well as a unit review, guided practice, or readiness check before an assessment. It includes a complete answer key so teachers can review procedural choices and address errors efficiently.

Worksheet settings

Font size

S
M
L
XL
Worksheets

Calculus II Unit 3 Review

Total questions: 15

Name
Class
Date
1.

Place the following limits in ascending order.

a)

lim⁡x→0 2ex\lim_{x\rightarrow0}\ 2e^x

b)

lim⁡x→0+ ln⁡(x)\lim_{x\rightarrow0^+}\ \ln\left(x\right)

c)

lim⁡x→∞ −3x2−2x+15x2+4x−2\lim_{x\rightarrow\infty}\ \frac{-3x^2-2x+1}{5x^2+4x-2}

d)

lim⁡x→∞ sin⁡(1x)\lim_{x\rightarrow\infty}\ \sin\left(\frac{1}{x}\right)

e)

lim⁡x→−∞ ln⁡∣x∣\lim_{x\rightarrow-\infty}\ \ln\left|x\right|

1)
2)
3)
4)
5)
2.

lim⁡x→0 xex\lim_{x\rightarrow0}\ \frac{x}{e^x}

a)

−∞-\infty

b)

0

c)

1

d)

∞\infty

3.

lim⁡x→∞ xex\lim_{x\rightarrow\infty}\ \frac{x}{e^x}

a)

−∞-\infty

b)

0

c)

1

d)

∞\infty

4.

lim⁡x→∞ 25x5ex\lim_{x\rightarrow\infty}\ \frac{25x^5}{e^x}

a)

−∞-\infty

b)

0

c)

1

d)

∞\infty

5.

With the following limit, when plugging in the value of "a" to the function, we get an indeterminate product. lim⁡x→∞ xsin⁡(1x)\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?

a)

lim⁡x→∞ sin⁡(1x)1x\lim_{x\rightarrow\infty}\ \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}}

b)

lim⁡x→∞ xcsc⁡(1x)\lim_{x\rightarrow\infty}\ \frac{x}{\csc\left(\frac{1}{x}\right)}

c)

lim⁡x→∞ x1sin⁡(1x)\lim_{x\rightarrow\infty}\ \frac{x}{\frac{1}{\sin\left(\frac{1}{x}\right)}}

6.

With the following limit, when plugging in the value of "a" to the function, we get an indeterminate product. lim⁡x→−∞ x2ex\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?

a)

lim⁡x→−∞ ex1x2\lim_{x\rightarrow-\infty}\ \frac{e^x}{\frac{1}{x^2}}

b)

lim⁡x→−∞ exx−2\lim_{x\rightarrow-\infty}\ \frac{e^x}{x^{-2}}

c)

lim⁡x→−∞ x−2ex\lim_{x\rightarrow-\infty}\ \frac{x^{-2}}{e^x}

d)

lim⁡x→−∞ x2e−x\lim_{x\rightarrow-\infty}\ \frac{x^2}{e^{-x}}

7.

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

a)

Right/Left Riemann Sum

b)

Simpson's Rule

c)

Midpoint Rule

d)

Trapezoid Rule

1)
2)
3)
4)
8.

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

∫0π245cos⁡(x) dx\int_0^{\frac{\pi}{2}}45\cos\left(x\right)\ dx

a)

45π2\frac{45\pi}{2}

b)

452 or 22.5\frac{45}{2}\ or\ 22.5

c)

4545

d)

π2\frac{\pi}{2}

9.

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

∫12e1x dx\int_1^2e^{\frac{1}{x}}\ dx

a)

ee

b)

5e16\frac{5\sqrt[]{e}}{16}

c)

e\sqrt[]{e}

d)

3e3e

10.

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

∫45 2x−3 dx\int_4^5\ \frac{2}{x-3}\ dx

a)

4848

b)

11

c)

22

d)

485\frac{48}{5}

11.

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

a) 13[f(1)+4f(2)+2f(3)+4f(4)+2f(5)+f(6)]\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) 16[f(1)+4f(2)+2f(3)+4f(4)+2f(5)+4f(6)+f(7)]\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) 13[f(1)+4f(2)+2f(3)+4f(4)+2f(5)+4f(6)+f(7)]\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) 16[f(1)+4f(2)+2f(3)+4f(4)+2f(5)+f(6)]\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]

a)

a

b)

b

c)

c

d)

d

12.

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

a)

True

b)

False

13.

Consider the improper integral ∫14 1x2−4dx\int_1^4\ \frac{1}{x^2-4}dx . What would be the correct way to write this as a limit?

a)

lim⁡t→4 ∫t4 1x2−4dx\lim_{t\rightarrow4}\ \int_t^4\ \frac{1}{x^2-4}dx

b)

lim⁡t→2 ∫14 1x2−4dx\lim_{t\rightarrow2}\ \int_1^4\ \frac{1}{x^2-4}dx

c)

lim⁡t→2− ∫1t 1x2−4dx + lim⁡t→2+ ∫t4 1x2−4dx\lim_{t\rightarrow2^-}\ \int_1^t\ \frac{1}{x^2-4}dx\ +\ \lim_{t\rightarrow2^+}\ \int_t^4\ \frac{1}{x^2-4}dx

14.

Find the following limits:

​ ​ lim⁡t→−∞et2\lim_{t\rightarrow-\infty}e^{t^2} ​ ​ (a)  

lim⁡t→−∞e−t2\lim_{t\rightarrow-\infty}e^{-t^2} ​ ​ (b)  

lim⁡t→∞e−t2\lim_{t\rightarrow\infty}e^{-t^2} ​ (c)  

lim⁡t→−∞e−t3\lim_{t\rightarrow-\infty}e^{-t^3} ​ ​ (d)  

Choose from the below words

∞\infty  

00  

−∞-\infty  

15.

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

a)

∫xln⁡(x)dx\int x\ln\left(x\right)dx

b)

∫ ln⁡(x)xdx\int\ \frac{\ln\left(x\right)}{x}dx

c)

∫xex dx\int xe^{x\ }dx

d)

∫xex2 dx\int xe^{x^2\ }dx

Answer Key

Calculus II Unit 3 Review

Total questions: 15

1.
b, c, d, a, e
2.
b)

0

3.
b)

0

4.
b)

0

5.
a)

lim⁡x→∞ sin⁡(1x)1x\lim_{x\rightarrow\infty}\ \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}}

6.
d)

lim⁡x→−∞ x2e−x\lim_{x\rightarrow-\infty}\ \frac{x^2}{e^{-x}}

7.
a, d, c, b
8.
c)

4545

9.
d)

3e3e

10.
a)

4848

11.
c)

c

12.
a)

True

13.
c)

lim⁡t→2− ∫1t 1x2−4dx + lim⁡t→2+ ∫t4 1x2−4dx\lim_{t\rightarrow2^-}\ \int_1^t\ \frac{1}{x^2-4}dx\ +\ \lim_{t\rightarrow2^+}\ \int_t^4\ \frac{1}{x^2-4}dx

14.

∞\infty  

,

00  

,

00  

,

∞\infty  

15.
b)

∫ ln⁡(x)xdx\int\ \frac{\ln\left(x\right)}{x}dx

, d)

∫xex2 dx\int xe^{x^2\ }dx

Browse by subjects

Browse by grades

Browse by format

FAQs

What does the Calculus II Unit 3 Review worksheet cover?

This 15-question worksheet reviews several connected Calculus II skills: rewriting indeterminate products for L'Hopital's Rule, ordering limits and numerical-integration methods, finding K values for error bounds, identifying valid Simpson's Rule sums, representing improper integrals as limits, evaluating limits, and selecting integrals for u-substitution. Ordering, multiple-choice, drag-and-drop, and multiple-select formats require students to compare methods, recognize correct setups, and apply procedures rather than simply recall definitions.

How can I use this worksheet to teach the Unit 3 calculus concepts?

Use the two ordering items to begin a class discussion about how limit behavior and approximation accuracy are compared, then model one error-bound question by identifying the derivative-based K value before students work independently. Review the L'Hopital quotient-form items and the improper-integral limit item by asking students to explain why each setup is valid, and use the Simpson's Rule and u-substitution questions as short application checks.

What mistakes should I watch for when students complete this Calculus II review?

Watch for students rewriting an indeterminate product as a quotient in an unhelpful form, confusing the accuracy ranking of Left or Right Riemann Sums, the Midpoint Rule, the Trapezoid Rule, and Simpson's Rule, or choosing the wrong derivative quantity when finding K for an error bound. Students may also write an improper integral without its limiting process, accept an incorrect Simpson's Rule summation, or select u-substitution without checking whether the integrand contains a suitable inner function and derivative.

How do I assign and grade this Calculus II Unit 3 Review worksheet?

Wayground provides this worksheet as a printable PDF and as a digital quiz, and it includes a complete answer key for the ordering, multiple-choice, drag-and-drop, and multiple-select items. For printable submissions, scan student work with the Wayground for Teachers app to support grading and review.

How can I differentiate this limits and numerical integration worksheet for my students?

In the worksheet's Advanced Settings, adjust font spacing or choose a larger font size to make the symbolic limit expressions, error-bound choices, and Simpson's Rule options easier to read. You can also enable a dyslexia-friendly font or translate the worksheet into another language while keeping its mixed question format.

Where can I find more worksheets like this on Wayground?

Wayground offers a broad collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to help students build essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

© 2026 Wayground Inc.

Accessibility and Inclusion