Practice exam · Calculus II · Units 3B–4B

Calculus II Practice Final Exam with Complete Solutions

A balanced twenty-five-question Calculus II final with study map and complete worked key.

25 problems120 min estimated timeCumulative progression

What is included

Prepare for a Calculus II final with integration methods, applications, convergence, power series, and Taylor series.

Skills assessed

method selection
convergence proof
series construction
course synthesis

Prerequisites

Calculus II Units 3B through 4B

Exam conditions

Suggested time: 120 minutes

Points: 4 points per question; 100 points total

Calculator: Scientific calculator permitted; computer algebra is not required.

This is an original BetterGrades practice exam, not a released institutional exam.

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Evaluate $\int xe^x dx$ .
Evaluate $\int\sin^3x\cos xdx$ .
Evaluate $\int1/(x^2-1)dx$ .
Determine convergence of $\int_1^\infty x^{-2}dx$ .
Find area between y=x and y=x² on [0,1].
Rotate y=x on [0,2] about x-axis. Find volume.
A force F(x)=4x acts from x=0 to 3. Find work.
Find $\lim_{n\to\infty} n/(n+1)$ .
Sum $\sum_{n=0}^\infty3(1/2)^n$ .
Test $\sum_{n=1}^{\infty} n/(n+1)$ for convergence.
Test $\sum_{n=1}^{\infty}1/n^{3/2}$ .
Test $\sum_{n=1}^{\infty}1/(n^2+4)$ .
Test $\sum_{n=1}^{\infty}(3n+1)/(n^2+2)$ .
Test $\sum_{n=1}^{\infty} n!/4^n$ .
Test $\sum_{n=1}^{\infty}(2n/(3n+1))^n$ .
Test $\sum_{n=1}^{\infty}(-1)^{n-1}/n$ .
Classify $\sum_{n=1}^{\infty}(-1)^n/n^2$ .
Find R for $\sum_{n=0}^{\infty}x^n/5^n$ .
Find the interval for $\sum_{n=1}^{\infty}(x-1)^n/n$ .
Write the Maclaurin series for cos x.
Find T_3 for e^x at 0.
Bound alternating-series error after four terms of $\sum_{n=1}^{\infty}(-1)^{n-1}/n$ .
For x=t²,y=t³ find dy/dx.
Convert r=2cosθ to Cartesian form.
Why must power-series endpoints be tested separately?

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

01

Problem 1: Evaluate $\int xe^x dx$ .

Answer: $e^x(x-1)+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

With $u=x$ , $dv=e^x dx$ , $I=xe^x-\int e^x dx=e^x(x-1)+C$ .
The verified result is $e^x(x-1)+C$ .

02

Problem 2: Evaluate $\int\sin^3x\cos xdx$ .

Answer: $\sin^4x/4+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Let $u=\sin x$ , $du=\cos xdx$ ; then $\int u^3du=u^4/4+C=\sin^4x/4+C$ .
The verified result is $\sin^4x/4+C$ .

03

Problem 3: Evaluate $\int1/(x^2-1)dx$ .

Answer: $\frac12\ln|\frac{x-1}{x+1}|+C$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

$1/(x^2-1)=\frac12/(x-1)-\frac12/(x+1)$ ; integrating gives $\frac12\ln|x-1|-\frac12\ln|x+1|+C$ .
The verified result is $\frac12\ln|\frac{x-1}{x+1}|+C$ .

04

Problem 4: Determine convergence of $\int_1^\infty x^{-2}dx$ .

Answer: $\text{converges to }1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

$\lim_{b\to\infty}[-x^{-1}]_1^b=\lim_{b\to\infty}(1-1/b)=1$ , so the improper integral converges.
The verified result is $\text{converges to }1$ .

05

Problem 5: Find area between y=x and y=x² on [0,1].

Answer: $1/6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

On $[0,1]$ , $x\ge x^2$ , so $A=\int_0^1(x-x^2)dx=[x^2/2-x^3/3]_0^1=1/6$ .
The verified result is $1/6$ .

06

Problem 6: Rotate y=x on [0,2] about x-axis. Find volume.

Answer: $8\pi/3$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

$V=\pi\int_0^2x^2dx=\pi[x^3/3]_0^2=8\pi/3$ .
The verified result is $8\pi/3$ .

07

Problem 7: A force F(x)=4x acts from x=0 to 3. Find work.

Answer: $18$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

$W=\int_0^3 4x\,dx=[2x^2]_0^3=18$ .
The verified result is $18$ .

08

Problem 8: Find $\lim_{n\to\infty} n/(n+1)$ .

Answer: $1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Divide by $n$ : $n/(n+1)=1/(1+1/n)\to1$ .
The verified result is $1$ .

09

Problem 9: Sum $\sum_{n=0}^\infty3(1/2)^n$ .

Answer: $6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Here $a=3$ , $r=1/2$ , and $|r|<1$ , so $S=3/(1-1/2)=6$ .
The verified result is $6$ .

10

Problem 10: Test $\sum_{n=1}^{\infty} n/(n+1)$ for convergence.

Answer: $\text{diverges}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The terms satisfy $n/(n+1)\to1\ne0$ , so the nth-term test proves divergence.
The verified result is $\text{diverges}$ .

11

Problem 11: Test $\sum_{n=1}^{\infty}1/n^{3/2}$ .

Answer: $\text{converges}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

This is a p-series with $p=3/2>1$ , so it converges.
The verified result is $\text{converges}$ .

12

Problem 12: Test $\sum_{n=1}^{\infty}1/(n^2+4)$ .

Answer: $\text{converges}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

For $n\ge1$ , $0<1/(n^2+4)<1/n^2$ ; comparison with the convergent p-series proves convergence.
The verified result is $\text{converges}$ .

13

Problem 13: Test $\sum_{n=1}^{\infty}(3n+1)/(n^2+2)$ .

Answer: $\text{diverges}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

With $b_n=1/n$ , $a_n/b_n=n(3n+1)/(n^2+2)\to3$ ; limit comparison with the harmonic series gives divergence.
The verified result is $\text{diverges}$ .

14

Problem 14: Test $\sum_{n=1}^{\infty} n!/4^n$ .

Answer: $\text{diverges}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

$a_{n+1}/a_n=(n+1)/4$ , which eventually exceeds 1, so the positive terms do not approach zero and the series diverges.
The verified result is $\text{diverges}$ .

15

Problem 15: Test $\sum_{n=1}^{\infty}(2n/(3n+1))^n$ .

Answer: $\text{converges}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The root-test limit is $\lim 2n/(3n+1)=2/3<1$ , so the series converges absolutely.
The verified result is $\text{converges}$ .

16

Problem 16: Test $\sum_{n=1}^{\infty}(-1)^{n-1}/n$ .

Answer: $\text{converges conditionally}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

$1/n$ decreases to zero, so the alternating-series test gives convergence; $\sum1/n$ diverges, hence convergence is conditional.
The verified result is $\text{converges conditionally}$ .

17

Problem 17: Classify $\sum_{n=1}^{\infty}(-1)^n/n^2$ .

Answer: $\text{converges absolutely}$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The absolute series $\sum1/n^2$ is a p-series with $p=2>1$ , so the original series converges absolutely.
The verified result is $\text{converges absolutely}$ .

18

Problem 18: Find R for $\sum_{n=0}^{\infty}x^n/5^n$ .

Answer: $R=5$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

This is geometric with ratio $x/5$ ; convergence requires $|x/5|<1$ , or $|x|<5$ , so $R=5$ .
The verified result is $R=5$ .

19

Problem 19: Find the interval for $\sum_{n=1}^{\infty}(x-1)^n/n$ .

Answer: $[0,2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The ratio test gives $|x-1|<1$ . At $x=0$ , $\sum(-1)^n/n$ converges; at $x=2$ , $\sum1/n$ diverges, so the interval is $[0,2)$ .
The verified result is $[0,2)$ .

20

Problem 20: Write the Maclaurin series for cos x.

Answer: $\sum_{n=0}^{\infty}(-1)^nx^{2n}/(2n)!$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The derivatives at zero cycle $1,0,-1,0$ , leaving even powers with alternating signs: $\sum_{n=0}^{\infty}(-1)^nx^{2n}/(2n)!$ .
The verified result is $\sum_{n=0}^{\infty}(-1)^nx^{2n}/(2n)!$ .

21

Problem 21: Find T_3 for e^x at 0.

Answer: $1+x+x^2/2+x^3/6$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Every derivative of $e^x$ equals 1 at zero, so $T_3=\sum_{n=0}^3x^n/n!=1+x+x^2/2+x^3/6$ .
The verified result is $1+x+x^2/2+x^3/6$ .

22

Problem 22: Bound alternating-series error after four terms of $\sum_{n=1}^{\infty}(-1)^{n-1}/n$ .

Answer: $\le1/5$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

The terms $1/n$ decrease to zero; after four terms the first omitted magnitude is $1/5$ , so $|R_4|\le1/5$ .
The verified result is $\le1/5$ .

23

Problem 23: For x=t²,y=t³ find dy/dx.

Answer: $3t/2$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

$dx/dt=2t$ and $dy/dt=3t^2$ , so $dy/dx=(3t^2)/(2t)=3t/2$ for $t\ne0$ .
The verified result is $3t/2$ .

24

Problem 24: Convert r=2cosθ to Cartesian form.

Answer: $(x-1)^2+y^2=1$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Multiply by $r$ : $r^2=2r\cos\theta$ , so $x^2+y^2=2x$ , which completes the square to $(x-1)^2+y^2=1$ .
The verified result is $(x-1)^2+y^2=1$ .

25

Problem 25: Why must power-series endpoints be tested separately?

Answer: $\text{The ratio/root test is inconclusive when }|x-a|=R.$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

At $|x-a|=R$ , the limiting ratio or root equals 1, so the test is inconclusive and each endpoint series requires its own convergence test.
The verified result is $\text{The ratio/root test is inconclusive when }|x-a|=R.$ .

Common errors

Choosing an integration or convergence test from surface appearance alone.
Skipping endpoint checks.
Claiming a Taylor representation without a convergence or remainder argument.