Calculus II · Unit 4B · answer key

Unit 4B Practice Exam B Answer Key

Finish an honest attempt first. Then compare one numbered response at a time, locate the first line where your reasoning diverged, and retry without the key open.

Answer 1

Problem 1

Radius $5$ ; at $x=3$ , terms are $n$ and diverge; at $x=-7$ , terms are $n(-1)^n$ and do not approach zero. Interval $(-7,3)$ .

Answer 2

Problem 2

$x^2+x^3+x^4+\cdots$ , $|x|<1$ .

Answer 3

Problem 3

Substitute $-x$ , differentiate, and obtain $1/(1+x)^2=\sum_{n=0}^\infty(-1)^n(n+1)x^n$ .

Answer 4

Problem 4

$(x-1)-(x-1)^2/2+(x-1)^3/3$ .

Answer 5

Problem 5

$\sum(-1)^n x^{2n+1}/(2n+1)$ .

Answer 6

Problem 6

$1+0.02/2-(0.02)^2/8$ .

Answer 7

Problem 7

Four nonzero terms suffice: through $-x^7/7!$ . The first omitted magnitude is $0.5^9/9!\approx5.38\times10^{-9}<10^{-8}$ , and the terms decrease.

Answer 8

Problem 8

Using three nonzero terms,

\int_0^{0.25}\cos(x^2)\,dx\approx 0.25-\frac{0.25^5}{10}+\frac{0.25^9}{216}\approx0.2499023614.

Answer 9

Problem 9

Recurrence gives $c_n=2/n!$ , so $y=2e^x$ .

Answer 10

Problem 10

Uniform convergence uses one index working simultaneously for all points.

Answer 11

Problem 11

Small-angle pendulum: replace $\sin\theta$ by $\theta$ .

Answer 12

Problem 12

Answers vary but must name a valid remainder estimate and verify its hypotheses.

After the explanation

Use the section idea

Reading lens

Mixed work tests recognition: classify the task before selecting a convergence, construction, transformation, or error method.

Mental model

A complete solution preserves the series identity, interval, endpoint logic, and approximation claim as one chain.

Decision

Write the objective and conditions first, solve independently, then use one answer reveal to repair the earliest incorrect decision.

Common trap

Checking algebra while leaving the convergence domain or error certification unstated.

Check yourself

Can you reproduce the method and domain without the key?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.