Calculus I · Unit 3A · answer key

Unit 3A Practice Exam B Answer Key

Review the worked solutions and modeling rubrics for Unit 3A Practice Exam B. Check methods, units, and interpretation.

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Review the worked solutions and modeling rubrics for Unit 3A Practice Exam B. Check methods, units, and interpretation.

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A complete response carries setup, method, computation, bounds or constants, units, interpretation, and an independent verification.

Decide

Attempt the entire problem first, then use one answer at a time to locate the earliest reasoning decision that needs repair.

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Reading a key before modeling the problem or treating every mismatch as algebra when the first error was conceptual.

Unit 3A 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

6\sqrt{x}+4e^x-5\sin x+C.

Answer 2

Problem 2

$\Delta x=\frac12$ , with right endpoints $\frac12,1,\frac32,2$ :

R_4=\frac12\left(\frac14+1+\frac94+4\right)=\frac{15}{4}.

Answer 3

Problem 3

f_{\mathrm{avg}}=\frac13\int_0^3(x^2+1)dx =\frac13\left[\frac{x^3}{3}+x\right]_0^3=4.

Answer 4

Problem 4

Apply FTC to both bounds:

H'(x)=2\sqrt{1+(2x+1)^4}-2x\sqrt{1+x^8}.

Answer 5

Problem 5

Let $u=1+x^2$ :

\int\frac{x}{(1+x^2)^3}dx =\frac12\int u^{-3}du =-\frac{1}{4(1+x^2)^2}+C.

Answer 6

Problem 6

Integration by parts gives

\int x^2\ln x\,dx =\frac{x^3}{3}\ln x-\frac{x^3}{9}+C.

Answer 7

Problem 7

Write $\sin^3x=(1-\cos^2x)\sin x$ and use $u=\cos x$ :

-\frac{\cos^3x}{3}+\frac{\cos^5x}{5}+C.

Answer 8

Problem 8

Using $x=3\tan\theta$ , or the standard arctangent form,

\int\frac{dx}{x^2+9}=\frac13\arctan\left(\frac{x}{3}\right)+C.

Answer 9

Problem 9

With $h=\frac12$ ,

T_4=\frac{h}{2}\left[f(0)+2f(0.5)+2f(1)+2f(1.5)+f(2)\right] =\frac{17}{4}.

Answer 10

Problem 10

\int_0^1x^{-2/3}dx =\lim_{a\to0^+}\left[3x^{1/3}\right]_a^1=3.

The integral converges.

Answer 11

Problem 11

Indefinite integration reverses differentiation and therefore determines a family differing by constants. Definite integration subtracts endpoint values, so any antiderivative constant cancels.

Answer 12

Problem 12

Use integration by parts for $\int x\ln x\,dx$ ; substitution $u=x^2+7$ for $\int 2x/(x^2+7)dx$ ; and partial fractions after factoring $x^2-4$ for $\int dx/(x^2-4)$ .

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.

Unit 3A 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

6\sqrt{x}+4e^x-5\sin x+C.

Answer 2

Problem 2

$\Delta x=\frac12$ , with right endpoints $\frac12,1,\frac32,2$ :

R_4=\frac12\left(\frac14+1+\frac94+4\right)=\frac{15}{4}.

Answer 3

Problem 3

f_{\mathrm{avg}}=\frac13\int_0^3(x^2+1)dx =\frac13\left[\frac{x^3}{3}+x\right]_0^3=4.

Answer 4

Problem 4

Apply FTC to both bounds:

H'(x)=2\sqrt{1+(2x+1)^4}-2x\sqrt{1+x^8}.

Answer 5

Problem 5

Let $u=1+x^2$ :

\int\frac{x}{(1+x^2)^3}dx =\frac12\int u^{-3}du =-\frac{1}{4(1+x^2)^2}+C.

Answer 6

Problem 6

Integration by parts gives

\int x^2\ln x\,dx =\frac{x^3}{3}\ln x-\frac{x^3}{9}+C.

Answer 7

Problem 7

Write $\sin^3x=(1-\cos^2x)\sin x$ and use $u=\cos x$ :

-\frac{\cos^3x}{3}+\frac{\cos^5x}{5}+C.

Answer 8

Problem 8

Using $x=3\tan\theta$ , or the standard arctangent form,

\int\frac{dx}{x^2+9}=\frac13\arctan\left(\frac{x}{3}\right)+C.

Answer 9

Problem 9

With $h=\frac12$ ,

T_4=\frac{h}{2}\left[f(0)+2f(0.5)+2f(1)+2f(1.5)+f(2)\right] =\frac{17}{4}.

Answer 10

Problem 10

\int_0^1x^{-2/3}dx =\lim_{a\to0^+}\left[3x^{1/3}\right]_a^1=3.

The integral converges.

Answer 11

Problem 11

Indefinite integration reverses differentiation and therefore determines a family differing by constants. Definite integration subtracts endpoint values, so any antiderivative constant cancels.

Answer 12

Problem 12

Use integration by parts for $\int x\ln x\,dx$ ; substitution $u=x^2+7$ for $\int 2x/(x^2+7)dx$ ; and partial fractions after factoring $x^2-4$ for $\int dx/(x^2-4)$ .

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.

Unit 3A Practice Exam B Answer Key

Related resources

Original instruction with traceable references.

Unit 3A Practice Exam B Answer Key

Related resources

Original instruction with traceable references.