Calculus I · 2A · exam

Unit 2A Practice Exam B

Complete Unit 2A Practice Exam B as a cumulative Calculus I assessment, then use the separately published answer key for review.

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Section overview

Review, practice, exams, and reference

What this section is building

Complete Unit 2A Practice Exam B as a cumulative Calculus I assessment, then use the separately published answer key for review.

Notice

A dependable solution separates interpretation, rule selection, calculation, and verification.

Decide

Classify first, calculate second, and use answer reveals to diagnose the first incorrect decision.

Avoid

Reading the solution before making a complete attempt or treating every miss as mere algebra.

Answer key published

Finish first. Then check every answer.

The complete numbered key is online as its own easy-to-find route.

View the complete answer key →

Practice Exam B

Suggested time: 110 minutes. This version emphasizes synthesis.

Exercise

Derive the power rule for $x^4$ from the limit definition.

Exercise

A graph of $f$ has a corner at $x=0$ . Explain what must fail in the one-sided derivative limits.

Exercise

Differentiate $x^3\sqrt{x^2+1}/(x-2)^4$ .

Exercise

Differentiate $\sec^3(2x)$ .

Exercise

Differentiate $\log_5(1+e^{3x})$ .

Exercise

Differentiate $\arctan(\sqrt{x})$ .

Exercise

For $x^3+y^3=6xy$ , find tangent and normal lines at $(3,3)$ .

Exercise

If $f(2)=5$ , $f'(2)=-3$ , and $f"(2)=4$ , derive $(f^{-1})"(5)$ .

Exercise

Differentiate $x^{\sin x}$ for $x>0$ .

Exercise

Find $f"$ for $f(x)=e^{x^2}$ and factor the result.

Exercise

Find a formula for the $n$ th derivative of $e^{ax}$ .

Exercise

Determine where the derivative of $\arcsin(2x)$ exists as a finite real number.

Exercise

A calibration function is $S(V)=\ln(1+V^2)$ , where $V(P)=0.5\sqrt{P+4}$ . Find $dS/dP$ at $P=12$ .

Exercise

Explain how the chain rule becomes multiplication of local response factors in the previous problem.

Structured concept quiz

Unit 2A Practice Exam B

Write a response before revealing the model. These conceptual items use an honest attempt-and-reveal rubric rather than pretending an open response has one machine-provable wording.

Interactive checkexam-b-01

Derive the power rule for $x^4$ from the limit definition.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-02

A graph of $f$ has a corner at $x=0$ . Explain what must fail in the one-sided derivative limits.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-03

Differentiate $x^3\sqrt{x^2+1}/(x-2)^4$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-04

Differentiate $\sec^3(2x)$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-05

Differentiate $\log_5(1+e^{3x})$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-06

Differentiate $\arctan(\sqrt{x})$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-07

For $x^3+y^3=6xy$ , find tangent and normal lines at $(3,3)$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-08

If $f(2)=5$ , $f'(2)=-3$ , and $f"(2)=4$ , derive $(f^{-1})"(5)$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-09

Differentiate $x^{\sin x}$ for $x>0$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-10

Find $f"$ for $f(x)=e^{x^2}$ and factor the result.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-11

Find a formula for the $n$ th derivative of $e^{ax}$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-12

Determine where the derivative of $\arcsin(2x)$ exists as a finite real number.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-13

A calibration function is $S(V)=\ln(1+V^2)$ , where $V(P)=0.5\sqrt{P+4}$ . Find $dS/dP$ at $P=12$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-14

Explain how the chain rule becomes multiplication of local response factors in the previous problem.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

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

Practice Exam B

Suggested time: 110 minutes. This version emphasizes synthesis.

Exercise

Derive the power rule for $x^4$ from the limit definition.

Exercise

A graph of $f$ has a corner at $x=0$ . Explain what must fail in the one-sided derivative limits.

Exercise

Differentiate $x^3\sqrt{x^2+1}/(x-2)^4$ .

Exercise

Differentiate $\sec^3(2x)$ .

Exercise

Differentiate $\log_5(1+e^{3x})$ .

Exercise

Differentiate $\arctan(\sqrt{x})$ .

Exercise

For $x^3+y^3=6xy$ , find tangent and normal lines at $(3,3)$ .

Exercise

If $f(2)=5$ , $f'(2)=-3$ , and $f"(2)=4$ , derive $(f^{-1})"(5)$ .

Exercise

Differentiate $x^{\sin x}$ for $x>0$ .

Exercise

Find $f"$ for $f(x)=e^{x^2}$ and factor the result.

Exercise

Find a formula for the $n$ th derivative of $e^{ax}$ .

Exercise

Determine where the derivative of $\arcsin(2x)$ exists as a finite real number.

Exercise

A calibration function is $S(V)=\ln(1+V^2)$ , where $V(P)=0.5\sqrt{P+4}$ . Find $dS/dP$ at $P=12$ .

Exercise

Explain how the chain rule becomes multiplication of local response factors in the previous problem.

Structured concept quiz

Unit 2A Practice Exam B

Write a response before revealing the model. These conceptual items use an honest attempt-and-reveal rubric rather than pretending an open response has one machine-provable wording.

Interactive checkexam-b-01

Derive the power rule for $x^4$ from the limit definition.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-02

A graph of $f$ has a corner at $x=0$ . Explain what must fail in the one-sided derivative limits.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-03

Differentiate $x^3\sqrt{x^2+1}/(x-2)^4$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-04

Differentiate $\sec^3(2x)$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-05

Differentiate $\log_5(1+e^{3x})$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-06

Differentiate $\arctan(\sqrt{x})$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-07

For $x^3+y^3=6xy$ , find tangent and normal lines at $(3,3)$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-08

If $f(2)=5$ , $f'(2)=-3$ , and $f"(2)=4$ , derive $(f^{-1})"(5)$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-09

Differentiate $x^{\sin x}$ for $x>0$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-10

Find $f"$ for $f(x)=e^{x^2}$ and factor the result.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-11

Find a formula for the $n$ th derivative of $e^{ax}$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-12

Determine where the derivative of $\arcsin(2x)$ exists as a finite real number.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-13

A calibration function is $S(V)=\ln(1+V^2)$ , where $V(P)=0.5\sqrt{P+4}$ . Find $dS/dP$ at $P=12$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkexam-b-14

Explain how the chain rule becomes multiplication of local response factors in the previous problem.

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

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

Finish first. Then check every answer.

Practice Exam B

Related resources

Original instruction with traceable references.

Finish first. Then check every answer.

Practice Exam B

Related resources

Original instruction with traceable references.