Calculus I · 2B · exam

Unit 2B Practice Exam B

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

Notice

A complete response includes setup, calculus, candidates or rates, units, interpretation, and a reasonableness check.

Decide

Choose the governing relationship before calculating, then use the attempt-gated answer as feedback rather than as a shortcut.

Avoid

Reading an answer before modeling the problem or treating every miss as algebra when the first error was conceptual.

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: 130 minutes. This version emphasizes modeling and synthesis.

Exercise

A particle has velocity $v(t)=t^3-6t^2+8t$ . Determine direction and speeding behavior on $[0,6]$ , then describe a compatible position graph.

Exercise

A damped oscillator has $s(t)=8e^{-t/5}\cos(2t)$ . Find velocity and interpret the two contributions to it.

Exercise

Use differentials to estimate the percentage volume error of a sphere when radius has $0.4\%$ error.

Exercise

Water enters a cone of height $15$ and radius $5$ at $3$ m $^3$ /min. Derive $dh/dt$ as a function of $h$ .

Exercise

A response curve is $R(d)=100d/(d+12)$ . Analyze monotonicity, concavity, limiting response, and the practical meaning of diminishing returns.

Exercise

Perform a complete curve analysis of $f(x)=x/(x^2-1)$ .

Exercise

Minimize material for a closed cylinder with fixed volume $500\pi$ , and interpret the relationship between optimal height and radius.

Exercise

A cable must connect a point offshore to a point on land, with underwater installation costing four times as much per meter. Build and optimize a one-variable cost model.

Exercise

For $Q=12K^{0.4}L^{0.6}$ , interpret labor elasticity and explain diminishing marginal output with capital fixed.

Exercise

Evaluate $\lim_{x\to0^+}x^{\sin x}$ .

Exercise

Evaluate $\lim_{x\to\infty}x^4/e^x$ , and explain the growth-rate conclusion.

Exercise

Show that ordinary Newton iteration for $f(x)=x^3-2x+2$ cycles from $x_0=0$ , then explain what this teaches about algorithms.

Exercise

Prove that any critical point of a differentiable convex function is a global minimum.

Exercise

Write a brief critique of one modeling assumption in the stopping-distance or medication model from this unit.

Structured concept quiz

Unit 2B 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

A particle has velocity $v(t)=t^3-6t^2+8t$ . Determine direction and speeding behavior on $[0,6]$ , then describe a compatible position graph.

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 damped oscillator has $s(t)=8e^{-t/5}\cos(2t)$ . Find velocity and interpret the two contributions to it.

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

Use differentials to estimate the percentage volume error of a sphere when radius has $0.4\%$ error.

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

Water enters a cone of height $15$ and radius $5$ at $3$ m $^3$ /min. Derive $dh/dt$ as a function of $h$ .

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

A response curve is $R(d)=100d/(d+12)$ . Analyze monotonicity, concavity, limiting response, and the practical meaning of diminishing returns.

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

Perform a complete curve analysis of $f(x)=x/(x^2-1)$ .

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

Minimize material for a closed cylinder with fixed volume $500\pi$ , and interpret the relationship between optimal height and radius.

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

A cable must connect a point offshore to a point on land, with underwater installation costing four times as much per meter. Build and optimize a one-variable cost model.

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

For $Q=12K^{0.4}L^{0.6}$ , interpret labor elasticity and explain diminishing marginal output with capital fixed.

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

Evaluate $\lim_{x\to0^+}x^{\sin x}$ .

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-11

Evaluate $\lim_{x\to\infty}x^4/e^x$ , and explain the growth-rate conclusion.

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

Show that ordinary Newton iteration for $f(x)=x^3-2x+2$ cycles from $x_0=0$ , then explain what this teaches about algorithms.

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

Prove that any critical point of a differentiable convex function is a global minimum.

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

Write a brief critique of one modeling assumption in the stopping-distance or medication model from this unit.

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: 130 minutes. This version emphasizes modeling and synthesis.

Exercise

A particle has velocity $v(t)=t^3-6t^2+8t$ . Determine direction and speeding behavior on $[0,6]$ , then describe a compatible position graph.

Exercise

A damped oscillator has $s(t)=8e^{-t/5}\cos(2t)$ . Find velocity and interpret the two contributions to it.

Exercise

Use differentials to estimate the percentage volume error of a sphere when radius has $0.4\%$ error.

Exercise

Water enters a cone of height $15$ and radius $5$ at $3$ m $^3$ /min. Derive $dh/dt$ as a function of $h$ .

Exercise

A response curve is $R(d)=100d/(d+12)$ . Analyze monotonicity, concavity, limiting response, and the practical meaning of diminishing returns.

Exercise

Perform a complete curve analysis of $f(x)=x/(x^2-1)$ .

Exercise

Minimize material for a closed cylinder with fixed volume $500\pi$ , and interpret the relationship between optimal height and radius.

Exercise

A cable must connect a point offshore to a point on land, with underwater installation costing four times as much per meter. Build and optimize a one-variable cost model.

Exercise

For $Q=12K^{0.4}L^{0.6}$ , interpret labor elasticity and explain diminishing marginal output with capital fixed.

Exercise

Evaluate $\lim_{x\to0^+}x^{\sin x}$ .

Exercise

Evaluate $\lim_{x\to\infty}x^4/e^x$ , and explain the growth-rate conclusion.

Exercise

Show that ordinary Newton iteration for $f(x)=x^3-2x+2$ cycles from $x_0=0$ , then explain what this teaches about algorithms.

Exercise

Prove that any critical point of a differentiable convex function is a global minimum.

Exercise

Write a brief critique of one modeling assumption in the stopping-distance or medication model from this unit.

Structured concept quiz

Unit 2B 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

A particle has velocity $v(t)=t^3-6t^2+8t$ . Determine direction and speeding behavior on $[0,6]$ , then describe a compatible position graph.

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 damped oscillator has $s(t)=8e^{-t/5}\cos(2t)$ . Find velocity and interpret the two contributions to it.

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

Use differentials to estimate the percentage volume error of a sphere when radius has $0.4\%$ error.

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

Water enters a cone of height $15$ and radius $5$ at $3$ m $^3$ /min. Derive $dh/dt$ as a function of $h$ .

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

A response curve is $R(d)=100d/(d+12)$ . Analyze monotonicity, concavity, limiting response, and the practical meaning of diminishing returns.

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

Perform a complete curve analysis of $f(x)=x/(x^2-1)$ .

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

Minimize material for a closed cylinder with fixed volume $500\pi$ , and interpret the relationship between optimal height and radius.

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

A cable must connect a point offshore to a point on land, with underwater installation costing four times as much per meter. Build and optimize a one-variable cost model.

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

For $Q=12K^{0.4}L^{0.6}$ , interpret labor elasticity and explain diminishing marginal output with capital fixed.

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

Evaluate $\lim_{x\to0^+}x^{\sin x}$ .

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-11

Evaluate $\lim_{x\to\infty}x^4/e^x$ , and explain the growth-rate conclusion.

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

Show that ordinary Newton iteration for $f(x)=x^3-2x+2$ cycles from $x_0=0$ , then explain what this teaches about algorithms.

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

Prove that any critical point of a differentiable convex function is a global minimum.

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

Write a brief critique of one modeling assumption in the stopping-distance or medication model from this unit.

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.