Calculus I · Unit 3A · exam

Unit 3A Practice Exam A

Answer key published

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Practice Exam A

Exercise

Evaluate

\int\left(4x^3-\frac{2}{x}+\sec^2x\right)\,dx.

Exercise

Use four midpoint rectangles to estimate

\int_0^2(x^2+1)\,dx.

Show the subinterval width and the four midpoint sample values.

Exercise

Explain the difference between an indefinite integral and a definite integral. Your answer must mention both the type of output and the role of the constant of integration.

Exercise

Differentiate

G(x)=\int_1^{\sin x}e^{t^2}\,dt.

Exercise

Evaluate

\int_0^1 2x(1+x^2)^4\,dx.

Exercise

Evaluate

\int xe^{2x}\,dx.

Exercise

Evaluate by partial fractions:

\int\frac{3x+5}{x^2+3x+2}\,dx.

Exercise

Determine whether

\int_1^\infty\frac{1}{x^2}\,dx

converges. If it converges, find its value. Show the limit that defines the improper integral.

Exercise

A tank's net inflow rate is $r(t)$ liters per minute. Explain the meaning and units of

\int_0^5r(t)\,dt.

State how a negative value should be interpreted.

Exercise

Compare midpoint, trapezoidal, and Simpson estimates. State one reason Simpson's Rule is often more accurate for a smooth function and one reason no numerical estimate should be accepted without a reasonableness check.

After the explanation

Use the section idea

Reading lens

Mixed integral work tests recognition: classify the output and structure before committing to a method.

Mental model

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

Decision

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

Common trap

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

Check yourself

Can you reproduce the reasoning without the key and explain why the final form fits the question?

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