Calculus I · Unit 2B · review

Higher Derivatives and Motion Review

Review

Summary

v=s',\qquad a=v'=s''.

Velocity gives direction; speed is $|v|$ . Speed increases when $v$ and $a$ share a sign. The second derivative measures the change of the first derivative and controls concavity.

Exercise

For $s(t)=t^4-8t^2$ , find $v$ , $a$ , and all rest times.

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Exercise 1 answer

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Exercise

For $s(t)=t^3-3t^2$ on $[0,4]$ , determine direction changes.

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Exercise 2 answer

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Exercise

Determine speeding-up intervals for the previous problem.

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Exercise 3 answer

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Exercise

Sketch a position graph whose velocity is positive but decreasing.

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Exercise 4 answer

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Exercise

A company's revenue $R(q)$ has $R'(100)=80$ and $R"(100)=-0.6$ . Interpret both values.

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Exercise 5 answer

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Exercise

Estimate $f'(2)$ and $f"(2)$ from a table of your own design and explain the needed spacing.

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Exercise 6 answer

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Exercise

Match plausible graphs of $f$ , $f'$ , and $f"$ for a cubic with two turning points.

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Exercise 7 answer

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After the explanation

Use the section idea

Reading lens

Use the sign, size, units, and zeros of derivatives to tell a time-aligned story about motion or another changing quantity.

Mental model

Position, velocity, and acceleration are synchronized views: amount, rate of amount, and rate of the rate.

Decision

Separate direction from speed, and compare the signs of velocity and acceleration before describing speeding behavior.

Common trap

Treating negative velocity as slowing down or confusing a function's height with the slope of its graph.

Check yourself

Can you interpret a first and second derivative at the same input without mixing their units or meanings?

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