Worksheets — Set 11

Calculus III · Partial derivatives and optimization

Print and write your answers. Use diagrams to show thinking.

Warm-up

1. Find lim(x→14) (x² − 196)/(x − 14) = ___ 2. If f(x) = 14x³ − 15x, find f′(13) = ___ 3. ∫₀^14 (15x + 1) dx = ___

_See printable PDF for diagram._

Written practice

1. Find lim(x→13) (x² − 169)/(x − 13) = ___ 2. If f(x) = 13x³ − 14x, find f′(12) = ___ 3. ∫₀^13 (14x + 1) dx = ___ 4. Find critical points of f(x) = x² − 26x + 16 ___ 5. Related rates: radius grows 12 cm/s. Find dV/dt when r = 14. ___ 6. State the meaning of f′(a) in context. ___ 7. Find lim(x→16) (x² − 256)/(x − 16) = ___ 8. If f(x) = 16x³ − 17x, find f′(15) = ___

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More written practice

1. ∫₀^16 (17x + 1) dx = ___ 2. Find critical points of f(x) = x² − 32x + 19 ___ 3. Related rates: radius grows 15 cm/s. Find dV/dt when r = 17. ___ 4. Find lim(x→19) (x² − 361)/(x − 19) = ___ 5. If f(x) = 19x³ − 20x, find f′(18) = ___ 6. ∫₀^19 (20x + 1) dx = ___

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Patterns and rules

1. Find lim(x→30) (x² − 900)/(x − 30) = ___ 2. If f(x) = 30x³ − 31x, find f′(29) = ___ 3. ∫₀^30 (31x + 1) dx = ___

_See printable PDF for diagram._

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Parent tip: Sketch graphs to check signs and behavior