Applications of Derivatives

Use derivatives to study gradients, turning points, optimization, and motion contexts.

Focus points

✓Interpret first and second derivatives in context.

✓Classify stationary points with clear reasoning.

✓Translate worded optimization problems into functions.

Orin

The Wise Owl

Guides study flow, revision planning, and calmer focus sessions.

Practice set

Question lineup

Mark each one done as you go — finish them all to claim your chips.

6 revision prompts

Question 1OpenStax Calculus Volume 1, Sec. 4.1 Review Exercises

Find the critical points of f (x) = x^{3} - 3 x + 1 .

Question 2

A rectangle has perimeter 40 . Express the area as a function of one side and find the maximum area.

Question 3

The position of a particle is s (t) = t^{3} - 6 t^{2} + 9 t . Find the velocity and acceleration.

Question 4

Determine the intervals where f (x) = x^{4} - 4 x^{2} is increasing and decreasing.

Question 5

Use the second derivative test to classify the stationary points of y = x^{3} - 6 x^{2} + 9 x .

Question 6

A box with square base and no lid has volume 108 cm^{3} . Find dimensions that minimize surface area.