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.

Session plan

Start with the first two questions to warm up before moving into the harder ones.

Use the calculator only after you have set up the working or estimated the answer.

Work through the page like a focused study sheet instead of jumping around.

How to use this page

A cleaner maths workflow

1. Read the prompt carefully

These pages are meant to feel consistent with the rest of the site, but the maths stays front and center.

2. Solve in order

The question list works best as a progression rather than a random set of cards.

3. Use support tools when needed

You can jump back to the calculator or home without losing the overall flow.

Practice set

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.