Related Rates & Optimization

Apply derivatives to real-world dynamics. Solve related rates of changing variables using implicit differentiation, and maximize enclosure areas using extreme value optimization.

Interactive Sandbox

Rates & Extrema Visualizer

Toggle between Related Rates (Sliding Ladder) and Optimization (Parabolic Fenced Yard). Drag handles directly on the canvas or adjust sliders to see calculus in action.

Related Rates: Sliding Ladder

Pythagorean Rate Equation

x² + y² = L² ⇒ 2x(dx/dt) + 2y(dy/dt) = 0

Base Distance (x) 3.00 m

Sliding Speed (dx/dt) 0.80 m/s

Drag the ladder base horizontally. Observe how dy/dt shoots up to infinity as the ladder approaches the floor.

Algebraic Solver

Implicit & Extreme Steps Solver

Trace exact rates differentials and derivative test optimization bounds.

Analytical Steps Solution

Values Summary

Ladder Base Speed (dx/dt)

dx/dt = 0.80 m/s

Ladder Top Speed (dy/dt)

dy/dt = -0.60 m/s

Evaluation

Rates & Extrema Quiz

Test your conceptual understanding of related rates, optimization criteria, and Jacobians.

Question 1 of 5 Score: 0/0

A ladder sliding down a wall has dy/dt which depends on what ratio of side lengths?

Correct Answer!

Explanation text...

Rates & Extrema Rules

Ladder Rates: dy/dt = -(x/y) dx/dt
Pen Area (3-sided): A = x(P - 2x)
Extremum Peak: dA/dx = 0 ⇒ x = P/4