Course 4 · Lesson 10

Capstone: Math in the Real World

Bring it all together. See how trigonometry, statistics, and calculus solve real engineering and science problems — and build your own mathematical model.

Key Concepts

Projectile Motion

Height: h(t) = h₀ + v₀t − ½gt². Velocity: v(t) = h'(t) = v₀ − gt. At max height, v = 0. Uses derivatives, trig (launch angle), and quadratics all together.

Exponential Growth/Decay

P(t) = P₀ · eᵏᵗ. Population, compound interest, radioactive decay. Derivative: P'(t) = kP(t) — the rate of change is proportional to current size.

Optimization

Many real problems ask: 'What x maximizes/minimizes f(x)?' Find critical points where f'(x) = 0, check second derivative to confirm max or min.

Data & Regression

Fit a mathematical model to real data. Linear regression finds the best-fit line y = mx + b. Residuals measure how well the model fits. R² measures goodness of fit.

Live Python Practice

Interactive Lab

Speed v₀ (m/s): Angle (°): Height h₀ (m):

Projectile motion simulator. Yellow dot = max height. Try 45° for max range. Try non-zero h₀ for cliff launches.

Check Your Understanding

At the maximum height of a projectile, the vertical velocity is:

In optimization, you find critical points by setting:

P(t) = P₀eᵏᵗ models exponential growth when:

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