Unit 4: Contextual Applications of Differentiation
Work through these 7 topics in order. Clear the gatekeeper and the rest of the course opens up.
4.1 Interpreting the derivative in context
In a real situation, f'(a) is the INSTANTANEOUS RATE of change of the quantity at the input a — and it carries UNITS: (units of f) per (unit of the input).
4.2 Straight-line motion: position, velocity, acceleration
For motion on a line, velocity is the derivative of position and acceleration is the derivative of velocity.
4.3 Rates of change in applied contexts
Any derivative is a rate of change of one quantity with respect to another — cost per item, people per year, liters per second.
4.4 Introduction to related rates
When several changing quantities are tied together by an equation, their RATES are tied together too.
4.5 Solving related-rates problems
Model the relationship, differentiate with respect to time, THEN substitute the values true at that instant, and solve for the unknown rate — reporting units.
4.6 Local linearity and linearization
Zoomed in near a point, a smooth curve looks like its tangent line.
4.7 L'Hôpital's Rule for indeterminate forms
When a limit gives the indeterminate form \tfrac{0}{0} or \tfrac{\infty}{\infty}, compare how fast the numerator and denominator change — replace each by its derivative.