AP®MathsCollege BoardCalculus ABRevision NotesUnit 4: Contextual Applications of DifferentiationUnit 4 OverviewUnit 4 Summary

Unit 4 Summary (College Board AP® Calculus AB): Revision Note

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 6 May 2026

Contextual applications of differentiation summary

Key facts & definitions

$\frac{d y}{d x}$ is the rate at which $y$ is changing per unit increase in $x$
Displacement is the position relative to a fixed point
Velocity is the rate of change of displacement
Speed is the absolute value of velocity
- An object is speeding up if its velocity and acceleration have the same sign
- An object is slowing down if its velocity and acceleration have different signs
Acceleration is the rate of change of velocity
A linear approximation gives:
- an overestimate if $f$ is concave down at the point of tangency
- an underestimate if $f$ is concave up at the point of tangency
An indeterminate form is an expression that does not tell you the value of a limit
- $\frac{0}{0}$ and $\frac{\infty}{\infty}$ are indeterminate forms

Key theorems

The rates of change between the variables $x$ , $y$ and $u$ are related by the formula
- $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
L'Hospital's rule states that
- $\lim_{x \to a} \frac{f (x)}{g (x)} = \lim_{x \to a} \frac{f^{'} (x)}{g^{'} (x)}$
- Provided one of the following is true
  - $\lim_{x \to a} f (x) = 0$ and $\lim_{x \to a} g (x) = 0$
  - $\lim_{x \to a} f (x) \to \pm \infty$ and $\lim_{x \to a} g (x) \to \pm \infty$

Key formulas

Let $x (t)$ , $v (t)$ and $a (t)$ represent the position, velocity, and acceleration of an object at time $t$
- $v (t) = x' (t)$
- $a (t) = v' (t) = x'' (t)$
The linear function that approximates values of $f (x)$ close to $x = a$ is
- $L (x) = f (a) + f' (a) (x - a)$

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