Unit 4 Summary (College Board AP® Calculus AB): Study Guide

Contextual applications of differentiation summary

Key facts & definitions

• $\frac{dy}{dx}$ 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

  • $\frac{dy}{dx}=\frac{dy}{du}·\frac{du}{dx}$

• L'Hospital's rule states that

  • $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\lim }\frac{f^{\text{'}}\left(x\right)}{g^{\text{'}}\left(x\right)}$

  • Provided one of the following is true

    • $\underset{x\to a}{\lim }f\left(x\right)=0$ and $\underset{x\to a}{\lim }g\left(x\right)=0$

    • $\underset{x\to a}{\lim }f\left(x\right)\to \pm \infty$ and $\underset{x\to a}{\lim }g\left(x\right)\to \pm \infty$

Key formulas

• Let $x\left(t\right)$, $v\left(t\right)$ and $a\left(t\right)$ represent the position, velocity, and acceleration of an object at time $t$

  • $v\left(t\right)=x\text{'}\left(t\right)$

  • $a\left(t\right)=v\text{'}\left(t\right)=x\text{'}\text{'}\left(t\right)$

• The linear function that approximates values of $f\left(x\right)$ close to $x=a$ is

  • $L\left(x\right)=f\left(a\right)+f\text{'}\left(a\right)\left(x-a\right)$