Unit 5 Summary (College Board AP® Calculus AB): Revision Note

Analytical applications of differentiation summary

Key definitions

• An extremum (plural extrema) is a maximum or minimum point of a function

  • A global extremum is the max/min over the function's whole domain

  • A local (relative) extremum is the max/min over some open interval around the point

• If $f\text{'}\left(x\right)>0$ at a point, then $f$ is increasing at that point

• If $f\text{'}\left(x\right)<0$ at a point, then $f$ is decreasing at that point

• A critical point of a function $f$ is a point where $f\text{'}\left(x\right)=0$ or $f\text{'}\left(x\right)$ does not exist (provided $f\left(x\right)$ is defined there)

• If $f\text{'}\text{'}\left(x\right)>0$ at a point, then $f$ is concave up at that point

• If $f\text{'}\text{'}\left(x\right)<0$ at a point, then $f$ is concave down at that point

• A point of inflection is a point where the concavity of the graph changes

Key theorems

• The mean value theorem states that if $f$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists a $c$ in $\left(a,b\right)$ such that

  • $f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

• Rolle's theorem is the special case of MVT when $f\left(a\right)=f\left(b\right)$, guaranteeing a $c$ with $f\text{'}\left(c\right)=0$

• The extreme value theorem states that if $f$ is continuous on $\left[a,b\right]$, then $f$ has at least one global maximum and one global minimum on $\left[a,b\right]$

Key facts

• The first derivative test classifies a critical point $x=a$ (where $f\text{'}\left(a\right)=0$) by checking the sign of $f\text{'}\left(x\right)$ on either side:

  • positive → negative: local maximum

  • negative → positive: local minimum

  • no sign change: point of inflection

• The second derivative test classifies a critical point $x=a$ (where $f\text{'}\left(a\right)=0$) by checking the sign of $f\text{'}\text{'}\left(a\right)$

  • $f\text{'}\text{'}\left(a\right)>0$ → local minimum

  • $f\text{'}\text{'}\left(a\right)<0$ → local maximum

  • $f\text{'}\text{'}\left(a\right)=0$ → test inconclusive (use first derivative test)

• The candidates test finds global extrema on a closed interval $\left[a,b\right]$ by comparing the values of $f\left(x\right)$ at all critical points and at the endpoints

  • The largest is the global max

  • The smallest is the global min

• All local extrema occur at critical points, but not all critical points are local extrema (some are points of inflection)

• For an implicitly-defined curve, the tangent is

  • horizontal where $\frac{dy}{dx}=0$

  • vertical where $\frac{dx}{dy}=0$