Curve Sketching and Optimization

For multivariable functions, see surface sketching.

Abstract

  1. Find x-intercepts
  2. Find y-intercepts
  3. Perform the first derivative test
    • Find the local max, min
    • Find the intervals where the function is increasing/decreasing
  4. Perform the second derivative test if necessary
    • Find POIs
    • Find concave up/down

For rational functions:

Increasing, Decreasing

Definition

  • is increasing on an interval if whenever
  • is decreasing whenever

Monotonicity

Definition

If is increasing or decreasing on , then is monotonic on . That is, it stays increasing or decreasing throughout the entire interval.

More formally, given the function and variables and , if , then .

Critical Values

Definition

A critical value occurs if:

  1. A value of is excluded from the domain of (e.g VA, hole)
  2. A value of is excluded from the domain of (the function is not differentiable) at
  3. Values of where
Important

All extrema inside the domain occur at critical points (not all critical points are extrema, but all extrema are located critical points).

Extrema

Definition - Absolute Extrema

  • A function has an global (absolute) maximum at if for all in the domain of
  • A function has an global (absolute) minimum at if for all in the domain of

Definition - Local Extrema

  • A function has a local (relative) maximum at if there exists a number such that for all
  • A function has a local (relative) minimum at if there exists a number such that for all

First Derivative Test

If changes from positive to negative at , then has a local maximum at

If changes from negative to positive at , then has a local minimum at

If does not change sign at , then has no max or min at

Concavity

A graph is concave up on an interval if it lies above all of its tangents

A graph it concave down on an interval if it lies below all of its tangents

Second Derivative Test

Definition

  1. Determine the critical numbers where
  2. If , then has a local minimum at (concave up)
    if , then has a local maximum at (concave down)

Concave up means SMILING, so is positive
- Walker

Important

The second derivative test does not apply when or does not exist

Point of Inflection

Definition

A POI is a point on which the graph changes concavity

A point of inflection occurs at if: or does not exist.

Locating POIs

  1. Determine and state values of where is not defined
  2. Let and solve for
  3. If changes sign at , then has a POI as