12-03: Curve Sketching

Maxima & Minima

  • ==Local Maximum==: if the y coordinate of all points are less than the y
    • For a ==local== interval, it is the highest point, but it ==isn’t the highest point for the whole graph==
  • ==Local minimum==: if the y coordinate for all points in the vicinity are greater than the y coordinate of the point
    • For a ==local== interval, it is the lowest point, but it ==isn’t the lowest point for the whole graph==
  • ==Local extrema==: local maximum and minimum values of a function, also called ==turning points==
  • ==Absolute maximum==: the ==highest y coordinate== on the function
    • As high as the function goes overall
  • Absolute minimum: the ==lowest y coordinate== on the function
    • As low as the function goes overall

Critical Numbers

Critical numbers: value a in the domain of the function for which either ^^f’(a) = 0 or f’(a) = DNE^^

  • Critical numbers are ^^x values^^, to find a point, substitute the x values in and solve for y
    • Critical points: ^^(a, f(a))^^

 

First Derivative Test

The first derivative test: %%indicates whether a critical number yields a local maximum, a local minimum, or neither%%

  • If f’(x) changes from %%positive to negative%%: Local max
  • If f’(x) changes from %%negative to positive%%: Local min
  • If the sign of f’(x) does not change: Not a local max or min, we have a horizontal tangent
    • Local extrema occur when the sign of the tangent changes
To determine critical numbers (x-values) of the critical points:
  • Find an expression for f’(x)
  • Solve for the roots of the derivative and find where the derivative equals 0

==Steps:==

  1. Find f’(x)
  2. Set to 0 and solve for x
  3. Use critical numbers as x values, sub in for y coordinate

   

  1. Use these to graph
  2. If there is a mention of endpoints, use these as x values and solve for a coordinate point (y value) as well

Increasing and Decreasing Functions

Afunctionincreasesoveranintervalif:itrisesfromlefttorightA function increases over an interval if: it rises from left to right

Afunctiondecreasesoveraninterval:itfallsfromrighttoleftA function decreases over an interval: it falls from right to left

The first derivative of a continuous function, f(x), can be used to determine the intervals over which the function is increasing and decreasing

  • ^^When f’(x) is greater than 0, positive and above the x axis: the function is increasing^^
  • ^^When f’(x) is less than 0, negative and below the x axis: the function is decreasing^^
    • Critical numbers occur when f’(x) = 0 and local extrema occur when the sign of the derivative changes

==Steps== - to solve for increasing and decreasing intervals:

  1. Determine the first derivative
  2. Solve for the roots of the derivative (factor if needed)
  3. Solve for where the derivative of x is greater than 0 (increasing), or where the derivative of x is less than 0 (decreasing)

   

  1. Use an interval table to do this

       1. Set up (-∞, #) (#, ∞) with as many columns are needed to touch upon all x intercepts; zeros separate the columns 2. Set up all factors in the rows of the chart 3. Pick a number in the boundaries of the column headers (a number) - substitute this into the rows and record the sign overall

          1. From here, multiply the signs (whether they be positive or negative) and see what you get as a final 4. The final verdict of sign (positive or negative) reflects whether the function is increasing (+) or decreasing (-) at a given time

Concavity and the Second Derivative Test

%%Concave up%%: all tangents on the interval are below the curve (slope increasing)

  • Graph curves upwards (like a parabola opening up)

%%Concave down%%: all tangents on the interval are above the curve (slope decreasing)

  • Graph curves downward (like a parabola opening down)

%%Point of inflection%%: the point at which the graph changes concavity

  • Changes from concave up to concave down

   

The second derivative, f”(x), of a function can be used to determine its concavity

  • If f”(x) is ^^greater than 0: Concave up^^
  • If f”(x) is ^^less than 0: Concave down^^
  • If ^^f”(x) = 0: Possible point of inflection (f”(x) must change signs over the zero)^^

The Second Derivative Test

Evaluate f”(x) at the critical numbers (where f’(x) = 0)

 

  • First derivative test finds critical points and whether they’re local max/min/neither. Second derivative test helps classify critical points but also can identify points of inflection

Steps to solve second derivative test questions:

  1. Find first derivative
  2. Find second derivative
  3. Interval table
  4. Sub x values in for y values to get points for the points of inflection

 

Simple Rational Functions

Rational function: %%a function in the form y=(f(x))/(g(x))%%

  • Restriction on the denominator where ^^g(x) ≠ 0^^ because then it would be undefined
Summary of properties & characteristics:

Asymptotes

  • %%Vertical asymptote%%, VA: zeros of the denominator, solve the denominator
  • %%Horizontal asymptote,%% HA:
    • \     
    1. If the degree of the numerator is less than the degree of the denominator → y=0
    • \     
    1. if the degree of the numerator is equal to the degree of the denominator → y = LC/LC
  • Oblique asymptote, OA: when the degree of the numerator is greater than the degree of the denominator
    • OA y = the quotient of the numerator and denominator

Intercepts

  • %%x int%%: sub y=0, zeros of the numerator (solve the numerator)
  • %%y int%%: sub x = 0

Holes

  • Hole (open point): if a factor cancels out

◊ Rational functions can change from increasing to decreasing (or vice versa) across a VA, the concavity can also change as well. CA must be considered and included in intervals of increase/decrease or concavity on account of this.

Putting It All Together

Steps to sketching the graphs of polynomial and rational functions

  1. Determine the domain of the function

   

  1. Is it a rational function, are there asymptotes, limit behaviour
  2. VA: lim x→VA- ; lim x→VA+
  3. HA: lim x→±∞
    1. Determine intercepts of the function

   

  1. x int: sub y=0
  2. y int: sub x=0
    1. Determine and classify the critical numbers of the function

   

  1. Solve for where f’(x) = 0 or where f’(x)=DNE
    1. Determine possible points of inflection

   

  1. Solve for where f”(x) = 0
    1. Set up intervals of increase/decrease and intervals of concavity & the points of inflection

   

  1. Set up interval tables
    1. Identify local extrema and points of inflection

   

  1. Look for where f’(x) and f”(x) change signs
    1. Sketch the function