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

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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

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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

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Increasing and Decreasing Functions

\ $A function increases over an interval if: it rises from left to right$

$A function decreases over an interval: it falls from right to left$

\ 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

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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)

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

• Changes from concave up to concave down

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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)^^

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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

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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

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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   * \          2. 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.

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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→±∞

2. Determine intercepts of the function

       1. x int: sub y=0    2. y int: sub x=0

3. Determine and classify the critical numbers of the function

       1. Solve for where f’(x) = 0 or where f’(x)=DNE

4. Determine possible points of inflection

       1. Solve for where f”(x) = 0

5. Set up intervals of increase/decrease and intervals of concavity & the points of inflection

       1. Set up interval tables

6. Identify local extrema and points of inflection

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

7. Sketch the function

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