8.1-8.3 Review: Functions and Max/Min Problems

A P C A L C U L U S

Analyzing Functions

       Critical Values: x coordinates of points at which derivative of f is 0 or undefined f(x) reaches relative max/min values when derivative is 0 or undefined (horizontal tangent/cusp) *** f ‘(x) must change sign for rel max/min Changes in concavity may occur when the second derivative f ’’(x) is 0 or undefined.

Function is concave up when f ’’ is > 0 Concave down when f ’’ < 0 The point of inflection occurs where the graph changes concavity.

Analyzing Functions

 Max/Min VALUE of a function: Y value of function.

 Absolute min/max: Highest or lowest value of function on an interval. Can take place where the derivative is undefined or 0, OR AT INTERVAL

ENDPOINTS!!!

Second Derivative Test

  At a point x, if f ‘(x) = 0 (possible rel. min or max – critical point) and f “(x) < 0 (concave down), f reaches a relative MAXIMUM at x.

If f ‘(x) = 0 and f “(x) > 0 (concave up), f reaches a relative MINIMUM at x.

VERTICAL ASYMPTOTES

 Vertical Asymptotes: Occur when denominator of function equals 0. Typically can factor or use the quadratic formula to determine.

(𝒙−𝟔)(𝒙+𝟕)  (𝟐𝒙+𝟏)(𝒙−𝟒) Vert. Asymptotes: x = -1/2, x = 4

Horizontal Asymptotes

 Horizontal Asymptotes: Value y approaches as x approaches infinity.

𝐸𝑥: 𝑓 𝑥 = 5𝑥 2 −4𝑥+2 3𝑥 2 −𝑥+9 lim 𝑥→∞ 𝑓 𝑥 = 5/3 So horizontal asymptote occurs at y = 5/3 4𝑥 2 −8𝑥+1 𝐸𝑥: lim 𝑥→∞ 5𝑥 3 −2𝑥 2 −7𝑥+4 = 0, so asymp. is y = 0.

Know how to:     Find derivatives of functions such as 𝑦 = 2𝑥 4 3 − 6𝑥 1 3 and factor the result to find solutions when f ‘(x) = 0.

Draw number lines illustrating f ‘(x) and f “(x) (to show intervals where graphs increase/decrease or are concave up/down. Use chart to identify graph features such as rel. min/max and points of inflection.

Draw a sketch of f(x) given f ‘(x) Sketch f(x) given number lines for f ‘(x) and f “(x)

MAX/MIN PROBLEMS

    Write equation of function to maximize or minimize. Typical examples are area, volume, distance, Pythagorean Theorem Be aware of any limitations. Often, a restriction function allows original function to be re-written using one variable.

Make sure function is written using one variable – max/min values occur when f ‘(x) = 0 (or possibly at interval endpoints).

Be careful! Draw/label diagrams!!!

Drawing f(x) given f ‘(x)

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Drawing f(x) given f ‘(x)

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Drawing f(x) given f ‘(x)

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