AP CALCULUS

8.1-8.3 Review: Functions and Max/Min Problems

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/minChanges in concavity may occur when the second

derivative f ’’(x) is 0 or undefined.Function is concave up when f ’’ is > 0Concave down when f ’’ < 0The 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.

Vert. Asymptotes: x = -1/2,x = 4

Horizontal Asymptotes

Horizontal Asymptotes: Value y approaches as x approaches infinity.

So horizontal asymptote occurs at y = 5/3

= 0, so asymp. is y = 0.

Know how to:

Find derivatives of functions such as 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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