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8.1-8.3 Review: Functions and Max/Min Problems. AP Calculus. 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) - PowerPoint PPT Presentation
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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)
ANSWER
Drawing f(x) given f ‘(x)
ANSWER
Drawing f(x) given f ‘(x)
ANSWER