CURVE SKETCHING

section 3-A

• Where the derivative is zero or the function does not exist.

Critical Values

• Use Extreme Value Theorem if on a closed interval [a,b] (f(x) has both a min and a max on the interval)

• Otherwise use the a) First derivative testb) Second derivative test

Extrema: Maxima and Minima

1) find all extrema on the interval [0,4]

xxxf 12)( 3

a) graphically

1) cont find all extrema on the interval [0,4]

xxxf 12)( 3

b) algebraically

2) Find the absolute extrema and the critical values

for on [-1,2] 32

)( xxf

a) graphically

2) Cont. Find the absolute extrema and the x-values of the critical numbers

for on [-1,2] 32

)( xxf

b) algebraically

3) Find the extrema forand determine the intervals where

increasing and decreasing

52

1)( xxxf

Analyzing the graph of a function a) Domain and Range: All real numbers except ___b) Extrema and the intervals where increasing and

decreasing (first derivative test)c) Intercepts: where the graph crosses the x-axis

and the y-axis d) Inflection points and the intervals where concave

up and concave down (second derivative test)e) Symmetry

1. About the y-axis if even function2. About the origin if odd function

)()( xfxf )()( xfxf

Find all critical values f) Asymptotes- rational functions• Vertical: set the denominator equal to zero and verify

the limit tends to infinity

• Horizontal: Take the limit of the function as x approaches ±∞

• Slant: occur when the degree of the numerator is one higher than the degree of the denominator. Use long division or synthetic division to find the line

g) Graph- put it all together

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