Review for Test 3

Glendale Community College

Phong Chau

Graphs of Functions Find the interval of

increase/decrease. f ’(x) > 0 , f(x) is increasing f ’(x) < 0 , f(x) is decreasing

Find the relative extrema. Use test values

Graphs of Functions Find the interval of concavity.

f ’’(x) > 0 , f(x) is concave up. + +

f ’’(x) < 0 , f(x) is concave down. – –

Find the inflection points. Inflection point occurs at the point where the

second derivative changes sign. Sketch the graph showing all important

features of the curve.

Absolute ExtremaFind absolute maximum/minimum of f(x) on a closed interval [a, b]1. Find the critical numbers of f(x)

2. Evaluating the functions at critical numbers and at the end points a and b.

3. The largest values in step 2 is the absolute maximum, and the smallest value is the absolute minimum

Implicit Differentiation

1. Differentiate both sides of an equation

2. Find an equation of a tangent line

Application Problems

1. Optimization Problem (14.2)

2. Related Rates Problem (14.5)

3. Elasticity of Demand (14.3)

'( )( )

( )

p f pE p

f p

Examples Find the interval of

increase/decrease.

relative extrema, intervals of concavity, and inflection points

3 22 9( ) 5 1

3 2f x x x x

4 2( ) 2 6g x x x 1/3 4/3( ) 4 2h x x x

Examples Find the absolute extrema for

The cost (in thousands) function of an auto parts is given by

Find the minimum value of the average cost.

2 0.5( ) , [2,5]xf x x e

2( ) 81 17 324 , 1 10C x x x x

Examples The demand for beer is given by the

demand function

Find the elasticity of demand function Find and interpret the elasticity of demand

at p = $6.

23q

p

Examples A fence must be build to enclose a

rectangular area of 20,000 sq. ft. Fencing material costs $2.50 per foot for the two sides facing north and south, and $3.20 per foot for the other two sides. Find the cost of the least expensive fence.