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Review for Test 3
Glendale Community College
Phong Chau
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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
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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.
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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
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Implicit Differentiation
1. Differentiate both sides of an equation
2. Find an equation of a tangent line
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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
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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
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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
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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
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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.