4.4 Class Notes

# 4.4 Class Notes - Part 1 Derivatives Derivatives product...

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Part 1: Derivatives Derivatives: product rule, quotient rule, chain rule. Partial derivatives Equation of tangent line. Exponential growth Maxima, minima, convexity, concavity, first and second derivative tests Max-min word problems Part 2: Integration Indefinite integrals: substitution, integration by parts Definite integrals Volume of solid of revolution Mean and variance Formula sheet on web page has details. A by now familiar problem (a) Find all the critical points (x values), (b) Use the second derivative test to determine if they are local max or min Find the open intervals on which the function is (c) increasing, (d) convex f(x) = 2 x 3/2 - 9 x + 12 x 1/2 x > 0 f(x) = 2 x 3/2 - 9 x + 12 x 1/2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 3/ 2 '( ) 3 9 6 (3 9 6) (3 3)( 2) so 1,4 ''( ) (3/ 2) 3 ''(1) (3/ 2) 3 3/ 2 0 maximum ''(4) (3/ 4) (3/8) 0 minimum increasing 0 1 and f x x x x x x x x x x f x x x f f x = + = + = = = = = − < = > < < 1/ 2 3/ 2 4 (3/ 2) 3 0 (convex) when 2 x x x x > > > Why do we learn calculus? (finale) You (or your wife) just gave birth to a baby.

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