SS_Mat135_T1

# Dy x 2 y 2 x y dx optimization problems step 1

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Unformatted text preview: y = ln (3 + x ) concave up? d2y SOLUTION: POWER RULE: EXAMPLE: SOLUTION: properties of logarithms. Step 3. Differentiate both sides with respect to x y ′ = sec 2 sec e sin ( x ) ⋅ sec e sin ( x ) tan e sin ( x ) ⋅ (( www.prep101.com dy dy + 2y ⋅ =0 dx dx dy (x + 2 y ) = −2 x − y dx dy 2x + y . =− dx x + 2y LOGARITHMIC DIFFERENTIATION: to differentiate a function with complicated exponent/a product of several functions, etc. It’s also used to differentiate functions that have an x in both the base and in the exponent. f(x) is concave down on that interval. A point of inflection occurs when f ( p ) changes sign (and thus concavity). Vertical asymptote: The line x = a is a vertical asymptote for the graph of the function f(x) if and only if lim f ( x ) = ±∞ or x→a + lim f ( x) = ±∞ . x→a − Horizontal asymptote: The line x = b is a horizontal asymptote for the graph of the function f(x) if and only if lim f ( x) = b or...
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## This note was uploaded on 03/15/2010 for the course MAT MAT135 taught by Professor Treung during the Fall '08 term at University of Toronto- Toronto.

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