21bpracticemt3solns

21bpracticemt3solns - MAT 21B Practice Midterm III 1 1)...

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Unformatted text preview: MAT 21B Practice Midterm III 1 1) (True/False) Label the following statements as either true or false. (No explanation required. Each correct label is worth two points.) a: ln(2 + 3) = ln 2 + ln 3 False. ln 2 + ln 3 = ln(2 3) = ln 6 negationslash = ln5 = ln 2 + ln 3 b: e 6 = ( e 3 ) 2 True. ( e 3 ) 2 = ( e e e ) 2 = ( e e e )( e e e ) = e 6 c: e ln( x y ) = e ln x e ln y False. Consider x = 1, y = 2. Then e ln x e ln y = e ln 1 e ln 2 = 1. But e ln(1 2) = e ln( 1) is not defined. It is, however, the case that if x > y , then e ln x y = e ln x e ln y = x y . d: integraldisplay (sec x + tan x ) 2 dx = (sec x + tan x ) 3 3 + C False. One can check that the derivative of the right hand side is (sec x + tan x ) 2 (sec x tan x +sec 2 x ), which is not the integrand on the left hand side. e: sin( m n ) = sin m + cos n tan x False. The left hand side does not involve x at all. 2) Find integraltext z +1 z 2 ( z 1 dz . We apply the method of partial fractions to write z + 1 z 2 ( z 1) = A z 1 + B z + C z 2 . 2 Finding a common denominator on the right hand side gives z + 1 z 2 ( z 1) = Az 2 + Bz ( z 1) + C ( z 1) z 2 ( z 1) = ( A + B ) z 2 + ( B + C ) z + ( C ) z 2 ( z 1) ....
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This note was uploaded on 04/16/2008 for the course MATH 21B taught by Professor Vershynin during the Spring '08 term at UC Davis.

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21bpracticemt3solns - MAT 21B Practice Midterm III 1 1)...

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