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StudySheetTest2_10

# StudySheetTest2_10 - Study Sheet for Test 2 Honors Calculus...

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Study Sheet for Test 2 Honors Calculus Keesling Test on 10/27/10 1. Suppose that f is an increasing function on the interval [ a , b ] . Subdivide [ a , b ] into n equal subintervals. Let S n ( f , a , b ) be the upper sum and let S n ( f , a , b ) be the lower sum. Show that S n ( f , a , b ) ! S n ( f , a , b ) = ( f ( b ) ! f ( a )) " ( b ! a ) n . Use this to show that lim n !" S n ( f , a , b ) # S n ( f , a , b ) \$ % & ' = 0 . 2. Suppose that F ( x ) and G ( x ) are differentiable functions such that ! F ( x ) " ! G ( x ) on an interval. Show that there is a constant C such that F ( x ) ! G ( x ) + C . 3. Suppose that f ( x ) is a continuous function and that F ( x ) = f ( t ) dt a x ! . Show that ! F ( x ) " f ( x ) . 4. Suppose that f ( x ) is a continuous function and that G ( x ) is a differentiable function having the property that ! G ( x ) " f ( x ) on the interval [ a , b ] . Show that f ( x ) dx a b ! = G ( b ) " G ( a ) . 5. Determine the following integrals. Be able to use the following methods and combinations of them to determine the integrals: Substitution or Integration by Parts. (a) tan x dx ! (b) sin 2 x dx ! (c) cos 6 x dx !

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