MAC3472StudyGuideTest3

MAC3472StudyGuideTest3 - Study Sheet for Test 23 Honors...

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Study Sheet for Test 23 Honors Calculus Keesling Test on 10/26/07 1. Suppose that F ( x ) and G ( x ) are differentiable functions on the interval a , b [ ] . Suppose that ! F ( x ) " ! G ( x ) on the interval a , b [ ] . Show that there is a constant C such that F ( x ) ! G ( x ) + C . 2. Consider the function f ( x ) = x 3 on the interval 1,5 [ ] . Give a formula for all the anti-derivatives of f on 1,5 [ ] 3. Let f ( x ) be a continuous function on the interval [ a , b ] . Define F ( x ) = f ( t ) d t a x ! . Show that ! F ( x ) " f ( x ) on [ a , b ] . 4. State and prove the Fundamental Theorem of Calculus. 5. Evaluate: (a) sin( x ) dx 0 ! " (b) sin( x )cos( x ) dx 0 ! " (c) sin 3 ( x ) dx 0 ! " (d) ( x 3 + 1) 5 x 2 dx 0 1 ! (e) x 2 ( x 3 + 1) 5 dx 0 1 ! 6. Evaluate: (a) cos 5 a b ! ( x ) dx (b) 2 x 3 ( x 2 + 1)( x 2 ! 1) a b " dx (c) x x 2 ! 5 a b " dx (d) tan 3 ( x )sec 2 ( x ) dx a b ! 7. Evaluate: (a) 1 x 2 ! 1 " dx (b) 1 x 3 ! 1 a b " (c) x + 1 3 x 2 + 2 a b ! dx (d) 2 x 2 ( x 2 + 1)( x 2 ! 1) a b " dx (e) 1 1 + x 2 dx a b ! (f) 1 (1 + x 2 ) 3 ! dx 8. Evaluate: (a) x sin x ! dx (b) e x cos x a b ! dx (c) sin 2 x a b ! dx (d) tan x a b ! dx (e) sin 4 x dx a b ! (f) tan 2 x ! dx 9. Determine the area under the graphs of the following functions over the given intervals using upper or lower sums.

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