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Name Student ID # Class Section Instructor Math 1210 Spring 2007 EXAM Dept. Use Only Exam Scores Problem Points Score 1. 20 2. 20 3. 20 4. 20 5. 20 TOTAL Show all your work and make sure you justify all your answers. Math 1210 Exam 1. Integrate the following problems R ( ( )) ( ) (a) f 0 g x g 0 x dx (b) R xcos x (c) R cos x sin sin x (d) R 2 3 x x (e) R x ( ) ( ) 2 dx ( [ + 5] 3 8 + +4 x ( )) dx (( + 5) ) sin 3 x 9 dx dx 1 2. These problems involve the second fundamental theorem of calculus. R (a) 1 0 xdx (b) R (c) R (2 0 1 0 ( )[ cos x x ( )] sin x 2 dx + 2)(2 + 4 + 99) 2 x x 95 2 dx 3. These problems are on the rst fundamental theorem of calculus. (a) Find dG dx Where ( ) = R (b) Find dG dx Where ( ) = R (c) Find dG dx Where ( ) = R G x G x G x x2 sin(x) cos ( ) (x2 +x+1)2 x x dx 2 x dx x2 cos(x2 ) tan 3 ( ) x dx 4. Estimate the area under the curve = from = 0 to = 1 with four rectangles. It may help to draw the picture. P 5. nd the exact number that the following sum equals Nk (1 5)k 6. These problems are on dierential equations. Either solve the separable dierential equation or show that is a solution to the dierential equation. (a) Show that = ( )+ ( ) is a solution to the dierential equation + =0 y 3 x x x =0 y y y=sp (b) Solve dy dx = yx3 (c) Solve dy dx = k1 cos x 00 k2 sin x y 2 3 x y 4 = 7. These problems are on the mean value thoerem for derivatives. (a) State and prove the mean value theorem for derivatives. (It is less likely that this question will be on the exam) (b) Let ( ) = nd all values c so that the statement of the mean value theorem for derivatives holds on [ 1 1] f x 3 x ; 5 8. This question is on section 3.4 and will come from your homework. In particular 14, 23, 29,50. 9. Find the maxium and minium value of each of the following functions. Us the rst or second derivative test to show it is a max or a min. (a) ( ) = x2x2 f x +1 (b) ( ) = f x 3 x 3 (c) ( ) = 4 x 2 (d) ( ) = 2 x f x f x x x + 4 + 4 on [ 10 10] x ; 6