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Name Student ID # Class Section Instructor Math 1210 Spring 2007 EXAM 1 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. these problems involve derivatives (a) State the product rule. ( ) ( + 1) (b) Dierentiate ( ) = cos x (c) Dierentiate ( ) = cos x f x f x 2 x ( ) ( + 1) 2 x ( ) sin x (d) State the chain rule. (e) Dierentiate ( ) = f x (( + 1) cos 2 x ( )) sin x (f) Dierentiate ( ) = sin(cos(sin ( )) ( + 1) sin( )) 2 f x x (g) State the quotient rule. (h) Dierentiate ( ) = f x x 2 ) cos(sin(x2 +1) cos(x) 1 2 x x 2. These problems involve implicit derivatives. Find equations. (a) tan(cos(sin( ) ) ( + 1) + = 0 x y (b) ( ) + xy (c) 3 x 5 3 y 4 4 x ( ) + ( + 1) = 0 sec xy + + y 3 2 x y + x y 2 y x 6 5 =( + ) 2 x y 2 2 3 dy dx for the following 3. This question will be on related rates and the test problem will be one the homework problems which I assigned. Also note that we discussed the solutions to all of these problems in class. 4. This problem will be on dierentials and approximations. Dene what a dierential is, Study example 3 on page 144, problem 26, 23,17. The test problem will be extreemly similar to one of two of these problems. 5. Find the maxium and minium value of each of the following functions if they exist. If they exist you must state why it exists and if it does not exist you must state why it does not exist. (a) ( ) = x2x2 x on [ 1 1] + 2 +1 f x ; (b) ( ) = x on [0 1] f x 2 4 (c) ( ) = cos x (d) ( ) = 2 x f x f x ; ( ) on [ 2 1] ; + 4 + 4 on [ 10 10] x ; (e) ( ) = x on [ 1 1] f x 1 ; 3 6. Find where the functions are increasing, decreasing, concave up and concave down. (a) ( ) = + 3 12 f x (b) ( ) = f x 3 x x 2 3 x + 12 x 4 df df exists and is continuous on the interval I, and if dx is 7. Prove that if dx nonzero for all points in I then either ( ) is increasing or decreasing on I. (Hint what does the intermediate value theorem say?) x f x 5