You must show your work in the space provided on this test Elgar. Partial credit will be given for any relevant, correct work. 1. (4 points) Find the points of intersection of y = $2 - 10 and y = x + 2. IO \\55 5 {Mo = Hz \\ :2 x 12 3. (4 points) Find the limit (if it exists) for 115.5.1 x _ 4 The graph of f, defined from [-5, 4], is given below. Use this for questions 4 and 5. Z values 4. (4 points) State the r values for which f is not differentiable, justify your choices. to I value 5. (4 points) State the r values for which f is not continuous, justify your choices. to 6. (5 points) Using the limit definition of the derivative of f, find f'(x) for f(x) = 2x + 1 f' ( x ) = f ( x +BX ) - fexs AX-20 toFor questions 7, 8, and 9 use the function f(z) z3 +1)2 7. (5 points) Find f'(x), using the appropriate derivative formulas. f ( x ) = 1 + ( x 3 + 1 ) 2 Is [ x 3 ] ' + f ( x) = #4 + ( 3 x 2 2 + 1 ) 2 f ( x ) = 1XA ( 6 * 2 + ()2 ((x) = 1x + 6*8 + 1) c( x ) = 1x+ 49 f( x 1 = 49 x fix ) at x = 1 + y 8. (2 points) Find the instantaneous rate of change of f at point (1, 5). lim AY lim - A* -20 * ( x + AX ) - f(Al = .' ( x ) Ax f ( 1 + 4 x ) >*# + ( 1) =8 ' ( x 1 2 A X 1/o or Vnegative 9. (2 points) Give the domain for f(x) and f'(x) in interval notation. f (* ) = 1 "+ ( x 3 + 1 ) 2 ( (x ) = 1 1210. (5 points) A company that manufactures fancy pet toys calculates that its cost and revenue can be modeled by C= 100 + , and R = 20x - " where x is the number of toys produced in 1 week. Production during one particular week is 5 toys and is increasing at a rate of 3 toys per week. Find the rate at which the revenue is changing this gelb ) - f ( a) 2 Ay [aib] b - a Ax finding only f ( 2x ) - f ( 2 x ) f (x) -f ( 2x ) average rate of change 100 - 20* 80 X were looking 2 for inst. rate of clause gog g 80 ~ 8 I 11. (4 points) Consider the function f(x) = 2 - 6x, its graph is given below: 101 X -10 -5 5 10 -10 Determine the open intervals on which f is increasing and decreasing. f ( x ) x 3 - 6 x f( x1 = 3 x" - 6* could use Inc. / dec test So find f ( x ) = 0 or f ( x) ONE12. (5 points) Find the critical numbers for h(x) = - 12 + 36 fex to first find h (x ), use quotient rule . X then *2+ 6 (is h' (x= o 01 6 ( ic ) hi (x ) ONE For questions 13, 14, and 15 suppose that f(I) = -13 + 3x2 -7, 13. (4 points) Determine the open intervals on which f is concave up and concave down. fix)so to run f ( x ) = -x3 + 3x2 - 7 on 4 - fl x) 2 - 3x 2 + 3 x 2 - 17 Signs ( ( x ) = fx+ 6x-$ look at (( X ) = (-6 , 6 Sec. 3. 2 and 3.3 14. (2 points) Does / have any inflection points? If so, state them below. for NO because the anumber is undefined # 13 , 14 , and 15 15. (4 points) Find any relative minimums or maximums of f. Hint: You could use either the first or second derivative test - ( x) = -* + 3x2 -7 min z - 3x7+ 3x2 BANKSa use product rule (A) ( 4 Extra Credit points) Find y' for y = 2xvx3 - 1 ( fg ) = fig + fg' to 4:21 110 13 -1 4 72 J 1 - 1 Y = O use implicit differentiation (B) ( 4Extra Credit points) Find y', dy/di, for Ary = 23 + 2 Sec- 2.7 4 xy 2 x 3 Ay 2 to 2x (Hay ) x 3 + 1 2 d. Y 28 ( 4 ) 28 1 2 x + 8 . 25