Calculus :

3-7

3, $5 point) The graph of the derivative of a function g is shown below. Which of the 5' ('5 point) Consider the function f(x) : x3' Since f(5) m 0.076 and f(6) m 0.302, ollowing is true about 97 and at) is continuous everywhere, we know by the Intermediate Value Theorem, this function must have a zero between ac = 5 and z = 6, Use Newton's Method to nd :52, 12 ' ' ' the second iteration of the approximation of that zero starting with $1 = 5, (Round to two decimal places) 10 (a) 1:1 m 5.17 5 . . . (b) 361 z 5.18 (c) 051% 5.19 (d) 112: 5.20 g (e) 051% 5.21 6. (.5 point) Which is the minimum surface area of a rectangular box with an open top, whose base is a square, and whose volume is 108 cm3 (a) g is increasing on the interval (00, 0) (b) g is decreasing on the interval (0, 3) (c) g has a turning point at x : 3 (d) .'I has a turning point at I = '2 (a) 6cm2 (b) 18cm2 (c) 36cm2 (d) 720m2 (e) 108cm2 (e) None of these is true about g. X 7. (.5 point) Consider the function f (x) : 3x2 3. This function is continuous and differentiable everywhere, Therefore, the Mean Value Theorem applies on the interval 4' ('5 point) Suppose the second derivative of the function f is given by the rule: [2, 5]. Find the value c guaranteed by the Mean Value Theorem on the interval [2, 5]. = 3.5 f"(x) : 3x2(x 7m + 2)3 E2; :_ 4 5 Which of the following is true about f 7 (c) c : 5. 5 (a) The function is concave down at a: = 2 (d) c = 6.5 (b) The function is concave down at :c = 71 (e) c = 7.5 c) "he function has an inection point at x : O ( (d) The function is concave up at z = 2 (e) None of these is true about f, 8. (.5 point) The graph of a function f , its derivative, f' and its second derivative f\" are shown below. Determine which graph corresponds to the function, its derivative, and its second derivative 0)) Function: f f' f \" Graph: I II III Function: f f ' f' ' Graph: I III II Function: f f ' f" Graph: 11 I III m. Function: f f' f \" Graph: II III II Function: f f ' f \" Graph: III II I Function: f f ' f' ' Graph: III I II 9. (1.5 points) Consider the function f(:c) = x (a) (C) (d) (8) (x 5)2: 0.5 point) Find the critical numbers 0.2 point) List the intervals where the function is increasing. (02 point) List the intervals where the function is decreasing 0.2 point) Find the relative extrema. (It will sufce to list the ac-values only. Mark whether each is a relative maximum or relative minimum). 0.4 point) Find the absolute maximum value and absolute minimum value on the interval [2,4]