SOLUTIONS 1 (27) 2 (18) 3 (18) 4 (15) 5 (22) TOTAL (100) PROBLEM NUMBER SCORE MIDTERM 2. Form A. Recitation Instructor : Recitation Time :

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1 Math 5 March 8, 206 Form A Page of 8 Name : OSU Name.# : Lecturer:: Recitation Instructor : SOLUTIONS Recitation Time : SHOW ALL WORK in problems, 2, and 3. Incorrect answers with work shown may receive partial credit, but unsubstantiated correct answers may receive NO credit. You don' t have to show work in problems and 5. Give EXACT answers unless asked to do otherwise. Calculators are NOT permitted! PDA' s, laptops, and cell phones are prohibited. Do not have these devices out! The eam duration is 55 minutes. The eam consists of 5 problems starting on page 2 and ending on page 8. Make sure your eam is not missing any pages before you start. PROBLEM NUMBER SCORE (27) 2 (8) 3 (8) (5) 5 (22) TOTAL (00)

2 2 SOLUTIONS-A- SP 6.nb Form A, Page 2. (27 pts) (I) Find the derivatives of the following functions. NO NEED TO SIMLIFY! (a) ( pts) g () e a + tan * (a ) g ' () e a + a e a + + (a ) 2 a (b) (6 pts) y () (sin ) Since y () e ln sin e ln sin, it follows that y ' () e ln sin ln sin + (sin ) (ln sin + cot ) sin cos OR * using logarithmic differentiation Since y () (sin ), it follows that ln (y ()) ln (sin ) ln (y ()) ln (sin ). y' () By differentiating both sides, we get ln (sin ) + y () and y' () y () ln (sin ) + and, therefore, sin cos, sin cos y' () (sin ) (ln (sin ) + cot )

3 SOLUTIONS-A- SP 6.nb 3 (II) (5 pts) Given that h () ln (sec + tan ), show that h ' () h' () sec. sec + tan (sec tan + sec2 ) sec (sec + tan ) sec + tan sec Form A, Page 3 (III) (2 pts) Given that f () 2, f' () 3, and f'' () *, find the following values or state that the value cannot be determined. d d f' () f () f'' () f () * f' () f' () [f ()] 2 (a) f'' () f () * f' () f' () [f ()] 2 (*) (2) * (3) (3) [2] 2 *2 * 9 * d d f* () 2 f ' (f * (2)) f ' () 3 (b)

4 SOLUTIONS-A- SP 6.nb lim (c) f () f () 2 3 Since f' () 3, it follows that f is differentiable at a. Since f is differentiable at a, it is continuous at a. This means that lim 3 f () f () (d) lim h30 f ' ( + h ) * f ' () h This is the definition derivative of f ' at a. Therefore lim h30 f ' ( + h ) * f ' () h (f')' () f'' () * Form A, Page 2. (8 pts) A floor lamp (see figure ) is designed to have an adjustable height. The lamp has a shade in the shape of a cone with radius R 0 in and height H 20 in. Let h be the height of the lamp. Figure

5 SOLUTIONS-A- SP 6.nb 5 When the lamp is on, it lights up a circular region on the floor. (see figure 2 ). Figure 2 (a) Let r be the radius of the lit circle. In the figure above, label r, R and H. (b) Find an epression for r in terms of h. From the similar right triangle, it follows that r h R H, and, therefore r R H h 0 20 h h. 2

6 6 SOLUTIONS-A- SP 6.nb Form A, Page 5 (c) Let A be the area of the lit region. Assume that the height of the lamp is decreasing at the rate dh *2 in sec. dt Find the rate of change of A, the area of the lit circle, at the moment when h 36 in. A r 2 π 2 h 2 π π h2 da dt π 2 h dh dt da dt h36 2 π h dh dt h36 2 π (36) (*2) *36 π in2 sec or A r 2 π implies that da dt 2 r π da 2 r π dr dt h36 dt h36 dr dt, and, therefore Since r 36 h, it follows that r 8, when h 36, 2 2 and dr dt dh (*2) *. So h dt h36 da 2 r π dr 2 (8) π (*) *36 π in 2 sec dt h36 dt h36

7 SOLUTIONS-A- SP 6.nb 7 Form A, Page 6 3. (8 pts) Show your work. The position, s (t), of an object moving along a horizontal line (see the figure below) is given by s (t) 3 sin π t, t 0, where s is measured in feet and t in seconds.!!3!2! 2 3 s (a) (2 pts) Mark the position of the object at the time t 2 on the line above. s (2) 3 sin π (2) 3 sin π 2 3 (b) ( pts) Find the average velocity, v av, of the object over the interval [0, t]. v av s (t) * s (0) t * 0 3 sin π t *0 t * 0 3 sin π t t (c) ( pts) Using the epression found in part (b), evaluate the limit 3 sin π lim t30+ v av lim t t30+ t 3 π lim t30+ 3 π lim sin π t t30+ π t sin π t 3 π π t () 3 π Remember : lim 30 sin (d) (2 pts) What does the limit found in part (c) represent?

8 8 SOLUTIONS-A- SP 6.nb Initial velocity, v (0). (e) (6 pts) Find the velocity, v (t) and acceleration, a (t). v (t) s' (t) 3 sin π t ' 3 cos π t π NOTICE : v (0) 3 cos π (0) π 3 cos (0) π 3 () π 3 π a (t) v' (t) 3 π cos π t ' *3 π sin π t π *3 π 2 sin π t Form A, Page 7. (5 pts) (I) The graph of a function f is given below. y 3 y f ( ) tangent line %& i (3 pts) Circle the correct statement. (a) f () f (3); (b) f () > f (3); (c) f () < f (3) ;

9 SOLUTIONS-A- SP 6.nb 9 ii (3 pts) Circle the correct statement. (a) f ' () f' (3); (b) f' () > f' (3) ; (c) f ()' < f' (3); Notice that f is CONCAVE DOWN on the given interval. That means that f', the derivative of f, is DECREASING on the given interval. Therefore f' () > f' (3). iii (3 pts) Circle the best approimation of f' (3). (a) f' (3) 2 ; (b) f' (3) *2 ; (c) f' (3) ; (d) f' (3) * ; (e) f' (3) 2 ; f f' (3) * 2. It appears that the tangent line to the curve y f () at the point where 3 has a SLOPE 2 see figure. (II) The figure below shows the graphs of f, f', and another function, g. Which curve is which? y C!2! 2 B! A Fill in the blanks : f C ; f' A ; g B.

10 0 SOLUTIONS-A- SP 6.nb Form A, Page 8 5. (22 pts) EXPLANATION IS NOT REQUIRED, AND NO PARTIAL CREDIT WILL BE GIVEN. Assume that a function f is continuous on its domain, (*, 6). The graph of f', the derivative of f, is shown in the figure below. y y f ' ( )! !!2 (a) Find the * coordinates of all critical points of f (or write NONE). ANSWER : critical point (s) at 0, 3, 5 (b) Find the * coordinates of all local maima of f (or write NONE). ANSWER : local ma (s) at 5 (c) Find the * coordinates of all local minima of f (or write NONE). ANSWER : local min (s) at 3 (d) Find the interval (or intervals) on which f is increasing. ANSWER : f is increasing on (3, 5) (e) Find the interval (or intervals) on which f is concave down. ANSWER : f is concave down on (0, ) (f) Find the * coordinates of all inflection points (or write NONE). ANSWER : inflection point (s) at 0 and

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