1. Find the domain of the function f ( x )= x 2 x +2 1) ______ (A) D=( ,2 ) ( 2, ) (B) D=( ,2 ) (2, ) (C) D= ( 2, ) (D) D=(2, ) (E) none of the above 1 2. According to the Center for Disease Control and Prevention, Table 1 shows the growth rate of the deadly Ebola Virus and the number of deaths in a small city whose initial population was 120,000 would result from the Ebola virus if left untreated after x weeks. Table 1 P (infected) (weeks) 0 5714 0.5 11486 1 21968 1.5 38610 2 60128 2.5 81613 3 98185 3.5 108602 4 114332 4.5 117254 5 118687 5.5 119376 6 119705 Use the data to estimate the average growth rate of the disease from week 2 to week 4. Use the data to estimate the growth rate of the disease on week 4. x (i) ( ii ) 2) ______ (A) ( i ) (B) ( i ) (C) ( i ) (D) ( i ) (E) none 8,652 people/week; ( ii ) 27,102 people/week 54,204 people/week; ( ii ) 8,652 people/week 27,102 people/week; ( ii ) 8,652 people/week 27,102; ( ii ) 11,460 people/week of the above 2 3. Use the graph in Figure 2 to find each of the following, provided it exists. If it does not exist, explain why. Figure 2 3 V ( s )( b ) lim + s 3 V ( s ) ( c ) lim V ( s ) s 3 s 3 ( a ) lim 3a) ______ 3b) ______ 3c) _____ 4. Suppose L represents the limiting value of the following limit expression. Find L using the limit principles for infinite limits (You must show all your work to receive full credit; no short cuts ): lim 2 x 2 +1 L= x 9 x 4 +2 (A) L (B) L=2/9 (C) L=0 (D) L=2/3 (E) none of the above 4) ______ p ( x )=x 53 x 4 +2 x1 . Use the Intermediate Value Theorem to show there is a zero of the polynomial p ( x ) in the interval (2, 3). 5. Let 5) ______ 6. Find the derivative of the function 10 g ( x ) =( x 2+ x +1 ) tan 3 x 6) ______ (A) 9 10 ( x 2+ x +1 ) 3 ( tan 2 x ) ( sec 2 x ) 9 10( x 2 x 1) 9 3(tan 2 x)(sec 2 x) 10 10 ( x 2+ x +1 ) tan 3 x+ ( x 2 + x +1 ) 3 ( tan 2 x ) 9 (C) 10 ( x 2+ x +1 ) ( 2 x +1 )+ 3 ( tan 2 x )( sec 2 x ) 9 10 (D) 10 ( x 2+ x +1 ) ( 2 x +1 ) tan 3 x + ( x 2+ x +1 ) 3 ( tan 2 x ) ( sec 2 x ) (B) (E) none of the above 7. True or False (If your answer is true, explain your reasoning; if your answer is false give a counterexample). If f d dx is continuous on b ( a [ a , b] , then ) f ( x ) dx =f ( x ) 7) ______ 8. The graphs of C ) is f , f f , f ' , and f ' ' are shown. Identify which of the three graphs ( A , B , ' , and f ' ' . Explain your reasoning. A B C 8) ______ A = _______ B = _______ C = _______ 9. The graph of a function f is shown. Which graph is an antiderivative of f and why? 9) ______ A B f C D (A) Graph (B) Graph (C) Graph (D)Graph (E) none of A B C D the above 10. Find y' if x y= 4 t+9 dt 0 10) ______ of the above 3/ 2 2 1 (A) ( B ) 4 x+ 9 (C ) 4 x +9 9 ( D ) ( 4 x+ 9 ) ( E ) 6 4 x+ 9 11. A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m . If water is being pumped into the tank at a rate of 2 m3 /min , find the rate at which the water level is rising when the water is 1 2 3 m deep. (Hint: V = r h ) 3 11) ______ (A) 0.14 m/min (B) 0.28 m/min (C) 2.38 m/ min (D) 16.75 m/min (E) none of the above 12. Suppose f ( 0 )=3 and f ' ( x ) 5 for all values of x . How large can f ( 2 ) possibly be? 12) ______ (A) 3 (B) 5 (C) 7 (D) 9 (E) none of the above 13. Evaluate 4 2 1 +1 dx x 0 (A) 1/3 (B) -2/3 (C) 2 (D) 3 (E) none of the above 13) ______ 14. The cost to manufacture x computers per day is modeled by the following equation 14) ______ 2 C ( x )=20,000+25 x + x 20,000 The average cost C ( x )=C ( x ) / x is defined to be the total cost divided by the quantity produced. How fast is C changing if the current production rate is 2000 computers per day and is increasing at the rate of 200 units per day? (A) Increasing by $1.01 per day (B) Decreasing by $1.01 per day (C) Increasing by $125.01 per day (D) Decreasing by $125.01 per day (E) none of the above 15. Engineers use innovative designs to improve the ability of buildings to withstand earthquakes. For example, the San Francisco International Airport uses giant steel ball bearings built into each of the 267 columns (see Figure below) which support the weight of the airport. Each ball bearing measures 5 feet in diameter with a maximum error of 0.01 feet. What is the maximum error for this diameter in computing the volume of this ball bearing? 15) ______ B u i lB d a il n l B g a B s S e e u a p r p i o n r g t (A) 0.785 ft3 (B) 0.685 ft3 (C) 0.575 ft3 C o l u m n (D) 0.365 ft3 (E) None of the above 16. Sand falling from a hopper at 10 ft3/s forms a conical sand pile whose radius is always equal to its height. How fast is the radius increasing when the radius is 5 ft? 16) ______ (A) 5 ft/sec (B) 10 ft/sec (C) 2/5 ft/sec (D) not enough information (E) none of the above 17. Sketch the graph of the function, indicating all critical points and inflection points. Apply the second derivative test at each critical point. Show the correct concave structure and indicate the behavior of f ( x ) as x , for f ( x )=4 x 55 x 4 . 17) ______ 18. An aquarium has a square base made of slate costing 8/in2 and four glass sides costing 3 /in2. The volume of the aquarium is to be 36,000 in3. Find the dimensions of the least expensive such aquarium. 18) ______ (A) l = 30 inches; w = 30 inches; h = 40 inches (B) l = 20 inches; w = 20 inches; h = 90 inches (C) l = 30 inches; w = 40 inches; h = 30 inches (D) not enough information (E) none of the above 19. Find the open intervals on the x-axis on which the function increasing and those on which it is decreasing. f ( x )=x 4 + 4 x 3 is 19) ______ (A) f (B) f (C) f (D) f is increasing on ( ,3 ) ; f is decreasing (3, ) . is decreasing on (,3 ) ; f is increasing on (3, ) . is increasing on (, ) . is increasing on (,4 ) ( 0, ) ; f is decreasing on (4, 0 ) . (E) none of the above 20. Show that the function f ( x )= x satisfies the hypotheses of the mean value theorem on the interval [ 0, 4 ] . Find all numbers c in the interval that satisfy the conclusion of that theorem. 20) ______ (A) f (B) f (C) f (D) f is continuous on [ 0, 4 ] ; f ' is continuous on is continuous on [ 0, 4 ] ; f ' is continuous on is continuous on [ 0, 4 ] ; f ' is continuous on does not satisfy the Mean Value Theorem (E) none of the above 21. Evaluate: ( 0, 4 ) ; c=1/2 [ 0, 4 ] ; c=1 ( 0, 4 ) ; c=1 4 |x 2 x|dx 0 21) ______ (A) 14 (B) 16 (C) 50/3 (D) not possible (E) none of the above 22. A stone is dropped from the top of a building 960 ft high. What will its impact velocity be? (Note: You may ignore air resistance.) 22) ______ (A) 64 (B) -64 (C) 64 (D) -64 15 23. Evaluate: (E) none of the above 15 ( sin 4 x )6 ( cos 4 x ) dx 0 23) ______ (A) 0 (B) 1/2 (C) 1 (D) -1 (E) none of the above 24. Which of the following functions has neither a local or global extreme value? Explain. 24) ______ (A) f ( x ) = x above (B) f ( x )=x 2 (C) f ( x )=x 3 (D) f ( x )=x 4 (E) none of the 25. Use the following conditions to sketch a graph of the function f ( x ) . (1) f ( 0 )=0, f (1 )=f ( 1 )=3, f (2 )=f ( 2 )=5, f ' (2 ) =f ' ( 0 )=f ' ( 2 )=0 (2) f ' ( x ) >0 on ( ,2 ) ( 0,2 ) , f ' ( x ) <0 on (2, 0 ) ( 2, (3) f ' x>0 on (1,1 ) , f ' ' ( x )< 0 on (,1 ) (1, ) 25) ______