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Calculator Review #2 1. What is the average value of the function on the interval [1, 3]? A B C D E Use Average Value for Integrals Ans: C 1

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Calculator Review #2 2. Right triangle ABC has its right angle at A. Leg AB is decreasing at a constant rate of 3cm per minute. Leg AC is increasing at a constant rate of 4 cm per minute. How is the hypotenuse BC changing at the moment when AB = 12 and AC =5? A. It is decreasing at the rate of 16 cm per minute B. It is decreasing at the rate of 16/13 cm per minute C. It is not changing D. It is increasing at the rate of 33/13 cm per minute E. It is increasing at the rate of 16 cm per minute. This is a derivative/related rate problem C a b A c B 2

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Calculator Review #2 3. A B C D E The fundamental theorem of calculus Ans: D 3

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Calculator Review #2 4. The derivative of a function f is given by f(x) = sin (cos x) – 0.1x. How many critical points does f have on the open interval (0, 8)? A. None B. One C. Two D. Three E. Four Max/Min problem This function is the derivative of f(x) Graph f(x) then look for zeros, that is, where the function crosses x axis. 1/2 -1/ Ans: D 4

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Calculator Review #2 5. What is the area of the region bounded by the graphs of A B C D E Area - Integrals Ans: A 5

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Calculator Review #2 6.What is the x-value of the point at which the tangent line to the graph of A B C D E Tangent Lines - Deriv Ans: A 6

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Calculator Review #2 7. The region enclosed by the graphs of and the vertical lines x = 0 and x = 2 is rotated about the y = -3. Which of the following gives the volume of the generated solid? Volume – Integration outer inner y=-3 7

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Calculator Review #2 8. Which of the following is an equation of the tangent line to the graph of the function at the point where f(x) = 2? A. y = 2x – B. y = 2x C. y = 2x D. y = 2x E. y = 2x Graph And y = 2 Look for the points of intersection When x =.315, find f(.315)

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Calculator Review #2 9. The population of bacteria given by y(t) grows according to the equation where k is a constant and t is measured in minutes. If y(10) = 10 and y(30) = 25, what is the value of k? A B C D E Differential Equation Growth and Decay 9

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Calculator Review #2 10. The function is continuous on the closed interval [0, 3] and has values that are given in the table below. Using the subintervals [0, 1] [1, 2] [2, 3], what is the approximation to x 0123 f(x)2543

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Calculator Review #2 11. Because of rainfall from 9am on a given day, water enters an open reservoir at the rate of where R(t) is measured in gallons per hour and t is measured in hours after 9am (so 0 <= t <= 12). To prevent overflow, water is pumped out of the reservoir at the constant rate of k gallons per hour. The reservoir holds 500,000 gallons of water both at 9am and 9pm. A.What is k? Round your answer to the nearest whole number. 11

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Calculator Review #2 11. Because of rainfall from 9am on a given day, water enters an open reservoir at the rate of where R(t) is measured in gallons per hour and t is measured in hours after 9am (so 0 <= t <= 12). To prevent overflow, water is pumped out of the reservoir at the constant rate of k gallons per hour. The reservoir holds 500,000 gallons of water both at 9am and 9pm. B.To the nearest gallon, how much water is the reservoir at 7pm (t = 10)? Want to find the adding water and leaving water. The combined rate is W(t) = R(t) – k Know: W(0)= k =

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Calculator Review #2 11. Because of rainfall from 9am on a given day, water enters an open reservoir at the rate of where R(t) is measured in gallons per hour and t is measured in hours after 9am (so 0 <= t <= 12). To prevent overflow, water is pumped out of the reservoir at the constant rate of k gallons per hour. The reservoir holds 500,000 gallons of water both at 9am and 9pm. C.At what time t, for 0<= t <= 12, does the reservoir hold the greatest amount of water? W(t) = R(t) – k Graph this function and find the zeros. t = and t = These are between 0 and 12 Determine what happens at these points. At x = the function increases then decreases, so this is a max 13

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Calculator Review #2 12. The table below gives the velocity v(t) at selected times t of a car traveling along a straight road. A.Use the values of the table to approximate the acceleration of the car at time t = 6. Show the work that leads to your answer and indicate units of measure. Find the average acceleration by using 14 T V(t)

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Calculator Review #2 12. The table below gives the velocity v(t) at selected times t of a car traveling along a straight road. B. Use a right Riemann sum with the subintervals given in the table to approximate Indicate units of measure. What physical quantity does this integral represent? T V(t)

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Calculator Review #2 12. The table below gives the velocity v(t) at selected times t of a car traveling along a straight road. C.The function v(t) is twice differentiable on the interval [0,10]. Show that there must be a moment of time when the acceleration of he car is equal to zero. Choose a small interval on [5,8] V(7) = 70 V(8) = 65 V(5) = 60 V(7) is greater than this small interval of [5,8] Intermediate value theorem: find some c within [5,7] with v ( c ) = 65 The mean value theorem guarantees a point somewhere on [c,8] where v is zero. 16 T V(t)

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