Calculator Review #2

1. Calculator Review #2 • What is the average value of the function on the interval [1, 3]? • 0.146 • 0.914 • 0.964 • 0.987 • 1.928 • Use Average Value for Integrals Ans: C

2. Calculator Review #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? • It is decreasing at the rate of 16 cm per minute • It is decreasing at the rate of 16/13 cm per minute • It is not changing • It is increasing at the rate of 33/13 cm per minute • It is increasing at the rate of 16 cm per minute. • This is a derivative/related rate problem C a b A c B

3. Calculator Review #2 3. • 2.163 • 2.660 • 2.780 • 5.163 • 5.660 • The fundamental theorem of calculus Ans: D

4. 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)? • None • One • Two • Three • 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 -1/2 2 4 6 8 Ans: D

5. Calculator Review #2 • What is the area of the region bounded by the graphs of • 1.523 • 2.358 • 2.493 • 4.783 • 7.409 • Area - Integrals Ans: A

6. Calculator Review #2 6. What is the x-value of the point at which the tangent line to the graph of • -0.732 • -0.589 • -0.162 • 0.236 • 0.361 • Tangent Lines - Deriv Ans: A

7. Calculator Review #2 • 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

8. Calculator Review #2 • 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 – 0.630 B. y = 2x + 0.537 C. y = 2x + 0.839 D. y = 2x + 0.926 E. y = 2x + 1.469 Graph And y = 2 Look for the points of intersection When x = .315, find f(.315)

9. Calculator Review #2 • 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. -2.079 B. 0.046 C. 0.107 D. 0.125 E. 0.230 • Differential Equation Growth and Decay

10. Calculator Review #2 • 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 6 4 2 0 1 2 3

11. Calculator Review #2 • 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.

12. Calculator Review #2 • 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)=500000 k = 4338.07

13. Calculator Review #2 • 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 = 1.202 and t = 6.798 These are between 0 and 12 Determine what happens at these points. At x = 6.798 the function increases then decreases, so this is a max

14. Calculator Review #2 • 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

15. Calculator Review #2 • The table below gives the velocity v(t) at selected times t of a car traveling along a straight road. • 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? 80 70 60 50 40 30 20 10 2 3 2 1 2 1 2 3 4 5 6 7 8 9 10

16. Calculator Review #2 • 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.