1. [B1] The concentration of a drug, C(t), measured in [(mg)/ cc], in a patient's blood t hours following a dose is given by C(t) = 10 t e-0.7t. Graph the function C(t), label points on the t and C axes.

  2. [B2] See B1: Estimate the time of peak concentration. Estimate the peak concentration. (You can use the trace funtion on your calculator.)

  3. [B3] See B1: Write a formula for bioavailability of the drug during the 10-hour period following the dose. (The units of bioavailability are [(mg ·hr)/ cc].

  4. [B4] See B1: Find the bioavailability using your calculator.

  5. [B5] Assume that, during the first four minutes after a foreign substance is introduced into the blood, antibodies are made at the rate r(t), in thousands of antibodies per minute. he rate r(t) is given by
    r(t) = t
    t2+1
    		 ,
    where time t is measured in minutes, 0 £ t £ 4. Write a formula for the total number of antibodies made during the first four minutes.

  6. [I1] See B5: Find the total number of antibodies made during the first four minutes.

  7. [I2] Find the indefinite integral
    ó
    õ     		( 2t3 - 3
    t
    		 + cos(2t) - et -2) dt

  8. [I3] Find the exact value of the following definite integral using the Fundamental Theorem of Calculus.
    ó
    õ     		2

    0 
    		 (e3x - x) dx

  9. [I4] The graph below represents the density function for the amount of time spent waiting at a doctor's office. What is the maximum amount of time anyone will have to wait?

    GRAPH

  10. [I5] See I4: Approximately what percentage of patients end up waiting between 1 and 2 hours?

  11. [N1] See I4: Approximately what percentage of patients wait less than an hour?

  12. [N2] An experiment was done observing the time gaps between successive cars on a certain freeway on a weekday morning. The density function of these time gaps was found to be approximately
    p(x) = 0.122 e-0.122x
    where x is the time in seconds and 0 £ x £ 40. Find the probability that a gap between two successive cars is between 10 and 15 seconds.

  13. [N3] See N2: Find the mean time gap.

  14. [N4] See N2: Find the median gap.

  15. [N5] Test scores in a large class are normally distributed with mean, m = 70 points and standard deviation s = 8. This means that the density function is given by
    p(x) = 1
    s   æ
    Ö
    2 p
     
    		e- ( [((x - m)2)/( 2 s2)])
    That is, 1/(sÖ{2 p}) times e- ( [((x - m)2)/( 2 s2)]). Write a formula for the density distribution p(x) of test scores.

  16. [G1] See N5: Graph the density function showing units on the x axis.

  17. [G2] See N5: Write a definite integral that represents the fraction of students with test scores between 62 and 78.

  18. [G3] Evaluate the integral in G2.

  19. [G4] Is y = x2 +2 a solution to the differential equation [dy/ dx] = 3x2 y?

  20. [G5] Is y = ex3 a solution to the differential equation [dy/ dx] = 3x2 y?

  21. [O1] Is y = 2 ex3 a solution to the differential equation [dy/ dx] = 3x2 y?

  22. [O2] Which function from G4, G5, O1, above is a particular solution to the initial value problem, [dy/ dx] = 3x2 y, y(0) = 2?

  23. [O3] The amount of a drug in a patient's bloodstream, Q(t), in mg, decreases at a rate proportional at each instant to the amount of the drug present with the coefficient of proportionality -0.2. The time t is measured in hours. Write a differential equation for the function Q(t).

  24. [O4] See O3: Find the general solution to the equation.

  25. [O5] See O3: If the initial amount Q(0) = 200, how much drug is left after 10 hours?

  26. [B1] Find the general solution to the differential equation:
    dy
    dt
    		 = - 0.6(y-1)

  27. [B2] Find the solution to the initial value problem:
    dy
    dt
    		 = - 0.6(y-1),   y(0) = 4

  28. [B3] A yam has been heated to 170 degrees F, and is placed in a room whose temperature is maintained at 70 degrees F. In half an hour the yam cools to 100 degrees F. Let Y(t) be the temperature of the yam t hours after being placed in the room. Assuming that the temperature of the yam obeys Newton's Law of heating and cooling. Write a differential equation for Y.

  29. [B4] See B3: Solve the equation to find a formula for Y(t).

  30. [B5] Match the function z = e-(x2 + y2) with one of the graphs given below.

  31. [I1] Match the function z = x3 - 2y2 with one of the graphs given below.

  32. [I2] Match the function z = x3 + y2 with one of the graphs given below.

    4 graphs.

  33. [I3] Find fx for f(x,y) = 3x2 + 3x2y + 2y3.

  34. [I4] Find fy for f(x,y) = 3x2 + 3x2y + 2y3.

  35. [I5] Find fx for f(x,y) = xex2 + y2.

  36. [N1] Find fy for f(x,y) = xex2 + y2.

  37. [N2] Let f(x,y) = 2y3 + 3x2 - 6xy. Find all critial points and decide for each one if it is a local min, max, or saddle point.

  38. [N3] Let a point (a,b) be a critical point of the function function f(x,y). Suppose fxx(a,b) > 0, fyy(a,b) = 0, and fxy(a,b) > 0. What can you conclude about the behavior of f(x,y) near (a,b)?

  39. [N4] Is the value of fx(9,76) positive or negative? Use the contour map given below.

  40. [N5] Is the value of fy(9,76) positive or negative? Use the contour map given below.

    Contour graph.

  41. [G1] The wind-chill factor, C, in degrees F is a function C = f(w,T) of the wind speed w, in mph, and the air temperature, T in degrees F. What are the units of [(f)/( w)](w,T)?

  42. [G2] See G1: What is the practical meaning of the statement [(f)/( w)](10,25) = -0.7?

  43. [G3] What are the units of [(f)/( T)](w,T)?

  44. [G4] See G1: What is the practical meaning of the statement [(f)/( T)](10,25) = 1.2?

  45. [G5] See G1: Assume that f(10,25) = 10, [(f)/( T)](10,25) = 1.2, and [(f)/( w)](10,25) = -0.7. Estimate f(15,25).

  46. [O1] What is the average of g(t) = 2t over the interval [0,10]?

  47. [O2] Given a function f(t) which is concave up over the interval [0,4]. Which is larger, (a) The average of f(0) and f(4)? or (b) The average of f(t) over the interval, 0 £ t £ 4.

  48. [O3] The demand curve for a product is given by p = f(1) = 20 e-0.002q and the supply curve is given by p = g(q) = 0.02q+1 for 0 £ q £ 1000 where q is quantity and p is price in dollars per unit. What is the equilibrium quantity and price? Hint, graph both functions and use your calculator to find the point of intersection.

  49. [O4] See 03: If the quantity is 300, will the supply price or the demand price be higher?

  50. [O5] A small business expects an income stream of $5000 per year for a four-year period. Find the present value of the business if the annual interest rate, compounded continuously, is 3%.

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