Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 1 Math 131 - Final Exam Review 1. The following equation gives the rate at which the angle between two objects is changing during a game: lnx a(x) = radians per second x where x is the number of seconds after the game started, 1 x 86. For each of the following, interpret your answer where appropriate. (a) What is the change in this rate from 10 seconds to 20 seconds after the game started? (b) What is the percent change in this rate from 5 seconds to 12 seconds after the game started?
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 2 (c) How fast is this rate changing 13 seconds after the game started? (d) How fast is the angle between the two objects changing 13 seconds after the game started? (e) Find the total change in the angle between the two objects from 10 seconds to 20 seconds after the game started.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 3 (f) What is the average angle between the two objects from 3 to 9 seconds after the game started if the angle between the two objects was 5 radians 40 seconds after the game started? (g) What is the average rate of change of the angle between the two objects from 3 to 9 seconds after the game started? lnx (h) What is the slope of the secant line to a(x) = from x = 7 to x = 27? x
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 4 (i) What is the percentage rate of change of the angle between the two objects 7 seconds after the game started? (j) Using the same quantities that were used in the calculation of the answer to (i), estimate the angle between the two objects 10 seconds after the game started. lnx (k) What is the slope of the tangent line to a(x) = at x = 20? x
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 5 2. Use the graph below of f, g, and h to select the appropriate answer: A) The derivative graph of f(x) is g(x), and h(x) is an accumulation function of f(x). B) The derivative graph of h(x) is g(x), and f(x) is an accumulation function of h(x). C) The derivative graph of g(x) is h(x), and f(x) is an accumulation function of g(x). D) The derivative graph of f(x) is h(x), and g(x) is an accumulation function of f(x). E) none of these 3. Find the derivative of f(x) = 4 3x ln ( 5x 7) and simplify. 4. Water flows into a tank at a rate that is inversely proportional to the square of the depth of the water in the tank, and water flows out of the tank at a rate that is proportional to the depth of the water in the tank. Write a differential equation describing the rate of change of the depth D of the water in the tank with respect to time. 5. The heart rate of a person who has taken a particular drug is given by b(p) = 0.15p+77 beats per minute, where p is the amount of this drug (in ppm) in the person s bloodstream. The concentration of this drug (in ppm) in the person s bloodstream is p(t) = 1.5t +120 ppm, where t is the time in hours since the drug was administered, 0 t 80. Write a function that gives the person s heart rate as a function of the number of hours since the drug was administered.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 6 6. Find exactly x 5 3t 2 7t + π dt. ln3 7. The rate of change of a function f with respect to x is inversely proportional to the input x. (a) Write a differential equation describing the rate of change of f with respect to x. (b) Write a general solution for the differential equation. 8. The table below gives the percentage of students using the indoor recreation facilities (as opposed to the outside facilities) based upon the outside temperature in degrees Fahrenheit. Outside Temperature ( F) 40 45 48 53 58 66 Percent of students indoors 41 37.5 35 32 29 24 (a) Find a model for the percent of students using the indoor recreation facilities as a function of the outside temperature. (b) Find and interpret the slope. (c) If there are 535 students at this school, how many would you expect to use the indoor facilities if the outside temperature is 50 F?
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 7 9. Use the limit definition of the derivative to find the rate-of-change equation for f(x) = 3x 2 2x+4. 10. Find the derivative of h(t) = 4(5x ) (7 8x 11 ) 6 + cos(4x) sin(2x). 11. The graph below shows the slope field of a certain differential equation. Sketch the particular solution that satisfies y( 3) = 1. 4 3 y(x) 2 1 K4 K3 K2 K1 0 1 2 3 4 x K1 K2 K3 K4
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 8 12. The demand for computer programmers during certain years is given in the table. Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Thousands of 16.40 19.69 20.90 19.48 16.09 12.33 9.97 10.12 12.70 16.50 19.74 20.89 19.41 jobs per week (a) Construct a sine model for the data without using regression technology. (b) Use your model to estimate the weekly demand for computer programmers in 2010. Is this an interpolation or an extrapolation? (c) What is the average weekly demand for computer programmers from 1996 to 2005? (d) What is the average weekly demand for computer programmers from 2000 to 2006?
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 9 (e) Fit a sine model to the data using regression technology. (f) Using the regression model, find and interpret how quickly the weekly demand is changing in 2002. (g) Use calculus and the regression model to find the greatest weekly demand for computer programmers from 2000 to 2006. (h) Use calculus and the regression model to find the year between 2000 and 2006 in which the weekly demand for computer programmers was decreasing most rapidly.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 10 13. The function f(x) = 6 ( e 9x) + 6x2 + 5x 6 gives the growth rate (in cells per minute) of a population of x 3 bacteria cells, where x is the number of minutes after the cells were placed in a petri dish, 1 x 200. Find the total accumulated change in the size of this bacteria population from x = 1 to x = k minutes after the cells were placed in the dish. 14. At 2pm, a patient was administered a dose of cough syrup. The concentration of the cough syrup s active ingredient in the patient s body is given in the table. Time 2pm 3pm 4pm 5pm 6pm 7pm 8pm Concentration ( µg ) ml 10.0 8.5 7.3 6.2 5.3 4.5 3.9 (a) Find the best model for the data. (b) What is the percentage change? (c) Use the model to estimate the concentration of the cough syrup s active ingredient in the patient s body at 10pm.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 11 15. Find the particular solution to the differential equation dy dx = (2x+5) x 2 + 5x that satisfies y(4) = 200. 16. Write the general solution for the differential equation dy dx = 8x 5y.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 12 17. The population of a certain town can be modeled by P(t) = x 3 21x 2 + 135x+12 thousand people, where t is the number of years since 1990, 2 x 10. (a) Use calculus to find all extrema of P(t) on the interval [2,10]. (b) Use calculus to find the years in which the population was increasing and decreasing most rapidly from 1992 to 2000. Also give the population in those years and how fast it was changing.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 13 18. A population of a remote jungle village is given in the table below. Year 1880 1890 1902 1910 1920 1930 Population 2 12 60 112 172 182 (a) Find the best model P(t) for the data, where t is the number of years since 1880. (b) What will the eventual population of the jungle village be? (c) In what year was the population increasing most rapidly? 19. The table below shows the population of a city in California for different years. Year 1950 1960 1970 1980 1990 2000 Population (in thousands) 490 800 1000 1100 1050 930 (a) Find the best model P(t) for the data where t is the number of years since 1950. (b) Estimate the population of this city in 2005.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 14 20. $6,000 is invested in an account paying an APR of 6.79% compounded monthly. (a) How much is in the account after 5 years? (b) What is the APY for this investment? (c) When will the account have $10,000? 21. Find the derivative of y = (ln(2x)) 9. 22. The graph G shown below gives Bob s grade on an economics exam based on the number of hours he spends studying for this exam. Estimate and interpret the percentage rate of change at t = 10. (Figure source: Calculus Concepts, Latorre)
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 15 23. The table below shows the number of cars sold at a major dealership during the year 2006. Month Jan Feb Mar Apr May Jun Jul Aug Oct Nov Dec Number of cars sold 120 140 146 150 138 124 122 126 148 158 168 (a) Find the best model for the data. (b) Use the model to estimate the number of cars sold in September. 24. Compute (3x 4 6x) 5 (4x 3 2)dx. 25. The marginal profit for a bowling alley is given by P (x) = 50 8ln(x+10)+2x 1 5 dollars per bowler for x bowlers, 0 x 100. Using 4 rectangles of equal width, estimate the total accumulated change in profit given by 36 20 P (x)dx.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 16 21.801 26. On one particular day, the price of gold was G(x) = 1+0.031e 0.029x dollars per gram in the xth hour of the day, 0 x 24, and the amount of gold sold during the x th hour of that day was Q(x) = 0.1858x+5.69 kilograms, 0 x 24. Find a model for the hourly revenue from gold sales on that day. 27. Carbon-14 has a half-life of about 5,580 years. If a piece of pottery contains 17 mg of carbon-14, how long ago did it contain 40 mg of carbon-14? 28. The rate of change of the number of people who own a lawn mower L with respect to the number of people who live outside the city limits of a certain town S is jointly proportional to the cube root of the number of people who live outside the city limits and the inverse of the number of people who own a lawn mower. The constant of proportionality is 9. (a) Write a differential equation modeling this problem. (b) Find a general solution to this differential equation.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 17 29. Compute (3x 8 e x + 7 x + e 2x ) dx. 30. The growth rate of a population of bacterial cells in a petri dish can be modeled by f(t) = 0.2e 0.5t + 10 cells per day, where t is the number of days since the start of the experiment, 0 t 20. After 3 days, there were 30 bacteria cells. (a) What is the average rate of change of the population from day 10 to day 15? (b) What is the average number of bacteria cells in the population from day 10 to day 15? 31. Compute the following or show that the integral diverges: 1 5 3x 3 dx
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 18 32. Let y = 3sin(2x+ π 2 ) 1. (a) Find the amplitude of this sine curve. (b) Find the period of this sine curve. (c) Is this sine curve reflected across the horizontal axis? (d) State the vertical shift of this sine curve. (e) State the horizontal shift of this sine curve. (f) Sketch one cycle of this sine curve.
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 19 33. Find the general solution to the differential equation dy dt = ky t. 34. In the year 2000, 9.6% of all female elephants in a particular wildlife preserve gave birth to a live calf, while 10.5% gave birth to a live calf in 2005. (a) Find and interpret the rate of change of the percentage of live births to female elephants in this wildlife preserve, assuming the percentage of births increased at a constant rate. (b) Write a model that gives the percentage of live births as a function of time. Then estimate the percentage of female elephants in this preserve that will give birth to a live calf in 2008. 35. Which of the following differential equations produces the given slope field? A) dy dx = x+y B) dy dx = 2x y C) dy dx = x2 y D) dy dx = 2x y E) none of these y(x) 4 3 2 1 K2 K1 0 1 2 x K1 K2 K3 K4
Math 131 Spring 2008 c Sherry Scarborough and Heather Ramsey Page 20 x 36. On the axes provided in the second graph below, sketch the accumulation function A(x) = r(t)dt. Mark inflection points with and relative extrema with on the accumulation function graph. a Rate of Change Function r(t) Accumulation Function A(x) a b c d e f g a b c d e f g 37. Let f(x) = (0.6) x + 3 and g(x) = x 2 3x+1. (a) Find the area between f and g on the interval [ 2,1]. (b) Suppose that f and g are rate-of-change functions. Find the difference in accumulated change of f and g on the interval [ 2,1]. 38. The table shows the relationship between the number of items sold and the advertising cost. Number of items sold 2,000 6,200 9,500 10,300 16,800 Cost of advertising (thousands of dollars) 28.35 32.19 33.64 33.92 35.58 (a) Find the best model for the data. (b) Use the model to estimate the number of items sold when the advertising cost is $38,500.