project 8 pre calc

16-21

Take this test as you would take a test in class. After you are finished, check your work against the answers given in the back of the book. In Exercises 1-4, write the first five terms of the sequence. 1. a, = (-1)"(Begin with n = 1. ) 2. a, = 12 and a + 1 = a, + 4 3. b = - (Begin with n = 1. ) 4. b = (2n + 1)! (Begin with n = 1.) 11! . 4! n! 2n! 5. Simplify 4 . 7! 6. Simplify- (n + 1)!' 7. Simplify- (n - 1)!' 8. Write an expression for the apparent nth term of the sequence 2, 5, 10, 17, 26, . . .. (Assume n begins with 1). In Exercises 9 and 10, find a formula for the ath term of the sequence. 9. Arithmetic: a, = 5000, d = -100 10. Geometric: a, = 4, a.4 1 = 10, 2 2 11. Use sigma notation to write- 3(1) + 1 3(2) + 1 3(12) + 1 12. Use sigma notation to write 2 + 7 + + 12 + 128 + . . . In Exercises 13-15, find the sum. 13. (8n - 5) 14. > 24(2)"- 15. 5- 2 + 3 - # + 73-' 16. Use mathematical induction to prove the formula 3+6+9+ . + 301 = 3n(n + 1) 2 17. Use the Binomial Theorem to expand and simplify (20 - 56)4. In Exercises 18-21, evaluate the expression. 18. ,C3 19% 203 20. 9P2 21. 70 3 22. Solve for n in 4 . ,P3 = a+1P4. 23. How many distinct license plates can be issued consisting of one letter followed by a three-digit number? 24. Four students are randomly selected from a class of 25 to answer questions from a reading assignment. In how many ways can the four be selected? 25. A card is drawn from a standard deck of 52 playing cards. Find the probability that it is a red face card. 26. In 2006, six of the eleven men's basketball teams in the Big Ten Conference were to participate in the NCAA Men's Basketball Championship Tournament. If six of the eleven schools are selected at random, what is the probability that the six teams chosen were the actual six teams selected to play? 27. Two integers from 1 to 60 are chosen by a random number generator. What is the probability that (a) both numbers are odd, (b) both numbers are less than 12, and (c) the same number is chosen twice? 28. A weather forecast indicates that the probability of snow is 75%. What is the probability that it will not snow?Take this test as you would take a test in class. After you are finished, check your work against the answers given in the back of the book. In Exercises 1-3, use a graphing utility to graph the function and approximate the limit (if it exists). Then find the limit (if it exists) algebraically by using appropriate techniques. x - 1. lim 2. lim -x+ 5x - 3 3. lim - VX - 2 2x In Exercises 4 and 5, use a graphing utility to graph the function and approximate the limit. Write an approximation that is accurate to four decimal places. Then create a table to verify your limit numerically. 4. lim sin 3.x 5. lim par - 1 -+0 X 6. Find a formula for the slope of the graph of fat the point (x, f(x)). Then use it to find the slope at the specified point. (a) f(x) = 3x3 - 5x - 2, (2, 0) (b) (x) = 2x3 + 6x, (-1, -8) In Exercises 7-9, find the derivative of the function. 7. f (x) = 5 - =x 8. f(x) = 2x2 + 4x - 1 9. f(x) = - 104 In Exercises 10-12, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. 10. lim 1 - 3x2 I -1 5x - 1 11. lim - *hoe X2 - 5 12. lim = 1 9- 3 3x + 2 In Exercises 13 and 14, write the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, explain why. Assume n begins -2 with 1. n' + 3n - 4 Figure for 15 13. an = = 14. a, = + (-1) 2n2 + n - 15. Approximate the area of the region bounded by the graph of f(x) = 8 - 2x3 shown at the right using the indicated number of rectangles of equal width. Time Height (seconds), x (feet), y In Exercises 16 and 17, use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. 16. f(x) = x + 2; interval: [-2, 2] 17. /(x) = 3 - x2; interval: [- 1, 1] 23 18. The table shows the height of a space shuttle during its first 5 seconds of motion. 60 (a) Use the regression feature of a graphing utility to find a quadratic model 115 y = ax + bx + c for the data. 188 (b) The value of the derivative of the model is the rate of change of height with respect to time, or the velocity, at that instant. Find the velocity of the shuttle after 5 seconds. Table for 18