P(z) =.0.2X2 + 22x - 400

Transcription

1 Survey ofcalcu1us I (Math 121 Exam 3 November 13, 2002 Part I. Multiple Choice. (2 points each) P(z) =.0.2X2 + 22x Find the marginal profit at a production level of 50 clocks. numerical answer, not the units. 0 :S: x :S: 100 For this problem, just find the (A) (F) 22 (B) 0.04 (G) 23 (C) (H) (D) 2 (I) 87 (1) (E) (A) clocks (B) dollars (E) clocks per dollar per dollar 3 (C) clocks per dollar (D) dollars per clock (F) dollars per clock per clock

2 4. Let f be a continuous function, and suppose that on some interval, its derivative I' is increasing. What else is certain to be true on this interval? (A) I is positive (B) I is increasing (C) f is concave upward (D) f' is positive (E) I' is concave upward (F) f" is increasing (G) f" is concave upward For problems 5 through 7, assume f is a continuous function. s. Suppose!,(c) = 0 and!,'(c) < O. Then which of the following must be true? (A) I bas a local maximum at c. (B) I bas a local minimum at c. (C) I bas a saddle point at c. (D) I has none of the above at c. (E) The behavior of I at c cannot be determined from the given infomlation 6. Suppose /' (c) = 0 and /" (c) = O. Then which of the following must be true? (A) / has a local maximum at c. (B) / has a local minimum at c. (C) / has a saddle point at c. (D) / has none of the above at c. (E) The behavior of / at c cannot be determined from the given infom1ation. 7. Suppose J'(c) = 1 and J"(c) = O. Then which of the following must be true? (A) f has a local maximum at c. (B) f has a local minimum at c. (C) f has a saddle point at c. (D) f has none of the above at c. (E) The behavior of f at c cannot be detennined from the given information 8. How long (to the nearest year) will it take an investmento triple in value at annual interest compounded continuously? (A) 6 years (E) 10 years (I) 14 years (B) 7 years (F) 11 years (J) 15 years (C) 8 years (G) 12 years (D) 9 years (H) 13 years

3 Part ll. True-False (1 point each) Mark your answer card "Att if the statement is true and "Btt if the statement is false. 9. Let f be a function. If I' does not exist at x = c, then f does not exist at x = c. 10. If a function f is concave downward on an interval, then f must also be decreasing on that interval. 11. If lim_f(x) = 00 or -00, and if lim f(x) = 00 or -00, then the line x = c is a vertical x-c x-c+ asymptote for f. 12. If Jim /(3:) = b and if lim /(3:) = b, then the line y = b is a horizontal asymptote for /. x-+-oo x-+oo 13. If j(z) = tt is a rational function and d(c) = 0, then the line x = c is a vertical asymptote for j. 14. A rational function cannot have more than one horizontal asymptote. 15 The graph of a rational function can never cross one of its asymptotes. 16 If / bas a local minimum at c, then / bas an absolute minimum at c, 17. A continuous function always has an absolute maximum and an absolute minimum. 18 1im(1+8)~ =e The function /(%) = ezgrows more rapidly than the function g(%) = % The function f(x) = lnx grows more rapidly than the function g(x) = Xl0.

4 Part ill. Computational Follow directions and work carefully on each of the following problems. 21 (6 points) Suppose f is a function, and suppose its derivative f' is given as follows. /,(x) = :r;2 + 4x - 12 (a) Construct a sign chart for I' At what x-value(s), ifany, does f has a local minimum? Construct a sign chart for 1". At what x-values. ifany. does / have an inflection point?

5 22. (7 points) You do not need to show any work on this problem. (a) Find the y-intercept, if there is one. Write it as an ordered pair. (b) Find the x-intercept(s), if there are any. Write each as an ordered pair. (c)

6 23 (8 points) Suppose you know the following about a function f (i) f is continuous with domain all real numbers. The only x- or y-intercept is the origin (0,0). f has no vertical asymptotes, but it has a horizontal asymptote, namely, y = O. The sign charts for I, I', and I" are as follows. f ~ 0, /' -/ 0, 1" J Carefully graph f to be consistent with the above information.

7 24 (4 points) Let /(%) = z3-9z2 + 15x. Note that /'(x) = 3%2-18% + 15 = 3(x -1)(% - 5). Suppose we wish to find the absolute maximum value off on the interval [0,4]. We have two potential strategies for finding an absolute maximum. Choose one of these two strategies and do the following. (a) Explain why it is valid to apply your strategy in this situation. In other words, state what conditions must be met in order for the strategy to be applicable. (b) Apply the strategy to find the x-value where the absolute maximum value of f on [0, 4] is found.

8 25. (7 points) An adventure company offers balloon rides at $300 each and sells an average of 2000 rides per season. Market research shows that for each dollar the company decreases the price, it will gain 10 customers. At what price should the company offer balloon rides in order to maximize its revenue from those rides? (Hint: The single critical value which you will get does lead to maximum revenue. Therefore you do not need to work through steps 4 and 5 of the optimization process, namely, finding the interval of possible values for the independent variable and testing to see whether the critical value truly leads to an absolute maximum.)

9 26. (4 points) Let /(:r:) = ~ 2eZ Find f'(x). You do not need to simplify your answer.