Name: Mathematics E-15 Exam I February 21, 28 Problem Possible 1 10 2 10 3 12 4 12 5 10 6 15 7 9 8 8 9 14 Total 100 Instructions for Proctor Please check that no student is using a TI-89 calculator, a TI-Nspire CAS calculator, or a calculator with a QWERTY keyboard. If any student leaves the room before completing the exam, that student must leave his or her phone with you. You must show all work to receive credit. Instructions for Student Give exact answers (such as ln 5 or e 2 ) unless requested otherwise. Calculators with symbolic differentiation capabilities or QWERTY keyboards (including, but not limited to, the TI-89 and TI-Nspire CAS) are not permitted. No electronic devices (including phones) other than calculators are permitted. Please write your answers in the spaces provided on the pages containing the questions. Use extra paper only if you need additional space. If you write more than one answer to a given question, please cross out or erase all those except the one you wanted counted as your final answer.
3 1. Consider the function f(x) = 4 5x 2. (a) Compute the average rate of change of f on the interval [1, 1.] correct to three decimal places. (b) Using the limit definition of the derivative, find f (x). (c) Find the equation of the tangent line to f at x = 1. 2. For the following, assume that t is measured in hours. In each case, write an exact formula for a function that has the properties listed. (a) The function f(t) is 88 at t = 0 and grows 1% every hour. (b) The function g(t) is 7 at t = 0 and grows 300% every hour. (c) The function h(t) is 6666 at t = 0 and triples every 10 minutes. (d) The function j(t) is 555 at t = 0 and decays 20% every day. (e) The function p(t) is 44.4 at t = 28 and decays 1% every 2 minutes. (f) The function r(t) is 3.33 at t = 22 and doubles four times per day. (g) The function s(t) is a sinusoidal function that oscillates from a high of 11 to a low of 1 with a period of 9 hours and is at 1 at t = 0.
3. Suppose that the function f(x) gives the number of minutes Sonya Thomas needs to eat x pounds of cheesecake. Carefully interpret each of the following, including units in your answers. (a) f(11) = 9 [These numbers are real, by the way.] (b) f 1 (12) Your answer should take the form f 1 (12) is... (c) f (11) = 0.8 (d) Using the information in (a) and (c) above, estimate f(13) and explain its meaning. 4. Find all exact solutions to each of the following. If an equation has no solution, say so explicitly. (a) e 28x = 7 (b) ln(3x) = 28 (c) 5 log 5 x2 = 36 (d) x 3 e 2x 17xe 2x = 0 (e) Simplify tan(arcsin(7x)) by rewriting it as an expression that contains no trigonometric or inverse trigonometric functions. Hint: drawing a triangle may help.
5. For the graph of f shown, carefully sketch a graph of f. f(x) semicircle f (x) 6. Shown is a graph of f (x), not f(x). The entire domain is visible. If for one or more of the questions below, we do not have enough information to answer, write that the answer cannot be determined. Note that the graph of f (x) shown is concave down on (c, e) and (g, i) and concave up elsewhere. At which labeled x-value(s) is (a) f greatest? (b) f least? (c) f zero? (d) f greatest? (e) f least? (f) f zero? (g) f greatest? On which interval(s) is (a) f positive? (b) f decreasing? (c) f concave down? (d) f positive? (e) f positive? (f) f decreasing? f (x) a b c d e k0 1m g 0 1 f h i j (h) f zero? (i) f decreasing most rapidly?
7. Shown is a graph of y = f(x). It has exactly two x-intercepts and exactly one horizontal asymptote. One other point is also labeled. (-7,13) y=f(x) y=8 (-4,0) (6,0) (0,-5) For each function below, give the x-intercept(s), y-intercept, and y-value of the horizontal asymptote. (a) g(x) = f(x) 13 i. x-intercept(s) ii. y-intercept iii. horizontal asymptote (b) h(x) = 3f(2x) i. x-intercept(s) ii. y-intercept iii. horizontal asymptote (c) j(x) = f(x 4) i. x-intercept(s) ii. y-intercept iii. horizontal asymptote 8. Suppose that the domain of f(x) is all real x and that f (x) = e x 28 for all real x. (a) Is it possible to determine the x-value at which f achieves its lowest y-value? If so, give this exact x-value; if not, explain why it is not possible. (b) Is it possible to determine the lowest y-value that f achieves? If so, give this exact y-value; if not, explain why it is not possible. (c) For what x-values is f concave up? Briefly justify your answer.
9. The domain of y = f(x) is (, 300) (that is, x < 300), the range of f(x) is ( 100, 200) (that is, 100 < y < 200), and f(x) is always decreasing (which means that f 1 (x) will be a function). Below are some known values of f(x) and f (x) at various x-values. f( 4) = 37 f( 2) = 28 f(0) = 6π f(2) = e f(4) = 17 f ( 4) = 5 f ( 2) = π f (0) = 2e f (2) = ln 7 f (4) = 2 For each of the following expressions, state the numerical value of the expression, or that the value of the expression is known to be undefined, or that more information is needed in order to determine the value of the expression. (a) f 1 (2) (b) f 1 (222) (c) f 1 (f( 8675309)) (d) lim f(x) x (e) lim h 0 f( 2 + h) 28 h (f) g (0) if g(x) = 5f(x + 4) (g) h (1) if h(x) = f(2x) 1 BONUS (2 points - no partial credit.) Find the exact value of 1 + 1 1+ 1+ 1+... 1 expression continues infinitely in this same pattern. where the... indicates that the