APPM 1235 Exam 2 October 23, 2013 ON THE FRONT OF YOUR BLUEBOOK write (1) your name, (2) your instructor’s name, (3) your section number and (4) a grading table. Textbooks, class notes and all electronic devices are NOT permitted. Fully simplify all solutions. Box your answers. Please start a new page for each numbered problem. For problems #2 -­‐ #5 you must show a complete and valid solution method for full credit. 1. (5 points each, 40 points total) (a) The graph of a 5th-­‐degree polynomial function is shown below. The polynomial has real coefficients and has a zero at 2i. Find the equation of the polynomial. Leave your equation in factored form. (b) Find all of the solutions to the equation x 2 ln x − 3x ln x − 10 ln x = 0 . 50
(c) Find all of the solutions to the e€quation = 4 . 1+ ex
ln A
(d) True or false: log 3 A =
ln 3
€
x
(e) True or false: lne ln e = x €
For equations (f) through (h), find all asymptotes for the functions given below. You must label each asymptote as €
horizontal, vertical or slant. €
(g) g(x) = 2 − e x (f) f (x) = log(x + 2) €
2. (15 points) Consider the function f (x) =
−x 2 + 4 x − 4
x2 −1
(h) h(x) =
x 3 + 5x 2 − 3
x2 − x − 2
€
. (a) What are the horizontal and vertical asymptotes? Label your asymptotes as vertical or horizontal. €
(b) What are the x-­‐ and y-­‐intercepts? (c) Sketch the function. 3. (15 points) (a) Solve for x: ln(x 2 − 4) − 3 = ln(x + 2) (b) Write the following as a single logarithm: 5log 2 x + log 2 4 − log 2 (x 2 + 1) − 3 €
(c) Evaluate the following and fully simplify. [Hint: Your answer should be an integer.] €
1
log 25 5+ log 4 36 − log 4 16 − 2 log 4 3+ log 9 27 2
4. (15 points) The 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union sent several hundred kilograms of radioactive Cesium-­‐137 into the atmosphere. The area will not be considered safe until there is € material remaining. The function 100 kg or less of radioactive ⎛ 1 ⎞t / 30
f (t) = 1000⎜ ⎟
⎝ 2 ⎠
describes the amount f (t) in kilograms of Cesium-­‐137 remaining in Chernobyl t years after 1986. €
(a) What is the half-­‐life of Cesium-­‐137? €
€ Cesium-­‐137 will remain in the year 2046? (b) How much ⎛ 1 ⎞
(c) When will it be safe for humans in Chernobyl? Use the fact that log⎜ ⎟ ≈ −0.30 to get an approximate value for the ⎝ 2 ⎠
answer. [Hint: Your answer should be an integer.] 5. (15 points) The figure below shows a rectangle inscribed in Quadrant I of the coordinate plane. The left side of the € The upper right corner of the rectangle is the point (x, y) rectangle is on the y-­‐axis and the bottom side is on the x-­‐axis. on the line y = − 1 x + 2 . Answer the following questions to find the value of x such that the area A of the rectangle is a maximum. 2
€
(a) Find an equation for area A as a quadratic function of x. BUY-­‐OUT: If you are unable to find the function for area A in answer to part (a), you may request that the function be given to you so that you can complete the rest of the problem, at a cost of 6 points on this problem. If you would like to take this “buy-­‐out” option for part (a), stay in your seat, raise your hand, and an exam proctor will come to you. This offer expires at 8:15 pm on the night of the exam. (b) Find the value of x that maximizes the area A. (c) Find the maximum area A, and give the length and width of the rectangle with the maximum area. END OF EXAM