M Assignment Functions, Exponential, Logarithms Dr. Taylor Submit your typed up work at the assignment upload page. Support your answers with your work, indicating any tools that you used and uploading any support files. Do at least 8 of the following 11 problems. Computational devices are strongly recommended for many of these problems. 1. Find two real numbers that satisfy: six times the smaller one plus twice the larger one is equal to 36; the product of the numbers is 18. 2. Find all local maxima, local minima, and zeros of x3 5x2 + x + 3. 3. Find the extrema (x and y values, type) and zeros of .01(x + 1.5)3 (x 2)2 (x + 3)4 30 x2 3 . What is the domain of f ? What do the values of 4. We are given f (x) = (x + 1)(x 10) f tend to as x becomes massive (|x| very large)? 5. Solve for x in terms of : x2 = x . Assume that x > 0 and > 0. 6. A factory can clean a percentage P of its output. The cost is given by the function 30 P + . The domain of this function is 0 < P < 100. C is in millions C(P ) = 2 (P 100) P of dollars. (a) What is the minimum cost and what percent of output is cleaned at that level? (b) If the company wants to keep costs below 3 million, what range of P is allowed? 7. A test is given to a large group of students. The results are given by the function 2 2 f (x) = 4e(x5) + 3e2(x8) where x is the score in tens (e.g., x = 5.6 represents 56%) and f is in hundreds of students. f has domain 0 < x < 10. (a) How many students received a 73%? (b) What range of scores had more than 250 students receiving that score? (c) What is the most common score? (d) Find all local maxima and minima of this function. (e) (Bonus) How many students took this test? 8. Submit a graph of x3 3x2 + 2x. Make sure the graph contains the zeros, maxima, and minima with labels for each. 9. Describe, with pictures and text, the massive x behavior of 2x (a) f (x) = 2 x +1 5x3 + 2x2 (b) g(x) = x2 + 2 QM Assignment Functions, Exponential, Logarithms Dr. Taylor 1 at the points x5 x = 0, 0.5, 1, 1.5, ...., 10, 20, 30, ..., 90, 100. For each x, sum the five values at each point and indicate which function(s) dominate in the sum for that x value. If we stopped at 10, would the data tell us which function would eventually have the dominant values? 10. Compute a table of values that evaluates x, ex , ln(x), x10 , and 11. To understand the power of logarithm, one must deal with large numbers. 2 (a) Graph f (x) = ex 7x+5 + 70x for x from 10 to 10. Try to fit as much of the function graph as possible in the picture by varying the y-axis range. (b) Do a separate graph of g(x) = ln(f (x)) for x from 10 to 10. Try to fit as much of the function graph as possible in the picture by varying the y-axis range. (c) Which one is easier to see the whole function over the given x range? Page 2