Rachel Cabral 10/31/2016. Mid-semester Exam. Find the average rate of change of the functions, below, on the indicated intervals:

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1 Rachel Cabral 10/31/2016 Mid-semester Exam Question 1. Find the average rate of change of the functions, below, on the indicated intervals: 1) f(x) = 5-2x 2 on [-2, 4] The answer is -4. 2) g(x) = x 2-4x + 1/x on [-3, 1] The answer is -7/12. 3) h(x) = 1/x + 3 on [1, 1+h] The answer is -1/4(h+4). 4) k(x) = 3x 2-2 on [x, x+h] The answer is h 2 + 2xh - 4/ h. For each of these problems I used the rate of change formula which is y 2 - y 1 / x 2 - x 1. The x values are given for each problem so by plugging each x into the original problem I found the y values to plug into the rate of change formula. By using the rate of change formula I could find and simplify the answer to each of these problems. Question 2. (a) Calculate f(g(h(x))) in terms of x, where: f(x) = x g(x) = 1/x h(x) = x + 3 The answer is (1/(x 2 + 6x + 9)) +1. I found this answer by first finding g(h(x)) which is 1/x +3 then I used that answer to find to f(g(h(x))) to make things easier on myself. (b) Write a formula for the function that results when the function f(x) = 1/x is vertically stretched by a factor of 8, shifted the right 4 units, and shifted up 2 units. The answer is f(x) = 8(1/x - 4) + 2. I referred back to the textbook to help me get my answer.

2 (c) For the function f(x) = x 2 + 1, find a domain on which f is one-to-one and non-decreasing, and then find the inverse of f restricted to that domain. The answer is that the domain = [0, ) and the inverse of f is f -1 (x) = x - 1. I referred back to the textbook to help me find the answers to this problem. Question 3. (a) At noon, a barista notices she has $20 in her tip jar. If she makes an average of $0.50 from each customer, how much will she have in her tip jar if she serves n more customers during her shift? The answer is f(x) = 0.5n I used the form of y = mx + b to plug in the given values to get this equation. (b) Find the point (x, y) at which the line y = 7/4x + 457/60 meets the line y = 4/3x + 31/5. The answer is (-17/5, 5/3). First I set the equations equal to each other and solved for the x value then plugged that answer into one of the original equations to find the y value. (c) Write an equation for a line parallel to g(x) = 3x - 1 and passing through the point (4, 9). The answer is g(x) = 3x I referred back to the textbook to help me get this answer using an example as a guide. So by acknowledging that the m would equal 3 and the x would equal 4 I plugged that into y = mx + b keeping y as 0 to solve for b, I got my answer. Question 4. (a) In 2003, the owl population in a park was measured to be 340. By 2007, the population was measured again to be 285. If the population continues to change linearly,

3 1) Find a formula for the owl population, P(t) in years t from The answer is P(t) = x. I found this answer by plugging in the given terms into the rate of change formula. I used (0, 340) and (4, 285) as my x and y values. 2) What does your model predict the owl population to be in 2016? The answer is that the owl population would be approximately 161 in I got this answer by plugging in 13 for x in the equation P(t) = x, I used 13 since = 13. (b) You are choosing between two different window washing companies. The first charges $5 per window. The second charges a base fee of $40 plus $3 per window. How many windows would you need to have for the second company to be preferable in terms of total cost? The answer is you would need 21 windows for the second company to be preferable in terms of total cost. I found this answer by finding the equation for each company then setting them equal to each other to solve for x, the x value would be when both the costs are equal. So by adding one to the x value of 20 (21) I found that by plugging 21 into each of the company equations the first company cost would be $105 and the second company cost would be $103. Question 5. (a) Find the vertical and horizontal intercepts of f(x) = 2x 2 -x -6. The answer is the horizontal intercepts are x = 2 and x = -3/2, and the vertical intercept is y = -6. To get the horizontal intercepts I replaced y for 0 and solved for x. To get the y intercept I replaced the x values with 0 and solved for y. (b) What is the long run behavior of g(x) = -2x 3 + x 2 - x + 3? The answer is as x, f(x) -, and as x -, f(x).

4 I referred to the textbook to help me get this answer. (c) A farmer wishes to enclose three pens with fencing, as shown below: If the farmer has 700 feet of fencing to work with, what dimensions will maximize the area enclosed? The answer is the dimensions will have to be 175 ft by 87.5 ft. I found this answer by manipulating the formula A = LW to find the variable L then using that value to find W. Question 6. (a) A rectangle is inscribed with its base on the x axis and its upper corners on the parabola y = 3 - x 2. What are the dimensions of such a rectangle that has the greatest possible area? The answer is that the dimensions for the greatest possible area would be 2 by 2 so the area would be 4. I got this answer by manipulation the formula A = BH with the given information and with the help and mathematica and the textbook. (b) A scientist has a beaker containing 30 ml of a solution containing 3 grams of potassium hydroxide. To this, she mixes a solution containing 8 milligrams per ml of potassium hydroxide. 1) Write an equation for the concentration in the tank after adding n ml of the second solution. The answer is that the equation would be y = n. I got this answer by plugging the values into y = mx + b 2) Find the concentration if 10 ml of the second solution has been added. The answer is 77 mg/ml. I got this answer by plugging it into the previous equation.

5 3) How many ml of water must be added to obtain a 50 mg/ml solution? The answer would be 30 ml of water. 4) What is the behavior a n and what is the physical significance of this? The answer would be as x f(x), the significance of this is that is shows the concentration of the solution. Question 7. (a) The amount of area covered by blackberry bushes in a park has been growing by 12% each year. It is estimated that the area covered in 2009 was 4,500 square feet. Estimate the area that will be covered in The answer is 5.33 * feet squared. I used the equation f(x) = a(1+r) x to plug in the given values to find the answer to this problem. (b) A population of bacteria is growing according to the equation P(t) = 1600e 0.17t, with t measured in years. Estimate at what time t the population P(t) will exceed The answer is t will exceed 3443 at approximately 4.51 years. I referred to an example in the book to help me solve this problem. Question 8. (a) The population of Algeria was 34.9 million in 2009 and has been growing by about 1.5% each year. If this trend continues, when will the population exceed 45 million? The answer is the population will exceed 45 million when it is approximately years. I used the equation f(x) = a(1+r) x and referred to the textbook to help me solve this answer.

6 (b) A wooden artifact from an archeological dig contains 15 percent of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.) The answer is the artifact was made approximately 15,679 years ago. I used an example from the textbook to help me through this problem to get the answer.