MAT170 - Exam 2 Review - Fall_2023 (updated)

MAT-170 Exam 2 Review Name: _____________________________ Exam 2 Review Page 1 © 2023 Carlson and Oehrtman 1. The population of Northfield, New Jersey is expected to increase by 1.8% every three years. The population of Northfield in 2012 was 8,624 people. Let n be the number of years since 2009. a. What is the 3-unit growth/decay factor? b. What is the 1-unit percent change? c. What is the ½ unit growth/decay factor? d. Approximate the population of Northfield in 2009. e. Write a function P that models the population of Northfield as a function of the number of years since 2009. f. Determine the number of years it will take for the population of Northfield to reach 9,172 people. 2. Let f ( x ) = 3 x ( x - 2) 2 (3 x + 9)(2 x - 14) . a. What is the degree and the leading term of this function? b. What do the degree and the leading term tell you about the end behavior of the function? c. What are the roots of the function f ? d. Compute lim 𝑥 → ∞ ?(?) and lim 𝑥 → −∞ ?(?) e. Explain what happens to the behavior of the function if we change the 3? to −3? . f. What happens to the behavior of the function if we remove the power 2 on the factor (? − 2) 2 g. What happens to the behavior of the function if we make BOTH of these changes , i.e., if we change the 3? to −3? , and we remove the power 2 on the factor (? − 2) 2 3. Kelsie set p pennies next to a checkerboard. She then placed triple that number of pennies on the first square. Then she placed on the next square triple the number of pennies that were on the first square. She continued this pattern of always placing on the next square triple the number of pennies on the previous square. How many pennies will be on the 5 th square?

MAT-170 Exam 2 Review Name: _____________________________ Exam 2 Review Page 2 © 2023 Carlson and Oehrtman 4. Use the graph of the function, f , shown below, to answer the following questions. a. Given the graph of f as shown above, what are the roots of f , and what are their multiplicities? b. What is the value of f (0), and what is the significance of this value in terms of the graph of the function? c. Approximate the inflection points of the function. d. Approximate on what interval(s) the function is concave down. e. Approximate on what interval(s) the function is concave up. f. Approximate on what interval(s) the function is decreasing g. Approximate on what interval(s) the function is increasing h. Approximate on what interval(s) the function is negative i. Approximate on what interval(s) the function is positive j. Compute lim 𝑥 → ∞ ?(?) and lim 𝑥 → −∞ ?(?) k. Does this polynomial have an odd degree or an even degree ? l. What is the sign of the leading coefficient of this polynomial ? 5. Define a polynomial function g with the following characteristics: g has roots of multiplicity 2 at x = 4 and x = 1 , a root of multiplicity 1 at x = −3 , and passes through the point (0, −4 ). 6. Describe in words what log 2 (32) represents.

MAT-170 Exam 2 Review Name: _____________________________ Exam 2 Review Page 3 © 2023 Carlson and Oehrtman 7. Consider Segment A and Segment B shown below. a. The length of Segment A is _______ times as long as the length of Segment B. b. The length of Segment B is _______ times as long as the length of Segment A. c. The length of Segment B is ______ % of the length of Segment A. d. The length of Segment B is ______ % longer than the length of Segment A. 8. A customer buys a shirt that is discounted 23%. What percent of the original price will the customer pay for the shirt? 9. As the value of x increases from x = 3 to x = 5, the value of g(x) increases by 25%. (a) g(5) is ______ times as large as g(3) . (b) Assuming that the values of the function are growing exponentially, how many times larger will the value of g(7) be as compared to the value of g(3) ? (c) What is the one-unit growth factor for the values of g(x) ? (d) By what percent does the value of g(x) grow as x increases from x=7.25 to x=8.25 ? 10. Suppose h is an exponential function such that h (-3)=4 and the 3-unit growth factor of h is 2.5. a. What is the value of h (0)? b. What is the value of h (3)? c. What is the value of the 1-unit growth factor? d. Define the function h . 11. A rock is thrown upward from a bridge that is 20 feet above the surface of the lake. The rock reaches its maximum height above the surface of the lake 0.25 seconds after it is thrown and reaches the surface of the lake 1.50 seconds after it was thrown. Define a quadratic function, f , that gives the height of the rock above the surface of the lake (in feet) in terms of the number of seconds elapsed since the rock was thrown, t .

MAT-170 Exam 2 Review Name: _____________________________ Exam 2 Review Page 4 © 2023 Carlson and Oehrtman 12. Convert between log form and exponential form for the following expressions: Log form Exponential form ??? 5 (125) = ? 2 𝑥 = 64 ??? 2 (?) = −3 3 −4 = ? 13. A lab ran four different experiments involving insect populations. For each experiment described below, write an equation that describes the insect population in that experiment as a function of the number of days t since the start of the experiment . (a) There are initially 1000 insects, and the population f increases by 20% each additional day (b) There are initially 1000 insects, and the population g decreases by 20% each additional day (c) There are initially 1000 insects, and the population h increases by 20 insects each day (d) There are initially 1000 insects, and the population m doubles each day 14. A salt crystal is initially 3.2 inches long, and the length L of the crystal increases by 40% each day. (a) Determine the equation that describes the length L of the crystal t days after it started growing. (b) Solve for the exact number of days that are needed for the crystal to become 10 inches long. Give the exact expression for the value of t (i.e., use a log expression, not a decimal approximation). 15. E ach graph shown at the right represents the height of the water (in inches) in a bottle as a function of the volume of the water (in milliliters) in that bottle for one of the bottles shown below. For each of the graphs, determine which is the corresponding bottle. Graph 1 Graph 2

MAT-170 Exam 2 Review Name: _____________________________ Exam 2 Review Page 5 © 2023 Carlson and Oehrtman 16. Suppose the following graph of the function, f, represents the number of dollars in Meg’s bank account as a function of the number of days, t, since January 1, 2012. Number of days since January 1st a. Evaluate f (30). What does this value represent in the context of the problem? b. What is the meaning of f (60) = 1400 in this context? c. Determine whether the following statements are true or false. i. As the number of days since January 1 st increases from 60 to 75 days, for equal changes in the number of days elapsed, the changes in the number of dollars in Meg’s bank account are decreasing. ii. 80 days after January 1 st , the amount of money in Meg’s bank account increases at a constant rate of change. iii. As the number of days since January 1 st increases from 0 to 20 days, the amount of money in Meg’s bank account is increasing at an increasing rate. 17. The weight of a puppy is currently 5 pounds, and the puppy’s weight increases by 18% each month. After how many months will the puppy weigh 15 pounds? Round your answer to the nearest two decimal places. f 1800 1600 1400 1200 1000 800 600 400 200 20 40 60 80 100 Number of dollars in Meg’ s Bank Account