MATH 130: Exam 2 Review. 2.Find the equation of the piece wise function graphed below:

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1 Name: Section: MATH 130: Exam 2 Review Exam 2 covers Activities 9-15 in your course notebook. The purpose of this review is for you to customize your study time to focus on areas you need more practice. For more practice, please review your old homework assignments, quizzes, and activities. 1. Graph following piecewise defined functions: a) ff(xx) = 2(xx + 1)2 3, xx 1 3 b) xx 1, xx > 1 4 ff(xx) = xx 3 2, xx < 2 5, 2 xx 0 3xx + 5, xx > 0 2.Find the equation of the piece wise function graphed below: a)

2 B) 3. For polynomial, rational, exponential, and logarithmic functions, be able to explain or find: a. Domain b. Range c. Horizontal asymptote(s) d. Vertical Asymptote(s) e. Roots/x-intercept(s) f. Y-intercept(s) g. Graph 4. A) Given ff(xx) = 2xx 3 xx 2 15xx + 18 and that ff(2) = 0 is a factor of, find all the roots for ff(xx). B) Given that 2xx 1 is a factor of gg(xx) = 6xx 3 + 5xx 2 2xx 1, find all the roots for gg(xx). C) Given h(xx) = 4xx 3 10xx 2 + 6xx 1 and that ff 1 = 0 is a factor of, find all the roots for 2 h(xx). 2

3 5. For each polynomial given below find the following if they exist. If they do not exist you must say so i) Domain ii) x-intercept(s) iii) y-intercept iv) intervals where ff(xx) is positive or negative v) Graph a) ff(xx) = 2xx 3 + xx 2 9xx 2 b) ff(xx) = xx 4 xx 3 6xx 2 c) ff(xx) = xx 3 7xx 6 3

4 6. What do you need to know about a rational function in order to graph it? Why are the components you listed important? 7. Graph each of the rational functions a) ff(xx) = 3xx2 +1 xx 2 +xx 6 b) xx2 +xx 6 3xx 4 c) ff(xx) = 5xx+2 3xx 2 +2xx+6 d) 2xx 3 xx+3 4

5 e) xx 2 16 xx 2 xx Find an equation for a rational function that meets the properties given: a) A rational function with vertical asymptotes at xx = 1 and xx = 1, horizontal asymptote at yy = 0, and x-intercept at 4. b) A rational function with domain (, 4) ( 4, 2) (2, ), a double root at xx = 1, horizontal asymptote at yy = 1, and y-intercept at 1 8. c) A rational function with a hole at xx = 3 2, a vertical asymptote at xx = 1, x-intercepts at xx = 2 and xx = 2, no horizontal asymptote and y-intercept at 2. 5

6 d) 9. Graph Each of the following functions: What characteristics of the graph are essential to understand the behavior of the graph? a) ff(xx) = 4 ln(xx + 2) 3 6

7 b) ff(xx) = 3 ln(2xx + 1) + 1 c)ff(xx) = 2ee xx+2 3 d) ff(xx) = ee xx

8 10. Find the inverse of the functions given above. 11. In what ways is the inverse related to the original functions? 12. Justify whether the following graphs are rational. exponential or logarithmic functions. Use the Domain, Range, Asymptotes, and X- and Y- intercepts in your answer. a) b) c) d) 8

9 e) f) 13. Use the properties of logarithms to expand the expression as sum, difference, and/or constant multiples of logarithms. (You may assume all variables are positive). a) log(xxxxzz 2 ) b) ln xx2 yy 2 zz 3 3 c) log( xxzz2 ) d) ln xx yy yyzz Condense each logarithmic expression as one logarithm. a) log 7 + log xx 3 log yy b) 2 ln xx 7 ln yy ln zz c) 5 log xx log yy 2 log zz d) ln xx + 1 ln(xx + 2) 3 ln(xx 7) 3 2 9

10 15. Solve for xx exactly. (Do not give a decimal approximation- do not use your calculator) a) ee xx2 +6 = ee 3xx b) ee 2xx 3ee xx = 0 c) 5(3 xx 1 ) 2 = 3 d) 5(ee) xx+1 = 2(3) xx e) (3) xx+4 = (5) 3xx f) log(xx 2 ) 5 log(xx) = 0 g) 3 ln(2xx + 1) 5 = 1 h) log(5xx 1) log(xx 2) = 2 i) ln(xx + 5) = ln(xx 1) ln (xx + 1) j) log(xx 1) + log(xx 3) = log (8) 10

11 16. If you invest $5,000 at 5.6%, how much will you have at the end of 10 years, if the interest is compounded a) annually b) quarterly c) monthly d) continuously e) Find the effective interest rate for parts a)-d) f) compare your answers from a)-d) What do you notice? Provide an explanation why this might be. g) Notice your answers in part e) Is there a pattern? If so, why? 17. A) Write a formula for $500 invested in a savings account at a nominal rate of 4% compounded daily. b) How long will it take for the investment to reach $800? c) What is the effective interest rate. d) Write a formula for $500 invested in a savings account at a nominal rate of 4% compounded continuously. 11

12 e) How long will it take for the investment to reach $800 f) Find the effective interest rate. 18. Sarah invests $800 into a savings account compounded monthly at a 5% interest rate. Her friend Paul invests $700 compounded continuously. Who has made the better investment? Justify your answer. 19. After the release of radioactive material into the atmosphere from a nuclear power plant at Chernobyl (Ukraine) in 1986, the hay in Austria was contaminated by iodine-131. Iodine-131 has a half-life of 8 days. It is safe to feed the hay to cows when 10% of the iodine-131 remains. How many days will a farmer need to wait to use this hay? You may assume iodine- 131 decays continuously. Round your answer to the nearest tenth. 12

13 20. The Rhind papyrus, named after A. Henry Rhind, contains most of what we know about ancient Egyptian mathematics. An analysis of the papyrus shows that approximately 16% of its original Carbon-14 has decayed. Given that Carbon-14 decays continuously and has a halflife of 5730 years, find the age of the papyrus. 21. A population of mice is growing continuously, doubling every 60 days. How long would it take a population of mice to grow from 20 mice to 500 mice? 13