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Name__________________________________________________ Sec_____ Math 1005 Post-‐Exam2 ReviewFall 2013 The final exam covers material covered from the entire semester. This includes material from the guided notebook, homework, quizzes, and this review. You must show all your work to earn full credit. Calculators are allowed on the final exam, but phones or computers may not be used as a calculator. Use your notes, homework, quizzes, and previous review packets to study. These problems are from the sections covered after exam 2 which includes sec 4.4-‐4.6, 5.1-‐5.5, and 7.1. 1. Use synthetic division to divide 𝑓(𝑥) by 𝑥 − 𝑐 and then write 𝑓(𝑥) in the form 𝑓(𝑥) = (𝑥 − 𝑐)𝑞(𝑥) + 𝑟. a) 𝑓 𝑥 = −2𝑥! + 3𝑥! − 𝑥! + 7𝑥 + 2 , 𝑥 + 1 b) 𝑓 𝑥 = 6𝑥! − 2𝑥 + 3 , 𝑥 − 2 2. Use synthetic division and the remainder theorem to find the remainder when 𝑓 𝑥 = 𝑥! − 2𝑥! + 𝑥 + 6 is divided by 𝑥 − 3. 3. Use synthetic division and the factor theorem to determine whether 𝑥 + !
! is a factor of 𝑓 𝑥 = 2𝑥! + 5𝑥! − 8𝑥 − 5.
4. Find the remaining zeros of 𝑓 𝑥 = 3𝑥! − 5𝑥! − 2𝑥! + 6𝑥! − 𝑥 − 1 given that 𝑐 = 1 is a zero of multiplicity 3. Then rewrite 𝑓 𝑥 in completely factored form and sketch its graph.
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5. Use the rational zeros theorem to determine the potential rational zeros of the polynomial function 𝑓 𝑥 = 5𝑥! − 27𝑥! + 10𝑥! + 16 − 4𝑥. Do NOT find the zeros. 6. Find all complex zeros of the polynomial function, and write the polynomial in completely factored form. a) 𝑓 𝑥 = −6𝑥! + 19𝑥! − 15𝑥! − 3𝑥 + 5 b) 𝑓 𝑥 = 8𝑥! + 36𝑥! + 70𝑥! − 5𝑥! − 68𝑥 + 24 7. Solve the polynomial equation in the complex numbers. 𝑥! − 1 = 0 8. Write a polynomial 𝑓 𝑥 with real coefficients that satisfies the given information. Form a third-‐degree polynomial function with real coefficients such that 5𝑖 and 2 are zeros.
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9. Use the intermediate value theorem to show that the polynomial 𝑓 𝑥 = 2𝑥! + 5𝑥 + 1 has a real zero on the interval [−1,0]. 10. For 𝑓 𝑥 = −𝑥! − 5𝑥! − 3𝑥 + 9 , determine the end behavior, plot the y-‐intercept, find and plot all real zeros, and plot at least one test value between each intercept. Then connect the points with a smooth curve.
11. Determine the domain, the y-‐intercept if there is one, and any x-‐intercepts. 𝑓 𝑥 = !!!!!!!!!!!!!!!!!!
12. Find all vertical asymptotes, and create a rough sketch of the graph near each asymptote. 𝑓 𝑥 = !!!
!!!!!!!
13. Find the equation of all horizontal asymptotes (if any) of the rational function.
a) 𝑓 𝑥 = !!!
!!!!
b) 𝑓 𝑥 = !!!!!!!!!
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8x
-8-7-6-5-4-3-2-1
12345678y
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8x
-8-7-6-5-4-3-2-1
12345678y
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14. Use transformations of 𝑦 = !! or 𝑦 = !
!! to sketch the rational function. Label all intercepts, and find the equations of
all asymptotes. 𝑓 𝑥 = !!!!! ! + 3
15. Identify the coordinates of all removable discontinuities, and sketch the graph of the rational function. Label all intercepts, and find the equations of all asymptotes. 𝑓 𝑥 = !!!
!!!!!!
16. Find the slant asymptote of the graph of the rational function 𝑓 𝑥 = !!!!!!!!
17. Determine the correct exponential function of the form 𝑓 𝑥 = 𝑏! whose graph is given.
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8x
-8-7-6-5-4-3-2-1
12345678y
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8x
-8-7-6-5-4-3-2-1
12345678y
-5 -4 -3 -2 -1 1 2 3 4 5 x
-5
-4
-3
-2
-1
1
2
3
4
5f HxL
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18. Use the graph of 𝑦 = 4! or 𝑦 = 𝑒! and transformations to sketch each exponential function. Determine the domain and range, 𝑦-‐intercept, and the equation of the horizontal asymptote for each function. a) 𝑦 = 4!!! − 1
b) 𝑦 = −𝑒!! + 2
19. Solve each exponential equation by “relating the bases”. a) 3! = 81 b) !
!!= 2!
c) 𝑒!!!! = 𝑒!
d) 49! = !!
!!!
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8x
-8-7-6-5-4-3-2-1
12345678y
-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8x
-8-7-6-5-4-3-2-1
12345678y
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e) 𝑒!! ∙ 𝑒! = (!!!!)!
!!
20. Which of these investments will yield the greatest total amount? a) $2000 invested for 10 years compounded semiannually at 6% b) $3000 invested for 8 years compounded quarterly at 4.5% 21. An investment of $6000 earns 6.25% interest compounded continuously. What will the investment be worth in 15 years? Round your answer to the nearest cent. 22. Unbeknownst to the residents of northern Minnesota, a rare species of unicorn was introduced to the area in 1989. The growth rate for this species is 14.17%. By 2013, the unicorn population had grown to 150. a) How many unicorns were originally introduced in 1989? a) How many unicorns can be expected in the year 2049? 23. Write the exponential equation as an equation involving a logarithm. a) 2!! = !
!"
b) 𝑒!" = 𝑁
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24. Write each logarithmic equation as an exponential equation. a) log! 𝑥 − 1 = 3 b) ln 1 = 0 25. Evaluate without using a calculator. You must show your steps. a) log!
!( 4! )
b) 10!"# ! c) ln 𝑒 − ln(𝑒!) d) log!(6!") 26. Find the domain of each function. a) log!
!(2𝑥 + 8)
b) ln !!!!!!!!!"
27. Expand each of the following logarithms. Simplify if possible.
a. log! 𝑥!𝑦!
b. log! 27𝑥
c. log!!!!
!"!!!
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d. ln !!!!!!! !
!!
28. Write each expression as a single logarithm.
a. log! 80 − log! 5
b. 2 log! 𝑥 +!!log! 𝑦
c. log! 𝑥! − 5𝑥 + 6 − log! 𝑥! − 4 + log! 𝑥 + 2
29. Use the change of base formula and a calculator to approximate log! 21 to four decimal places. 30. Solve each of the following exponential or logarithmic functions.
a. 3! = 5
b. log! 𝑥! − 21 = log! 4𝑥
c. 4!!!!! = 64
d. 150𝑒!!! = 5
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e. log! 3 = −1
f. 7 2 − 10!!!! = 8
g. ln 3 + ln 𝑥! + !!!
= 0
31. Solve each of the following systems of equations using either substitution or elimination. If the system has infinitely many solutions, express the ordered pair in terms of x or y.
a. 5𝑥 + 4𝑦 = −6 8𝑥 − 4𝑦 = −72