Download EXPONENTIAL FUNCTIONS

Part I
Pre-Calculus
Name:________________________
Your Final Exam is __________________________________



This review will be graded as completion and will count as your curve for you final exam. If you do 50% of your
review you will receive 50% of the curve.
You will be given a formula sheet for the exam. BUT you need to know what formula to use and when to use it.
You will also be able to use a graphing calculator for the exam.
VECTORS-(Sections 6.3, 6.4)
Find (a) u+v, (b) u-v, (c) 3u, and (d) 2v+5u. Write your answers as a linear combination of i and j.
1. u = <3,1>
v = <-2,-2>
2. u = 4i - j
v = -6i + 5j
3. u = <5,2>
v = <1,3>
4. u = 2j
v = 3i + j
Find the magnitude of each vector.
5. 8(cos60i + sin60j)
6. <4,1>
7. 3i + 2j
Find (a) the dot product of u and v and (b) the angle between the two vectors.
9. u = <2,3>
v = <1,4>
10. u = <-2,-1> v = <3,5>
11. u = 2 2 i
12. How do you determine if vectors are (a) orthogonal? (b) parallel?
8. –5i + 5j
v=2 2 j
Determine whether u and v are orthogonal, parallel, or neither.
13. u = <-2,4> v = <2,-4>
14. u = -i
v = i + 3j
15. u = <1,3>
v = <-1,2>
v = <3,9>
16. u = <2,1>
CONICS-(Sections 10.2, 10.3)
Graph and analyze the following equations:
 x  4
2
 y  3

2
1
17. ( y  2)  ( x 1) 2
4
18.
Classification:____________
Vertex:
______
Focus:
______
Directrix:
______
p-value:
______
Classification:____________
Verticies:
______, _____
Center :
______
a-value:
______
b-value:
______
c-value:
______
Foci:
______, _____
12
16
1
19.
 x  1
4
2
 y  2

1
2
1
Classification:____________
Verticies:
______, _____
Center:
______
a-value:
______
b-value:
______
c-value:
______
Foci:
______, _____
Find the standard form of the equation with the given attributes. Then graph the conic.
20. A parabola w/ focus (-3 , 0) & vertex (0 , 0).
21. A parabola w/ vertex (-2 , 1) & directrix x = 2.
22. An ellipse w/ vertices (  6, 0) & foci (  2, 0).
23. An ellipse w/ center (2 , -1), vertex (2, 0.5),
and minor axis of length 2.
24. A hyperbola w/ vertices (0,  2) & foci (0,  4).
25. A hyperbola w/ vertices (  1, 0) & asymptotes
at y =  5x.
Polar Coordinates and Graphs (Sections 10.7, 10.8)
2/3
3/4
26. What are the equations used to convert points
between polar and rectangular coordinates?
(Hint: There are 4 of them).
Plot the points and then covert to rectangular coordinates.
27. (-1,  6 )
28. (2, 7 )
4
90o
/2
/3
/4
5/6
/6
180o, 
0o, 0
7/6
11/6
5/4
7/4
5/3
4/3
270o
3/2
Convert the point to polar coordinates. Use radians
29. (3, 0)
30. (2,3)
Convert the rectangular equation to
polar form.
31. x2 + y2 = 64
32. xy = 3
Convert the polar equation to rectangular form and describe the graph of the polar equation.
33. r = 4
34. r = 2 sin θ
35. r = 3 csc θ
Function Analysis (Sections 2.1, 2.2, 2.3)
36. Write the quadratic function in vertex form: f(x) = (x-h)2 + k; sketch its graphs, identify the
vertex, and x-intercepts;
a. f(x) = -x2 –6x –5
b. f(x) = x2-4x +3
c. f(x) = x2 +4x
Equation:_________
Equation:_________
Equation:_________
Vertex:_________
Vertex:_________
Vertex:_________
x-int:___________
x-int:___________
x-int:___________
37. Find all real zeros of the polynomial functions and determine if the function is even or odd
(E/O); describe the end behavior (EB), and then sketch the graph.
a. f(x) = x5 +8x4 +16x3
b. g(x) = -2x3 – 2x2
c. h(x) = x4 + 6x3 + 9x2
Zeros:____________
Zeros:____________
Zeros:____________
E/0:__________
E/0:__________
E/0:__________
EB:__________
EB:__________
EB:__________
Part II
COMPLEX NUMBERS and RATIONAL FUNCTIONS-(Sections 2.4, 2.5, 2.6)
1. Simplify the following. Write your answers in standard form.
a. (5 + i) + (6 – 2i)
b. -5i + 2i
c. 5i(13- 8i)
e. ( 7 + 5i) – ( 4 – 2i)
f.
6i
4i
2. Find all possible rational roots of the following.
a. -4x3 + 8x2 – 3x + 15
b. 3x4 + 4x3 – 5x2 - 4
d. (10 – 8i)(2 - 3i)
g.
3  2i
5i
c. 4x2 -2x + 1
3. For each of the following: (a) State the domain, (b) identify all intercepts, (c) all vertical and
horizontal asymptotes, (d) plot additional points if necessary, e) sketch the graph.
a.
4
x3
b.
8
x  10 x  24
2
c.
2 x  10
x  2 x  15
2
4. Find all zeros (both real and imaginary) of the following possible polynomials.
a. x3-2x2 -24x
b. x3 + 4x2 + 25x + 100
EXPONENTIAL FUNCTIONS - (Section 3.1)
5. Identify each function as modeling exponential growth or decay, and then find the percent of
increase or decrease.
a. y = 2(4)
x
16
b. y =  
25
x
c. y = 3(0.68)x
d. f(x) = 5(1.025)x
Growth / Decay
Growth / Decay
Growth / Decay
Growth / Decay
Percent: _____
Percent: _____
Percent: _____
Percent: _____
6. For each of the following write an exponential equation using the formula A = P(1  r)t to model the
situation and then use the equation to answer the question.
a. Write an exponential function to model a population of 450 squirrels increasing at a rate of 6.5% each
year.
Equation: ______________________ How many squirrels will there be after 5 years? ________
b. You just bought a new car for $18,000. It will depreciate (lose value) at a rate of 14% each year.
Write an exponential function to model the situation.
Equation: ______________________ What will the car be worth after 4 years? ___________
nt
r

7. Use the compound interest formulas A  P1   or A  Pert to solve the following.
 n
a. $12,000 is invested in an account that pays 8% interest compounded continuously. Find the amount in
the account after 6 years.
b. Your grandmother gave you $500 for graduation. You deposited the money in an account that pays 6%
interest compounded monthly. How much will be in the account after 4 years?
LOGARITHMIC FUNCTIONS - (Sections 3.2, 3.3, 3.4)
8. Write in logarithmic form:
a. 43 = 64
b. 253/2= 125
c. e 8 = 2.225
9. Evaluate the following logarithmic functions:
a. log 1000
b. log 9 3
c. log2 (1/8)
d. e0 = 1
e. log3 34
d. logb 1
10. Use transformations to graph the following functions.
a. –log3(x) + 2
b. log4(x – 3)
c. log3(x-1) + 4
11. Go to page 240 in your book and write the 3 properties of logarithms in the space provided below
Expand the following
12. Expand the following logarithms.
a. log5 (5x2)
b. ln(3xy2 )
c. log 7
 x 3
x
4

d. ln 
2 
 xy 
13. Write as a single logarithm
a. log2 5 + log2 x
b. 3lnx + 2 ln(x+1)
c.
1
ln 2 x  1  2 ln( x  1)
2
d. 5ln(x - 2) – ln(x + 2) – 3lnx
e. 3log3 x + 4log3 y – 4log3 z
14. Solve the following exponential and logarithmic equations.
a. 3x = 243
b. log5 x = -3
c. (3x) = 20
d. (46-2x) + 13 = 41
e. 2x = 32
f.
g. log2 3 + log2 (x+2) = log2 (x+6)
h. log2 3x = log2 27
i. log6 x – log6 7 = 1
j. 4x – 2 = 64
k. 4 x  2  8
l. 94t = 33
m. 2log x + log 4 = 3
n. log3(5x-1) = log3(x+7)
o. 274x-1 = 93x + 8
1I
F
G
H3JK= 9
x
Part III
SEQUENCES and SERIES - (Sections 9.1, 9.2, 9.3)
1. Find the first 4 terms of the following sequences.
a. an = 5n - 3
b. an = 2an-1 +3; a1 = 6
c. an = 5an-1 + (an-1)2; a1 = 6
d. an = 3n +4
2. Simplify the factorial expression.
a.
25!
23!
b.
10!5!
8!12!
c.
n!
(n  1)!
d.
(n  2)!
n!
3. Write an expression for the nth term of the following sequences
a. 0,3,8,15,24,…
b.
2 4 8 16
, , , ……
3 9 27 81
4. Write a formula for an for the following arithmetic sequences, then find the 6th term of the
sequence
a. d = -2; a1 = 7
b. a1 = 2; a12 = 46
c. a1 = -4
a5 = 16
d. 10, 5, 0, -5, -10…..
5. Write an expression for the nth term of the following geometric sequences then find the 8th
term.
a. a1 = 4
r = (1/2)
C. a5 = 40.5;
b. a1 = 3; a5 = 3/64
a7 = 364.5
d. 567,189, 63, 21, 7….
6. Evaluate the following series.
a.
4
 j3 1
b.
j 0
 1
d.  4  
2
i 1 
10
g.
50
 6n
i 1
n
i 1
1
2 

i 1  3 


4
10
 4 
e.  7 i 
i 1  10 
h.
100
 2n
i 1
c.
5
i
 1 
f.  6

i 1  10 
i.
400
 2n  1
i 1
i