Download Final Review Chapter 1 and 2

Final Review Chapter 1 and 2
Chapter 1
Find the domain of the function
1. f(x) = (x + 4)1/2
2. f(x) = (x)1/2/(x – 3)
identify the intervals on which the function is increasing and decreasing:
3. -(x +3)2 = f(x)
4. x2/ (x2 – 2)
Prove algebraically whether function is odd, even, or neither
5. g(x) = x2 – 2x + 3
6. x3/(x2 – 1)
Find all vertical and horizontal asymptotes of
7. y = 1/ (x2 – 4)
8. Y = x/ (x2 – 4x + 4)
find x and y intercepts where applicable
Find f(g(x)) and g(f(x)) state the domain of the composite functions
F(x) = 2x2 and g(x) = (x)1/2 - 1
F(x )= x2 , g(x) = 1/(x+1)
Consider the set of all ordered pairs defined the equations a. find the points
determined by t = -2,-1,0,1,2, and b. find an algebraic expression between x and y c.
Graph the relation in the xy plane
1. x = t + 2, y = 2t -1
2. X = t-1, y = t2
Find an inverse function f-1
F(x) = x / (x + 1)
f(x) = (x + 1) /(x-1)
F(x) = (2x – 3) ½
Find equations for reflections and stretches and shrinks
Find an equation for the reflection of each function across each axis
F(x) = (3x +2)/(3x2 – 5)
f(x) = ( x+ 2) ½ + 1
Find the equation for the following transformations
F(x) = x2 + 2x, vertical stretch by 3
f(x) = x2 – x, horizontal stretch by factor of 2
Describe transformations of parent graph y = x2
Y = (x+3)2 – 5
y –(3x)2 – 4
Chapter 2:
Writ an equation for the linear function f that satisifies the given conditions:
F(2) = -3 and f(6) = 3
F(0) = 2 and f(3) = 4
Find he vertex and the axis of symmetry. Write the equation of the function in vertex
form
F(x) = x2 + 2x
f(x) = x2 – 12x + 3
F(x) = x2 + 4x + 7
Find the zeros of the polynomials:
F(x)= x3 + x2 – 6x
F(x) = x3 -9x2 – 10x
Use the remainder theorem to divide f(x) by the given linear expression. Interpret
the answer.
F(x) = 2x2 + 4x + 3: x -2
F(x) = -x2 – 4x + 12; x -1
F(x) = 2x2 -4x + 2; x – 1
Factor the following polynomials with complex zeros. Write its linear factorization.
F(x) =x5 – 3x3 + 6x2 – 28x + 24
F(x) = x4 –x3 + x2 – x
f(x) = x4 - 16
Find the intercepts, end behavior, holes and asymptotes of each rational expression
F(x) = (x2 -1)/(x2 – 16)
f(x) = (2x2 + 5)/(x2 – 9)
Solve equation in one variable:
(2x-1)/x = 1
1/x + 1/(x+1) = 1
(2x +1) /x = 3x
(x+3)/(x + 4) = x
3/(x-1) – 2/(x-3) = 5/(x2 – 4x + 3)
2/x + 5/(x-1) = 8/(x2 – x)
solve each inequality
x2 < -x + 6
x3 < -2x2 + 3x
3/(x-2) – 2 < 0
(x + 1)/(x-3) > 3
Chapter 3 exponential, logistic, and logarithmic functions
Suppose the half -life of a radioactive substance is given and an initial amount is
given. Find the time when there will be 15 grams of he substance remaining.
Half -life 30 days, initial amount 40 g
Half -life 25 days, initial amount 20 g
Using properties of logrithms, write each expression as a sm or difference of logs or
multiples of logs
Ln (x2 + 1)/x
ln (x-2)2 /(x+2)2
Ln ( x2 – 100)1/2/(x2)
Using properties of logs, write each expression as a single logarithm
Lnx2 – lnxy
lnx2 – 4lnxy
3lnx2 -2ln(xy)
Solve the following equations:
24(.5)x/3 = 12
64(3/8)x/4 = 9
log x = 2
log 2x = 10
logx2 = 3
How long will it take for the given investment to grow to the given goal? Interest is
compounded monthly.
Principal: $1,000, rate: 7%, target amount $1,500
Principal: $1,600
rate: 9%, target amount $2,400
Interest is compounded continuously. Find the value of each investment after 5
years
Principal: 45,000, rate 74
Principal $3,000 rate 7%
Chapter 4
1. Let theta be an acute angle such that the cos theta = 4/7. Find the other five
trig functions.
2. Tan theta = 4/3. Find the other five trig functions
3. A right triangle with hypotenuse 10 includes a 650 angle. Find the measure of
the other two angles and the lengths of the other sides.
4. 4. From a point 60 feet from the base of the building, the angle of elevation to
the top of the building is 700. Find the height of the building.
5. Let theta be any angle in standard position whose terminal side
contains (-2,5) Find the six trig functions of theta.
6. Let theta be an angle whose terminal side contains (-2,-5) find the other six
trig functions.
7. Find the sin and tan of theta by using the given information
8. Cos theta = -4/5 and tan theta > 0
9. Cos theta is -3/5 and tan theta <0
10. Graph two periods of cos (2x)
11. Graph two periods of sin((1/2) x).
12. Describe the amplitude, phase shift, and period of f(x) = 2sin x
13. Describe the amplitude, phase shift, and period of f(x) = cos (2x – ∏)
14. Sin ∏/3 =
sin (2/3)∏ =
cos ((11/12)∏ =
Know all special angles
Chapter 5
1. tan theta = 1, and sin theta < 0, find sin and cos
2. tan theta = 4, cos theta > 0 , find cos and sin
3. simplify the epxressions
a. cos3 x + cosx sin2x
b. cos2x +cos2xtan2x
c. 2sec2x -2sec2xsin2x
d. sin2xcot2xsecx +sin2x
4. Prove the following identities
a. (1-cos2x) + 2cosx = sin2x + 2cos2x
b. (secx ÷cscx)cotx = 1
c. 1 –cos2x) = - sin4 x
1 –csc2x = cos2 x
d. __1_____ = 1 + sin t
e. 1 – sin t = cos2t
f. 1+ cost = (1+ cos t)2and
1 –cost = cos2t
Using sum and difference prove:
a. cos (x – 3∏/2) = -sinx
b. sin(x – 3∏/2) = cos x
c. sin (2∏ - x) = -sin x
d. sin(x + ∏) = -sinx
e. cos(x +∏) = - cos x
Prove ( double angle identities):
a. Cos 4x = cos22x – sin22x
b. Sin 4x = 2sin2xcos2x
c. cos 6x = cos23x – sin23x
Law of Sines
Solve ∆ABC given the following:
a.  and a 
b. and a = 30
c. a = 10, b= 11 and A = 410
d. a = 9, b = 11 and 
Remember that sin-1willnever recover an obtuse angle.(the ambiguous case)
a. a = 10, b = 12, and A = 350
b. a = 9, b = 11, and A = 340
Law of cosines:
Solve ∆ABC given the values
a. a = 15, b = 7, and C = 400
b. a = 20, b = 20, and C = 350
c. a = 6, b= 12, and C = 700
d. a = 10, b = 15, and c = 11
e. a = 24, b 20, and c = 16
Using Heron’s formula
Find area of triangle with given sides :
a. 15,15,25
b. 24,30,45
c. 18,18,30
Chapter 6 Vectors:
Find the unit vector in the direction of the given vector:
a. < 2,-1>
b. <2,3>
c. (-4,3>
find the components of the given vector:
a. u with a direction angle 1050 and magnitude 8
b. w with a direction angle 2080 and magnitude 4
c. v with a direction angle 750 and magnitude 7.5
find the magnitude and direction angle of
a. u = <1,4>
b. v = <-5,4>
c. c = < -3,2>
Find angle between the given vectors:
a. u = <2,5> and v = <3,1>
b. u = <4,3> and v = <-3,-1>
c. u = <6,2> and v = <-1,3>
Parametric motion
Eliminate the parameter and identify the graph of the parametric curve:
X=2–t
Y = 2t + 1
X=t–2
Y= 1 + 2t2
3cost = x
3sint = y
Projectile motion: Use parametric equations to model the motion of the object.
a. A gymnast throws a club straight up in the air, then catches it. The initial
speed of the club is 25 ft/ sec and the gymnast releases the club 4.5 above the
ground.
b. A rock falls off of a cliff 500 feet high
A cat tosses a toy up in the air; The toy’s initial position is 1 ft. above the ground and
its initial speed is 9 ft/sec.
Review problems 84,85 and 88 on pages 564 in textbook
polar coordinates and polar form
make the conversion given the coordinates
a. ( (4, 2∏/3)
b. b. (5, -750)
c. (3,3∏/4)
Find two polar coordinates for the points given
a. (1,2)
b. (-4,4)
Equation conversions rectangular to polar and polar to rectangular
Make the conversion: ϕ = theta
a. r = 4cos ϕ
b. b. r = 2cosϕ + sinϕ
c. r = 4cscϕ
d. x2 + (y - 4)2 = 16
e. (x+1)2 + (y -3)2 = 10
Find max r and number of petals (rose curve only). Find the smallest interval necessary to
get a complete graph.
a. 4sin3ϕ
b. 5cos2ϕ
c. -3cos5ϕ
d. 1 + 2cosϕ
e. 6 – sinϕ
Chapter 7
Solve the system
X+y+z=2
2x + 3y + z = 7
x –y + 2z = -3
2x – y =
x + 3y – z = -3
2x + 2y + z = 8
x+y+z=2
2x + 3y + z = 7
x+y–z=3
2x –y – 8z =-3
3x + 2y -5z = 8
Partial fractions:
Find the partial fraction decomposition of :
a. 2x – 1___
x2 – x – 6
b. 7x – 1____
2x2 – x – 1
c. 4x2 + 7x + 2
x3 + 2x2
8x2 + x + 2
(2x + 1)(2x2 + 1)
d. 2x2 -3x + 1
(x + 1) (x2 – x + 1)
Chapter 8: Conic sections
Graph the parabolas, ellipses, and hyperbolas
(y-1)2= 8(x + 1)
(y-2)2 = 4((x + 3)
25y2 + 9x2 =100
9x2 + 4y2 = 36
4x2 + y2 =4
x2 -25y2= 25
36x2 – 4y2=144
25y2 – 25x2= 625
Find the equation of the conic section in standard form and sketch a graph
Ellipse major axis endpoints : (-4,0) and (4,0), length of minor axis (2b) 3
Ellipse: length of major axis 8 (2a) minor axis endpoints ( 0,0) and (0,4)
x2 – 6x – y – 3 = 0
x2 + 4x + 3y2 – 5 = 0
x2 – y2 – 2x + 4y – 6 = 0
y2 – 6x – 4y – 13 = 0
2x2 – 3y2 – 12x – 24y + 60 = 0
9x2 + 4y2 – 18x + 8y -23 = 0
You must also be able to write equation of a conic section given its graph.
Chapter 9
Using multiplication principal:
How many different 6 letter words can be formed:
Peanut
Sundae
Apples
Cherry
The number of contestants is to be is to be narrowed to the given number of finalists. In
how many possible ways can the given number of finalists be selected?
50 contestants, 14 finalists
50 contestants, 18 finalists
Binomial theorem
Expand the following binomials:
(a+b)4
(a + b) 5
(a + b)8
find the coefficient in front of the given term:
x13, (x + 1)15
x3y14, ( x + y)17
Probability
A fair coin is tossed 6 times. Find the probability of the event HHTTTH
A coin is tossed five times. Find the probability of obtaining two heads and three tails
A fair coin is tossed four times, find the probability of obtaining 1 head and three tails
Suppose we have drawn a cookie at random from one of two identical cookie jars. Jar A
contains 3 chocolate chip cookies and 2 peanut butter cookies and jar B contains 1
chocolate chip cookie and 4 peanut butter cookies. What is the probability that it is
peanut butter and from jar B?
chocolate chip and came from jar A ?
given that it is chocolate chip, that it came from jar A
given tha it came from jar B, it is peanut butter
Sequences
Determine whether the sequence diverges or converges, if converges give the limit
1. ½,1/4,1/6,1/2n,
2. 5,10,15,20,25,30
3. 11/1,12/2,13/3,14/4….
4. 5n
For each of the following sequences find the explicit rule and the recursive rule for the
nth term arithmetic:
31,36,41,46,51,…
-9,-1,7,15,23…
1,11,21,31,41,….
Geometric sequences:
Find a recursive and explicit formula for the nth term:
7,21,63,189,567,…
2500,500,100,20,4,….
Find explicit formulas given the following:
The third and fifth term of a arithmetic sequence are -1 and 3 respectively. Find the first
term of the sequence and write an explicit formula for the sequence.
The fifth and tenth terms of an arithmetic sequence are 5 and 41. Find an explicit formula
for the nth term
The second and fifth term of a geometric sequence are 10 and 1250. Find an explicit
formula to find the nth term.
The fifth and seventh term of a geometric sequence are324 and 2916. Find an explicit
formula to find the nth term.
Series (sms)
Find the sum of the arithmetic and geometric series
-7,-3,1,5,9,13
117,110,103,…33
111,108,105,….27
3,6,12,…12,288
5,15,45,…98415
Determine whether series converges. If so, give the infinite sum
10,7.5,5.625,…
½,1/6,1/18,….
1/3,1/9,1/27,….