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Math Analysis
Final Exam Review Sheet
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Section 1.1: Real Numbers
Representations for sets of real numbers:
1. Listing e.g. The set A of all integers whose absolute value is less than 2 can be written as A = {-1, 0, 1}
2. Set-builder notation (same example): A = {x | x is an integer and |x| < 2}, or A = {x | x  Z and |x| < 2}
3. Interval notation (combined with set builder notation) A = {x | x  (-2, 2) and x  Z}
4. Union ( )and intersection (  )of sets…
Section 1.2: Exponents and Radicals
a  aa
Exponent Formulas:
a m  a n  a mn
a
n
n factors
a 0  1 for a  0
a n 
a 
am
 a mn
n
a
ab n  a n b n
a
 
b
n
ba
1
b
 
a
n
n
a
n
b
n
an 
a
m
n
n
m n
 a mn
n
an
a
   n
b
b
a  n bm

b m a n
n
 n a  bn  a
a
b
 a  b  
n
n
m n
 a
n
1
for a  0
an
n
 n am 
 a if a is odd
 a
n
m
1
1
a b
a b



a b
a b
a b a b
n
n
ab
a  mn a
an  a if a is even
1
a

a
a
1
1
a b
a b



a b
a b
a b a b
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Section 1.3: Algebraic Expressions
Distributive Law:
Product of differences:
Special Product Formulas
a( x  y )  ax  ay
( x  y)( x  y)  x 2  y 2
Square of a sum:
( x  y) 2  x 2  2 xy  y 2
Square of a difference:
( x  y) 2  x 2  2 xy  y 2
Product of two sums:
( x  a)( x  b)  x 2  (a  b) x  ab
Another product of two sums:
(ax  b)(cx  d )  acx 2  (ad  bc) x  bd
Cube of a sum:
( x  y) 3  x 3  3x 2 y  3xy 2  y 3
Cube of a difference:
( x  y) 3  x 3  3x 2 y  3xy 2  y 3
Sum of cubes:
( x  y)( x 2  xy  y 2 )  x 3  y 3
Difference of cubes:
( x  y)( x 2  xy  y 2 )  x 3  y 3
Factoring algebraic expressions:
1. Look for common factors
2. Look for expressions that match the product formulas
3. 2 other tricks…
Section 1.4: Fractional Expressions
1. To simplify fractional expressions, factor both the numerator and denominator, then cancel common
factors.
x 2  2 x  1  x  1 x  1  x  1


Example:
x2 1
 x  1 x  1  x  1
A C AC
 
B D BD
 x  1  1   x  1 
 x  1
1
Example:

2
2
 x  1 x  1  x  1  x  1  x  1 x  1 x  1  x  12
2. To multiply fractional expressions:
3. To divide fractional expressions:
Example:
A C A D AD
   
B D B C BC
 x  1  1   x  1  x 2  1   x  1 x  1 x  1   x  1
1
 x  1 x 2  1  x  1 1
 x  1
2
  x  1
2
4. To add or subtract fractional expressions, first convert each expression into equivalent expressions with
A B A B
common denominators, then:  
C C
C
 x  1  1   x  1  x  1  x  1   x  1   x  1  2 x
1
1
1



Example:
x2 1
x2 1
 x  1  x  1  x  1  x  1  x  1  x  1 x 2  1 x 2  1
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5. To simplify a compound fraction, multiply numerator and denominator by the L.C.D. of all fractions,
then simplify.
1 1 1 1
xy xy



yx
x y x y xy
x
y
Example:

 

1 1 1 1 xy xy xy y  x



x y x y
x
y
6. To rationalize the denominator, multiply numerator and denominator by the conjugate of the
denominator.
1
1
3
3
1
1
2 3
2 3
2 3
Example:
Example:




 2 3



2
3
43
2  3 2  3 2  3 22  3
3
3 3
 
Section 1.5: Equations
1. To solve linear equations, isolate the variable on one side.
1

4  y    y  6 5  y 
2

4 y  2  y  30  6 y
Example:
9 y  32
32
9
2
2. The solutions of x = c are x =  c .
y
1
3. The solutions of xm = c when m is even are x   m c  c m .
4. The solution of xm = c when m is odd is x  m c  c
1
m
.
1
5. The solution of m x  x m  c is x  c m .
6. If an equation can be factored to yield AB = 0, then either A = 0 or B = 0.
3x 2  2 x  1
3x 2  2 x  1  0
Example:  3 x  1 x  1  0
x
1
or
3
x 1
7. The solution to the quadratic equation ax2 + bx + c = 0 is x 
b  b2  4ac
.
2a
8. b2 – 4ac is called the discriminant
a. If b2 – 4ac > 0, then the equation has two distinct real solutions.
b. If b2 – 4ac = 0, then the equation has exactly one real solution.
c. If b2 – 4ac < 0, then the equation has two distinct complex solutions.
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x2  x 1  0
x
1  12  4 1 1
2 1
1  5
1  5
 0.618 or x 
 1.618
2
2
9. Procedure for Solving a Quadratic equation by Completing the Square:
x
b
c
x 0)
a
a
b. Complete the square by adding and subtracting the square of one-half the coefficient of x.
c. Write the equation as a square plus a constant
d. Solve for x.
3x 2  3x  7  0
a. Divide each side by a, the coefficient of the x2 term. (i.e., ax 2  bx  c  0  x 2 
7
0
3
1 1 7
 2
x  x    0
4 4 3

x2  x 
2
Example:
1  31

x   0
2  12

2
1  31

x  
2  12

x
1
31
1 31
1 93
1




93
2
12
2 3
2 9
6
1 1
x 
93
2 6
10. To solve equations with fractional expressions, multiply both sides of the equation by the L.C.D.
1
1
5


x 1 x  2 4
1 
5
 1


  4  x  1 x  2    4  x  1 x  2 
4
 x 1 x  2 
Example:
4  x  2   4  x  1  5  x 2  x  2 
5 x 2  3 x  14  0
 5 x  7  x  2   0
x
7
5
or
x2
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11. To solve equations with a radical, isolate the radical, then raise both sides to the appropriate power to
eliminate the radical.
5  x 1  x  2
5 x  x3

5 x

2
  x  3
2
5  x  x2  6x  9
Example:
x2  5x  4  0
 x  1 x  4   0
x  1 or x  4, but x = 1 does not work
12. To solve “quadratic-like” equations, make a variable substitution where u = the middle term, and u2 =
the other term.
x4  5x2  4  0
Let u  x 2
u 2  5u  4  0
Example:  u  1 u  4   0
u  1 or u  4
x 2  1 or
x 2  4
x  1 or x  2i
13. To solve absolute value equations, isolate the absolute value expression, then set that expression to  the
other side.
2 x  6  6  12
2 x  6  12
Example:
x6  6
x  6  6
x  12 or
x0
Section 1.6: Modeling with Equations
1. Draw a picture.
2. Label the picture with given quantities and the unknown.
3. Write down the equation that relates the unknown to the given information.
4. Solve the equation.
5. Check your answer.
6. Some geometric formulas for areas:
a. Rectangle: Arect  lw
b. Circle: Acirc  r 2
1
c. Triangle: Atri  bh
2
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d. Sphere: Asphere  4 r 2
7. Some geometric formulas for volumes:
a. Box: Vbox  lwh
b. Cylinder: Vcyl  r 2 h
4
c. Sphere: Vsphere  r 3
3
1
d. Pyramid: V pyr  Bh
3
1
1
e. Cone: Vcone  Bh  r 2 h
3
3
distance
8. Speed problems: speed 
time
9. The total is equal to the sum of its parts!
Section 1.7: Inequalities
1. To solve a linear inequality, isolate the variable using the Rules for Inequalities:
a. Adding or subtracting the same quantity to or from each side of an equality yields an equivalent
inequality.
b. Multiplying each side of an inequality by the same positive quantity yields an equivalent
inequality.
c. Multiplying each side of an inequality by the same negative quantity yields an equivalent
inequality.
d. Taking the reciprocal of each side of an inequality involving positive numbers reverses the
direction of the inequality.
e. Taking the reciprocal of each side of an inequality involving negative numbers reverses the
direction of the inequality.
4 x  7  19
Example:
4 x  12
x  3
2. To solve a nonlinear inequality, get all terms on the left hand side and factor them. Find the intervals of
sign changes, then make a table or diagram to show the signs of each factor and the overall quantity.
x 2  5 x  6
x2  5x  6  0
Example:  x  2  x  3  0
Interval:
 x  2 :
 x  3 :
 x  2  x  3
Solution: 2  x  3
x2 2 x3 x3









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4
  7  19
x
4
  12  0
x
4 12 x
 
0
x
x
4  12 x
0
x
4 1  3 x 
0
x
3x  1
0
x
1
1
 x0 x0
3
3



Interval: x  
 3x  1 :
 x :
Example:

3x  1

x
1
Solution: x    x  0
3




3. To solve absolute value inequalities, keep in mind that:
a. x  c is equivalent to  c  x  c
b.
x  c is equivalent to  c  x  c
c.
x  c is equivalent to x  c or x  c
d.
x  c is equivalent to x  c or x  c
Section 1.8: Coordinate Geometry
1. Distance Formula: d  A, B 
x2  x1 2   y2  y1 2
 x  x 2 y1  y 2 
2. Midpoint Formula: M   1
,

2 
 2
3. Know how to graph an equation by calculating and plotting points.
4. x intercepts are where a graph crosses the x-axis; calculate by setting y = 0 and solve for x.
5. y intercepts are where a graph crosses the y-axis; calculate by setting x = 0 and solve for y.
6. Know the graphs of some standard equations:
a. Line: y  mx  b
b. Vertical Parabola: y  ax 2  bx  c
c. Horizontal Parabola: y  ax  b  c
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d. Absolute Value: y  ax  b  c
e. Circle: x  h    y  k   r 2
7. If a graph has symmetry with respect to the x-axis, the equation is unchanged when y is replaced by –y.
8. If a graph has symmetry with respect to the y-axis, the equation is unchanged when x is replaced by –x.
9. If a graph has symmetry with respect to the origin, the equation is unchanged when x is replaced by –x
and y is replaced by –y.
2
2
Section 1.9: Solving Equations and Inequalities Graphically
1. Graph both equations, then read the intersection points off your calculator or hand-drawn graph.
Section 1.10: Lines
Slope:
Point-slope form of the equation of a line:
Slope-intercept form of the equation of a line:
Two-intercept form of the equation of a line:
General Equation of a line:
Horizontal line:
Vertical line:
Slope relation of parallel lines:
Slope relation of perpendicular lines:
y y 2  y1

x x2  x1
y  y1  mx  x1 
y  mx  b
x y
 1
a b
Ax  By  C  0
yb
xa
m1  m2
1
m1  
m2
m
Section 1.10: Modeling Variation
Direct Variation (Proportionality): y = kx.
Inverse Variation (Inverse Proportionality): y 
k
.
x
Joint Variation (Joint Proportionality): z  kxy or z 
k
kx
or z 
xy
y
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Section 2.1: What is a Function?
1. Definitions: A function f is a rule that assigns to each element x in a set A exactly one element f(x), in a
set B. The set A is the domain of the function. f(x) is the image of x under f. The range is the set of
all possible values of f(x) as x varies through the domain.
2. Know how to evaluate a function at a number and at an expression.
3. Know how to find the domain of a function:
a. Start with the domain being all real numbers.
b. Eliminate x values from the domain where a denominator is equal to zero.
c. Eliminate x values from the domain where a square root has a negative argument.
Section 2.2: Graphs of Functions
1. Definition: If f is a function with domain A, then the graph of f is the set of ordered pairs
{(x, f(x) | x  A}, or, in other words, the graph of f is the graph of the equation y = f(x).
2. Know how to graph a function by calculating and plotting points.
3. Know the graphs of some common functions:
a. Linear function: f ( x)  mx  b
b. Power functions: f ( x)  x n for n = even and odd integers greater than 1.
c. Root functions: f ( x)  n x for n = even and odd integers greater than 1.
d. Reciprocal functions: f ( x)  1 n for n = even and odd integers greater than 0.
x
e. Absolute value function: f ( x)  x
4. The vertical line test: A curve in the coordinate plane is the graph of a function if and only if no
vertical line intersects the curve more than once.
Section 2.3: Increasing and Decreasing Functions; Average Rate of Change
1. Average Rate of Change: The average rate of change of a function y = f(x) between x = a and x = b is
the slope of the secant line connecting the points (a, f(a)) and (b, f(b)):
y f (b)  f (a)

average rate of change =
.
x
ba
2. Increasing and Decreasing Functions:
a. A function f is increasing on an interval I if f(x1) < f(x2) whenever x1 < x2 in I.
b. A function f is decreasing on an interval I if f(x1) > f(x2) whenever x1 < x2 in I.
c. Note: intervals of increase/decrease change wherever a function has an extreme value…
Section 2.4: Transformations of Functions
1. Vertical shifts of graphs:
a. y = f(x) + c (c > 0) shifts the graph of y = f(x) upward by c units.
b. y = f(x) – c (c > 0) shifts the graph of y = f(x) downward by c units.
2. Horizontal shifts of graphs:
a. y = f(x – c) (c > 0) shifts the graph of y = f(x) to the right by c units.
b. y = f(x + c) (c > 0) shifts the graph of y = f(x) to the left by c units.
3. Reflecting graphs:
a. y = –f(x) reflects the graph of y = f(x) in the x-axis.
b. y = f(–x) reflects the graph of y = f(x) in the y-axis.
4. Vertical Stretching and Shrinking of Graphs:
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a. y = af(x) (a > 1) stretches the graph of y = f(x) vertically by a factor of a.
b. y = af(x) (0 < a < 1) shrinks the graph of y = f(x) vertically by a factor of a.
5. Even and Odd Functions:
a. f is even if f(–x) = f(x) for all x in the domain of f. (The graph of f is symmetric with respect to
the y-axis.)
b. f is odd if f(–x) = –f(x) for all x in the domain of f. (The graph of f is symmetric with respect to
the origin.)
Section 2.5: Quadratic Functions; Maxima and Minima
1. If you have a TI calculator, know how to find the extrema of a graph with it.
2. Some facts about the quadratic function f(x) = ax2 + bx + c:
a. Its graph is a parabola.
b. Its standard form is f(x) = a(x – h)2 + k
c. The vertex, or extremum of its graph is at the point (h, k).
d. Its extremum is a maximum if a < 0.
e. Its extremum is a minimum if a > 0.
b
b 2  4ac


f. It can be re-arranged into the form: y  a x 

2a 
4a

2
  b b  4ac 

g. Thus, (h, k) = 
,
2
a
4
a


2
Section 2.6: Modeling With Functions
10. Draw a picture.
11. Label the picture with given quantities and the unknown.
12. Write down the equation that relates the unknown to the given information.
13. Solve the equation.
14. Check your answer.
15. Some geometric formulas for areas:
a. Rectangle: Arect  lw
b. Circle: Acirc  r 2
1
c. Triangle: Atri  bh
2
16. Some geometric formulas for volumes:
a. Box: Vbox  lwh
b. Cylinder: Vcyl  r 2 h
4
c. Sphere: Vsphere  r 3
3
1
d. Pyramid: V pyr  Bh
3
1
1
e. Cone: Vcone  Bh  r 2 h
3
3
Math Analysis
Final Exam Review Sheet
17. Speed problems: speed 
Page 11 of 37
distance
time
Section 2.7: Combining Functions
1. Addition: (f + g)(x) = f(x) + g(x)
domain: A  B
2. Subtraction: (f – g)(x) = f(x) – g(x) domain: A  B
3. Multiplication: (fg)(x) = f(x)g(x)
domain: A  B
f
f ( x)
4. Division:  ( x) 
domain: A  B and g(x)  0
g ( x)
g
5. Composition: (f  g)(x) = f(g(x))
domain: all x  domain of g such that their images are in the range
of f
6. Be able to break a function down into the composition of two simple functions.
Section 2.8: One-to-One Functions and Their Inverses
1. One-to-One Function: A function with domain A is called a one-to-one function if no two elements of
A have the same image, that is, f(x1)  f(x2) whenever x1 x2.
2. Horizontal Line Test: A function is one-to-one function if and only if no horizontal line intersects its
graph more than once.
3. Algebraic Test: Start with f(x1) = f(x2). If this reduces to x1 = x2, then f is one-to-one.
4. Definition of the Inverse of a Function: Let f be a function with domain A and range B. Then its
inverse function f-1 has domain B and range A and is defined by f-1(y) = x  f(x) = y for any y in B.
5. Properties of Inverse Functions:
a. f-1(f(x)) = x
b. f(f-1 (x)) = x
6. Know how to find the inverse of a function:
a. Set y = f(x)
b. Solve for x
c. Switch variables x  y
d. y = f-1(x) now.
7. The graph of f-1 is obtained by reflecting the graph of f in the line y = x.
8. A point with coordinates (a, b) on the graph of f corresponds to a point with coordinates (b, a) on the
graph of f-1.
Math Analysis
Final Exam Review Sheet
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Section 3.1: Polynomial Functions and Their Graphs
1. Polynomial Functions: A polynomial of degree n is a function of the form
Px   a n x n  a n 1 x n 1  a n 2 x n 2    a 2 x 2  a1 x  a0
2. Know how to determine the end behavior of a polynomial (it’s determined by the anxn term…).
3. Know how to use zeros of a polynomial as a graphing aid (there are at most n of them).
4. Know how roots, zeros, and factors of a polynomial are related.
5. Know how a repeated root affects the graph of a polynomial function.
6. Intermediate Value Theorem for Polynomials: If P is a polynomial function and P(a) and P(b) have
opposite signs, then there exists at least one value c between a and b for which P(c) = 0.
7. Local Extrema of Polynomials: If P is a polynomial of degree n, then the graph of P has at most n – 1
local extrema.
Section 3.2: Dividing Polynomials
1. Division Algorithm: Px  D( x)  Q( x)  R( x) . P is the dividend, D is the divisor, Q is the quotient,
P( x)
R( x)
and R is the remainder. Can also be stated:
.
 Q( x) 
D( x)
D( x)
2. Know how to do long division of polynomials.
3. Know how to do synthetic division of polynomials.
4. Remainder Theorem: If the polynomial P(x) is divided by x – c, then the remainder is the value P(c).
5. Factor Theorem: c is a zero of P if and only if x – c is a factor of P.
Section 3.3: Real Zeros of Polynomials
1. All rational zeros of a polynomial P are of the form
p
, where p is a factor of the constant coefficient a0,
q
and q is a factor of the leading coefficient an.
2. Know how to list all possible zeros of a polynomial and test them using synthetic division.
3. Know how to use Descartes’ Rule of Signs.
4. Know how to use the upper and lower bounds theorem.
Steps for factoring a polynomial:
1. Graph the polynomial on your calculator to guide your work.
2. Make a list of all possible rational zeros using the rational zeros theorem.
3. Order the list and remove any redundant values.
4. Apply Descartes’ Rule of signs to predict how many positive and negative zeros you can expect; if you
find that no roots of a given sign are predicted, then your work is cut in half.
5. Perform synthetic division on the polynomial, starting with the smallest roots.
a. If the remainder is zero, then you found a root; check to see if it is a multiple root, and re-write
the polynomial with that root factored out. Only perform synthetic division on the lower-order
quotient.
b. If the remainder is not zero, look at the signs of the quotient row; the upper and lower bounds
theorem tells you when you can stop looking based on the signs of the numbers in the quotient
row.
6. Use the quadratic formula when you get down to a quadratic quotient.
Math Analysis
Final Exam Review Sheet
Page 13 of 37
Section 3.4: Complex Numbers
9. i   1
10. i 2  1
11. Know how to add and subtract complex numbers.
12. Know how to multiply complex numbers.
13. Know how to divide complex numbers (multiply numerator and denominator by the conjugate of the
denominator).
Section 3.5: Complex Zeros and The Fundamental Theorem of Algebra
1. Fundamental Theorem of Algebra: Every polynomial
Px   a n x n  a n 1 x n 1  a n 2 x n 2    a 2 x 2  a1 x  a0 with complex coefficients has at least one
complex zero.
2. Complete Factorization Theorem: There exist complex numbers such that any polynomial of degree
greater than 1 can be written as P( x)  a  x  c1  x  c2   x  cn 
3. The Zeros Theorem means that you have to keep factoring until you find n zeros  check any zero you
find again with synthetic division until the remainder is not zero.
4. Conjugate Zeros Theorem: If a polynomial has real coefficients and a complex zero z, then z is also a
zero.
5. Linear and Quadratic Factors Theorem: Every polynomial with real coefficients can be factored into
a product of linear and irreducible quadratic factors with linear coefficients.
Section 3.6: Rational Functions
1. To graph rational functions:
a. Factor numerator and denominator
b. Find x and y intercepts
c. Find vertical asymptotes by setting the denominator to zero
d. Find horizontal asymptotes by doing long division; the asymptote is the quotient you get
e. Graph
Math Analysis
Final Exam Review Sheet
Page 14 of 37
Section 4.1: Exponential Functions
1. For a > 0, the exponential function with base a  1 is defined by: f(x) = ax.
2. The domain is all reals.
3. The range (for a ≠ 1) is (0, ∞)
4. Common bases are a = 2, 10, and e.
5. The graph has one of the two following shapes (for a  1):
This is for a > 1
 r
6. Interest with discrete compounding: A(t )  P 1  
 n
7. Interest with continuous compounding: A(t )  Pert
This is for 0 < a < 1
nt
Section 4.2: Logarithmic Functions
1. If a is a positive number and a ≠ 1, then the logarithmic function with base a, denoted as loga, is
defined by loga x = y  ay = x. (i.e., loga x is the inverse function to the exponential function, and thus
is the exponent to which a must be raised to give x)
2. LOGARITHMS ARE EXPONENTS! loga x is the exponent on a that gives x
Math Analysis
Final Exam Review Sheet
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3. Properties of Logarithms:
a. log a1  0
b. log a a  1
c. log a a x  x
d. a log a x  x
4. Common Logarithm is the logarithm with base 10. It’s denoted by: log10 x = log x.
5. Natural Logarithm is the logarithm with base e. It’s denoted by: loge x = ln x.
Section 4.3: Laws of Logarithms
The “Laws of Logarithms”
1. “The log of a product is the sum of the logs”: log a  AB   log a A  log a B
 A
2. “The log of a ratio is the difference of the logs”: log    log a A  log a B
a B
3. “The log of a power is the product of the power and the log”: log Ac  c log a A
a
4. Change of base formula: log b x 
 
log a x
log a b
Exponent Rule
b0  1
Corresponding Logarithm Rule
logb 1  0
b1  b
logb b  1
bn  bn
log b b n  n
bu b v  bu  v
logb xy  logb x  logb y
u
b
 bu  v
v
b
 x
logb    logb x  logb y
 y
Math Analysis
Final Exam Review Sheet
b 
m n
 b mn
Page 16 of 37
log b  x n   n log b x
Section 4.4: Exponential and Logarithmic Equations
 An exponential equation has exponents that are variables.
o To solve exponential equations, take the logarithm of both sides of the equation.
 A logarithmic equation involves the logarithm of variables.
o To solve logarithmic equations, combine all logarithms into a single, isolated logarithm, then
write that resulting logarithmic equation in exponential form.
 Use your graphing calculator to check your work by graphing both sides of the equation as separate
functions, then finding the intersection point.
 Make sure that all answers you get are valid.
Section 4.5: Modeling with Exponential and Logarithmic Functions
1. Exponential growth: n(t )  n0e rt
2. Radioactive decay: m(t )  m0e  rt . The half-life h and the relative rate of growth r are related as:
ln 2
r
h
3. Newton’s Law of Cooling: T (t )  Ts  D0e  kt
4. The pH scale: pH = -log[H+]
I
5. The Richter Scale: M  log ;(S = 10-4 cm)
S
I
6. The Decibel Scale:   10 log ;(I0 = 10-12 W/m2)
I0
Section 5.1: The Unit Circle
1. The Unit Circle: x 2  y 2  1
2. The coordinates of the standard terminal points in the first quadrant are shown below.
3. To calculate terminal points in other quadrants, find the angle closest to the x-axis, then use the
coordinates of that angle with sign adjustments as needed.
Math Analysis
Final Exam Review Sheet
Page 17 of 37
A way to remember the trig functions for a few triangles:
Math Analysis
Final Exam Review Sheet
Page 18 of 37
Be able to label each of the 16 common terminal points shown in the image below with its angle in radians, its
angle in degrees, and its coordinates.
Section 5.2: Trigonometric Functions of Real Numbers
1. The Six Trig Functions defined on the Unit Circle:
t is the arc length to the terminal point; (x, y) is the coordinate of the terminal point.
opp.
a. sin t  y ;
;
domain: {t | t  }
sin t 
hyp.
adj.
b. cost  x ;
;
domain: {t | t  }
cos t 
hyp.
y

opp.
c. tan t  ;
;
domain: {t | t 
and t  + n for integer n}
tan t 
x
2
adj.
adj.
x
d. cot t  ;
;
domain: {t | t  and t  n for integer n}
cot t 
opp.
y
1

hyp.
e. sec t  ;
;
domain: {t | t 
and t  + n for integer n}
sec t 
x
2
adj.
hyp.
1
f. csct  ;
;
domain: {t | t 
and t  n for integer n}
csc t 
opp.
y
Math Analysis
Final Exam Review Sheet
Page 19 of 37
Be able to fill in the table below with exact values.
t
0

6

4

3

2
2
3
3
4
5
6

7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
sin(t)
cos(t)
tan(t)
cot(t)
sec(t)
csc(t)
Math Analysis
Final Exam Review Sheet
Page 20 of 37
2. Even-Odd Properties:
a. sin(-t) = -sin(t) (an odd function)
b. cos(-t) = cos(t) (an even function)
c. tan(-t) = -tan(t) (an odd function)
d. cot(-t) = -cot(t) (an odd function)
e. sec(-t) = sec(t) (an even function)
f. csc(-t) = -csc(t) (an odd function)
3. Reciprocal Identities:
1
a. csc t 
sin t
1
b. sec t 
cos t
1
cos t

c. cot t 
tan t sin t
1
sin t

d. tan t 
cot t cos t
e. Know how to calculate cotangent, secant, and cosecant on a calculator.
4. Pythagorean Identities:
a. sin 2 t  cos 2 t  1
b. tan 2 t  1  sec 2 t
c. cot 2 t  1  csc2 t
Section 5.3: Trigonometric Graphs
1. The curves y  a sin  k  x  b   and y  a cos  k  x  b   have amplitude |a|, period
2
,
k
and phase shift b.
2. To sketch these curves:
a. Factor the argument of the function so that it is of the form k(x – b).
2
b. Compute the period of the function using T 
.
k
c. Mark off the phase shift (i.e., horizontal shift of the graph b), two periods on either side of the
phase shift, and the quarter-points of each of those two periods.
d. Mark off the amplitude of the trig function.
e. Starting at the phase shift, sketch two periods of the function.
f. Use a graphing calculator or Winplot with the x-axis set in units of  to verify your work.
Math Analysis
Final Exam Review Sheet
Page 21 of 37
Section 5.4: More Trigonometric Graphs
3. Properties of the graphs of the tangent and cotangent functions:
a. The tangent and cotangent functions have period :
tan  x     tan  x 
cot  x     cot  x 
b. The curves y  a tan  k  x  b   and y  a csc  k  x  b   have amplitude |a|, period
shift b.
c. The tangent function has vertical asymptotes at
 2n  1 
2
for n  .
d. The cotangent function has vertical asymptotes at n for n  .
4. Properties of the graphs of the tangent and cotangent functions:

, and phase
k
Math Analysis
Final Exam Review Sheet
Page 22 of 37
a. The secant and cosecant functions have period 2:
sec  x  2   sec  x 
csc  x  2   csc  x 
b. The curves y  a sec  k  x  b   and y  a csc  k  x  b   have amplitude |a|, period
phase shift b.
 2n  1 
for n  .
2
d. The cosecant function has vertical asymptotes at n for n  .
c. The secant function has vertical asymptotes at
2
, and
k
Math Analysis
Final Exam Review Sheet


Page 23 of 37
y
y = csc(x)


    

  
x
  



Section 5.5: Modeling Harmonic Motion
Simple Harmonic Motion: Models quantities that vary as y  a sin t  or y  a cos t  . Examples
include masses supported by springs, sound waves,
Amplitude = |a|
2
Period = T 


Frequency = f 
( is called the “angular frequency”)
2
f is typically measured in units of inverse seconds, which is called a Hertz (1 Hz = 1 sec-1)
Damped Harmonic Motion: Models quantities that oscillate but die down, due to friction or some other
energy-draining force. y  kect sin t  or y  kect cos t  .
Damping coefficient = c
2
Period = T 

Math Analysis
Final Exam Review Sheet
Page 24 of 37
Section 6,1: Angle Measure
 180 
6. (radians)  
 = (degrees)



  
7. (degrees)  
 = (radians)
 180 
8. Arc Length: s = r.
9. Sector Area: A = ½r2.

10. Angular Speed:   .
t
11. Some units of angular velocity:
radians radians radians
a.
,
,
second minute
hour
degrees degrees degrees
b.
,
,
second minute
hour
revolutions revolutions
revolutions
c.
,
(rpm) ,
second
minute
hour
12. Speed of an object moving around a circle: v =  r.
Section 6.2: Trigonometry of Right Triangles
Big Idea: The six trigonometric functions we have studied can be applied to the ratios of sides in any right
triangle to find unknown quantities of those right triangles.
Big Skill: You should be able to identify known and unknown measurements on a right triangle and relate
those quantities using an appropriate trigonometric function.
SOH CAH TOA:
opp
hyp
opp
tan  
adj
1
hyp
sec  

cos 
adj
sin  
To solve a right triangle:
adj
hyp
1
adj
cot  

tan  opp
1
hyp
csc 

sin  opp
2
2
2
a b  c
cos 
Math Analysis
Final Exam Review Sheet
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1. Make a sketch of the triangle, label sides and angles consistently (a, b, and c for the legs and
hypotenuse; A and B for the complementary angles), and label the given information.
2. Find a way to relate the unknown parts to the given information using a trig function (sine, cosine, or
tangent) or the Pythagorean Theorem (a2 + b2 = c2). Try to use original given information to minimize
rounding errors.
3. Check your work:
a. Make sure the sides obey the Pythagorean Theorem.
b. Make sure the angles add up to 180.
c. Make sure unused trig functions give the right answers.
d. Make sure that the longest side is opposite the largest angle, and the shortest side is opposite the
smallest angle.
Section 6.3: Trigonometric Functions of Angles
Definition of the Six Trigonometric Functions: Let  be the angle to the terminal point P(x, y). If
r  x 2  y 2 is the distance from the origin to the terminal point P, then:
y
d. sin  
r
x
e. cos  
r
y
f. tan  
x
x
g. cot  
y
r
h. sec  
x
r
i. csc 
y
Area of a triangle: A 
1
ab sin 
2
Math Analysis
Final Exam Review Sheet
Page 26 of 37
Section 6.4: The Law of Sines
The Law of Sines
a
b
c


sin A sin B sin C
OR
sin A sin B sin C


a
b
c
The Law of Sines uses the sine function to relate the lengths of the sides of any triangle to its angles. The Law
of Sines is used when:
1. One side and two angles are given (the AAS, SAA, or ASA cases).
2. Two sides and an angle opposite one of the sides is given (the ASS or SSA cases).
This can sometimes lead to the so-called “ambiguous case.” The ambiguous case occurs when a given
side opposite the given angle is greater than the sine of that angle times the given side adjacent to the
angle, but less than the adjacent side. When this is the case, you get the other angle by subtracting the
angle you get from 180.
Section 6.5: The Law of Cosines
The Law of Cosines
a  b 2  c 2  2bc cos A
2
cos A 
OR
OR
b 2  c 2  a 2  2ca cos B
OR
a 2  c 2  b2
cos B 
2ac
c 2  a 2  b 2  2ab cos C
OR
cos C 
Heron’s Formula: A  s  s  a  s  b  s  c  , s 
abc
2
b2  c 2  a 2
2bc
a 2  b2  c 2
2ab
Math Analysis
Final Exam Review Sheet
Page 27 of 37
Section 7.1: Trigonometric Identities
1. Reciprocal Identities:
1
a. csc t 
sin t
1
b. sec t 
cos t
1
cos t

c. cot t 
tan t sin t
sin t
d. tan t 
cos t
2. Pythagorean Identities:
j. sin 2 t  cos 2 t  1
k. tan 2 t  1  sec 2 t
a. cot 2 t  1  csc2 t
3. Even-Odd Identities:
a. sin(t )   sin t
b. cos(t )  cos t
c. tan(t )   tan t
4. Cofunction Identities:


a. sin   t   cos t
2 


b. cos   t   sin t
2 


c. tan   t   cot t
2 
5. To simplify a trigonometric expression, use factoring, common denominators, and trigonometric
identities to convert the expression into one that has fewer trig functions and calculations.
6. To prove a trigonometric identity, try to show how the more complicated simplifies to the simpler side
using identities or by writing everything in terms of sines and cosines. If both sides are complicated, try
to simplify both sides.
Section 7.2: Addition and Subtraction Formulas
1. sin  s  t   sin s cos t  cos s sin t
2. sin  s  t   sin s cos t  cos s sin t
3. cos  s  t   cos s cos t  sin s sin t
4. cos  s  t   cos s cos t  sin s sin t
5. tan  s  t  
tan s  tan t
1  tan s tan t
Math Analysis
6. tan  s  t  
Final Exam Review Sheet
tan s  tan t
1  tan s tan t
7. A sin x  B cos x  k sin  x    , where k  A2  B 2 and cos  
Section 7.3: Double Angle, Half Angle, and Product-Sum Formulas
1. Double-Angle Formulas:
a. sin 2x  2sin x cos x
cos 2 x  cos 2 x  sin 2 x
b.
 1  2sin 2 x
 2 cos 2 x  1
2 tan x
c. tan 2 x 
1  tan 2 x
2. Power-Lowering Formulas:
1  cos 2 x
a. sin 2 x 
2
1

cos
2x
b. cos 2 x 
2
1

cos
2x
c. tan 2 x 
1  cos 2 x
3. Half-Angle Formulas:
u
1  cos u
a. sin  
2
2
u
1  cos u

2
2
u 1  cos u
sin u

c. tan 
2
sin u
1  cos u
4. Product-to-Sum Formulas:
1
a. sin u cos v  sin  u  v   sin  u  v  
2
1
b. cos u sin v  sin  u  v   sin  u  v  
2
1
c. cos u cos v  cos  u  v   cos  u  v  
2
1
d. sin u sin v  cos  u  v   cos  u  v  
2
5. Sum-to-Product Formulas:
x y
x y
cos
a. sin x  sin y  2sin
2
2
x y
x y
sin
b. sin x  sin y  2 cos
2
2
b. cos
Page 28 of 37
A
B
and sin  
k
k
Math Analysis
Final Exam Review Sheet
Page 29 of 37
x y
x y
cos
2
2
x y
x y
sin
d. cos x  cos y  2sin
2
2
c. cos x  cos y  2 cos
Section 7.4: Inverse Trigonometric Functions
1. All inverse trig problems have two answers; the second answer must be found by looking at the unit
circle, because the domains of the trig functions get restricted to make them one-to-one.
2. The inverse sine function: sin-1(x), or arcsin(x)
sin(x)
Domain: -/2  x  /2 (restricted)
Range: -1  y  1
sin-1(x)
Domain: -1  x  1
Range: -/2  y  /2
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3. The inverse cosine function: cos-1(x), or arccos(x)
cos(x)
Domain: 0  x   (restricted)
Range: -1  y  1
cos-1(x)
Domain: -1  x  1
Range: 0  y  
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4. The inverse tangent function: tan-1(x), or arctan(x)
tan(x)
Domain: -/2  x  /2 (restricted)
Range: -  y  
tan-1(x)
Domain: -  x  
Range: -/2  y  /2
5. Use a right triangle to compute the answer to a trig function composed with an inverse trig functions.
6. To calculate the inverse trig functions of the “black sheep” functions:
a. sec 1    cos 1  1 
b. csc1    sin 1  1 
c. cot 1    tan 1  1 
Section 7.5: Trigonometric Equations
1. To solve trig equations, try to combine all trig functions into one trig function, then isolate it and take
the inverse of both sides of the equation.
2. Also, don’t forget that polynomial equations of trigonometric functions can be solved by factoring.
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Section 9.1: Systems of Equations
1. Substitution Method:
a. Solve for one variable
b. Substitute
c. Back-Substitute to find value of other variable
2. Elimination Method:
a. Adjust coefficients so one variable has equal but opposite coefficients in both equations
b. Add the equations
c. Back-Substitute
3. Graphical Method:
a. Graph each equation
b. Find intersection points
Section 9.2: Systems of Linear Equations in Two Variables
1. A system of linear equations can have:
a. 0 solutions (called inconsistent; the two lines are parallel with different intercepts)
b. 1 solution (called consistent; the two lines have different slopes)
c. infinitely many solutions (called dependent; the two lines coincide (are the same))
 a x  b1 y  c1
b c bc
ac a c
2. For the linear system  1
, the solution is: x  2 1 1 2 and y  1 2 2 1 .
a1b2  a2b1
a1b2  a2b1
a2 x  b2 y  c2
3. To model with a system of linear equations:
a. Draw a picture and label it with known quantities.
b. Identify the variable or variables, and express all unknown quantities in terms of the variables.
c. Write a system of equations that relates all the information.
d. Solve the system.
Section 9.3: Systems of Linear Equations in Several Variables
Linear Equation: is of the form a1 x1  a2 x2   an xn  c
Example of a system of equations in triangular form:
x  2 y  z  1

y  2z  5


z  3

Formal statement of the “legal” operations behind the method of elimination:
1. The following operations yield an equivalent system of equations:
a. Add a nonzero multiple of one equation to another equation
b. Multiply an equation by a nonzero constant
c. Interchange the positions of two equations
Gaussian elimination: use the above operations to change the system to an equivalent triangular system, then
use back-substitution.
Number of solutions of a Linear System: For a system of linear equations, exactly one of the following is
true:
1. The system has exactly one solution.
2. The system has no solutions. This type of system is called an inconsistent system. Inconsistent systems
yield a false equation after applying Gaussian elimination.
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3. Dependent systems have infinitely many solutions. This type of system is called a dependent system.
Dependent systems yield one or more equations of 0 = 0 after applying Gaussian elimination.
Section 9.4: Systems of Linear Equations: Matrices
1. Definition of a matrix: An m  n matrix is a rectangular array of numbers with m rows and n columns,
with entries aij.
a1n 
 a11 a12 a13
a
a2 n 
 21 a22 a23
 a31 a32 a33
a3n 




 am1 am 2 am 3
amn 
2. Linear equation in n variables is converted to augmented matrix form as follows:
 x 3 y 3z  4
1 3 3 4 



 x 2 y 2 z  10  1 2 2 10 
3x  y 5 z  14

3 1 5 14
3. The operations behind the method of elimination also apply to matrices, but they are now called
elementary row operations:
a. Add a multiple of a row to another
b. Multiply a row by a constant
c. Interchange two rows
4. Converting a system of equations to triangular form is analogous to converting the augmented matrix to
row-echelon form:
a. The leading entry (first nonzero number) in each row is 1.
b. The leading entry in each row is to the right of the leading entry in the row immediately above.
c. Rows that are all zeros are at the bottom.
5. Back-substitution to solve a system of equations in triangular form is analogous to converting the
augmented matrix to reduced row-echelon form. Converting a matrix to reduced row-echelon form is
called Gauss-Jordan elimination.
a. Matrix is in echelon form.
b. Every number above and below each leading entry is a 0.
6. To solve a system of equations using a matrix:
a. Write the system as an augmented matrix.
b. Use row operations to convert the matrix to row-echelon form.
c. Use either back-substitution to finish solving, or do more row operations to put the matrix in
reduced row echelon form.
d. If the system is inconsistent, then one of the rows of the matrix will reduce to 0 = c.
e. If the system is dependent, then one or more of the rows of the matrix will reduce to 0 = 0.
Section 9.5: The Algebra of Matrices
1. Two matrices A = [aij] and B = [bij] are equal if and only if they have the same dimension (m  n)
(i.e., m rows  n columns) and each corresponding element is equal:
aij = bij] for i = 1, 2, …, m and j = 1, 2, …, n.
2. To add two matrices, add each corresponding element.
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3. To subtract two matrices, subtract each corresponding element.
4. To multiply a matrix by a scalar, multiply each element by the scalar.
 b1 
b 
2
5. If  a1 a2 ... an  is a row of matrix A, and   is a column of matrix B, then the inner product is the
 
 
bn 
number a1b1 + a2b2 + … + anbn.
6. The product of an (m  n) matrix A and an (n  k) matrix B is an (m  k) matrix whose elements are
formed by taking the inner product of each row of A with each column of B.
7. Properties of matrix arithmetic:
a. A+ (B + C) = (A + B) + C (associative property of addition)
b. A(BC) = (AB)C (associative property of multiplication)
c. A(B + C) = AB + AC (distributive property)
(B + C)A = BA + CA (distributive property)
d. A + B = B + A (commutative property of addition)
e. Note: matrix multiplication is not commutative for all matrices.
f. Additive identity matrix: [0]
g. Multiplicative identity matrix: The (multiplicative) identity matrix In is the n x n matrix that has
 1 0 0 ... 0 
 0 1 0 ... ...


all zero entries except for 1 on the main diagonal. I n   0 0 ... ... 0 


... ... ... 1 0 
 0 ... 0 0 1 
Section 9.6: Inverses of Matrices and Matrix Equations
1. The inverse of a square n x n matrix A is another square matrix A –1 that obeys the property
AA –1 = A –1A = In.
a b 
1  d b 
2. Inverse of a 2 x 2 matrix: If A  
, then A1 

ad  bc  c a 
c d 
3. Inverse of an N x N matrix:
a. Form an augmented matrix by writing the identity matrix next to the given matrix.
b. Do row operations to transform the left hand matrix into the identity matrix.
c. The resulting matrix on the right is the inverse of the original matrix.
4. Solving a matrix equation: If A is a square n x n matrix (representing the matrix of coefficients of a
linear system), and X is a column matrix with n rows (representing the matrix of variables), and B is
another column matrix with n rows (representing the matrix of constant terms), then the solution of
AX = B is given by X = A –1B. This is a third method for solving systems of linear equations.
5. Note: If a matrix is not invertible, then the system of equations it represents has no solution.
Section 9.7: Determinants and Cramer’s Rule
a b
1. Determinant of a 2 x 2 matrix:
 ad  bc
c d
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2. Determinant of a 3 x 3 matrix (not in the book):
3. Determinant of any n  n matrix:
a. First, the minor Mij of the element aij of a square n x n matrix A is the determinant of the matrix
obtained by deleting the ith row and jth column of A.
b. Next, the cofactor Aij of the element aij of a square n x n matrix A is Aij = (-1)i+jMij.
c. Finally, the determinant of a square matrix is calculated by multiplying each element in any row
or column by its cofactor, and then adding the results.
d. Strategy to make calculating determinants simpler: pick a row or column that has a lot of zeros.
4. Invertibility criterion: A square matrix A is invertible iff detA ≠ 0.
5. Adding a multiple of one row of a matrix to another row does not change its determinant.
a. Implied strategy: if you can add rows to get a row with a lot of zeros, then do it, because
calculating the determinant will be easier.
b. Note: exchanging rows of a matrix flips the sign of the determinant.
c. Note: multiplying a row by a constant divides the determinant by that same constant.
6. If a system of n linear equations in n variables x1, x2, …, xn is equivalent to the matrix equation DX = B,
Dx
and if |D|  0, then its solutions are xi  i , where Dxi is the matrix obtained by replacing the ith
D
column of D with the n  1 matrix B.
7. If a triangle has coordinates (x1, y1),
x1
(x2, y2), and (x3, y3), then its area is A   x2
x3
y1 1
y2 1 .
y3 1
Section 9.9: Systems of Inequalities
1. To graph an inequality:
a. Graph the equation.
i. Use dashed lines for > or <.
ii. Use solid lines for  or .
b. Choose a test point.
i. If the test point satisfies the inequality, then shade in that side of the plane.
ii. If the test point does not satisfy the inequality, then shade in the other side of the plane.
2. To graph a system of inequalities:
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a. Graph each inequality and shade in the appropriate side of each line.
b. The union of all the shaded regions satisfies the system.
c. Also find the intersection points.
Section 11.1: Sequences and Summation Notation
1. A sequence is a function f whose domain is the set of natural numbers. The values f(1), f(2), …, f(n), …
are called the terms of the sequence. Alternative notation: f(n) = an.
2. A partial sum of a sequence is Sn = a1 + a2 + … + an.
n
3. In sigma notation, Sn =
a
k 1
k
.
4. Properties of sums:
n
a.
k 1
n
b.
n
n
k 1
k 1
  ak  bk    ak   bk
 ca
k 1
k


 c   ak 
 k 1 
n
Section 11.2: Arithmetic Sequences
1. An arithmetic sequence is of the form a, a + d, a + 2d, … or an = a + (n – 1)d, where a is called the first
term, and d is called the common difference.
n
n
2. An arithmetic sequence’s partial sum is given by S n   2a  (n  1)d    a  an  .
2
2
Section 11.3: Geometric Sequences
1. A geometric sequence is of the form a, ar, ar2, … or an = arn-1, where a is called the first term, and r is
called the common ratio.
1 rn
2. A geometric sequence’s partial sum is given by Sn  a
.
1 r
a
3. If |r| < 1, then the infinite geometric series a + ar + ar2 + … + arn-1 + … has the sum S 
.
1 r
Section 11.5: Mathematical Induction
Principle of Mathematical Induction:
For each natural number n, let P(n) be a statement depending on n. Suppose that the following two conditions
are satisfied.
1. P(1) is true.
2. For every natural number k, if P(k) is true, then P(k+1) is true.
Then P(n) is true for all natural numbers n.
To apply the principle of mathematical induction:
1. Prove that P(1) is true.
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2. Assume that P(k) is true, then use that assumption to prove that P(k+1) is true. This is called the
induction hypothesis.
Sums of powers:
n
n
k 1
k 1
n
n
 k 0  1  n
n  n  1
2
k 1
k 1
n
n  n  1 2n  1
k2 

6
k 1
 k1   k 
n
k
k 1
3

n2  n  1
2
4
Section 11.6: The Binomial Theorem
Pascal’s Triangle is a shortcut for remembering the coefficients; every entry is obtained by adding the two
entries diagonally above it.
(a + b)0
1
1
(a + b)
1
1
(a + b)2
1
2
1
3
(a + b)
1
3
3
1
(a + b)4
1
4
6
4
1
5
(a + b)
1
5
10
10
5
1
(a + b)6
1
6
15
20
15
6
1
The binomial coefficients can also be calculated with the following formula:
 n
n!
 
 r  r !(n  r )!
Where the factorial operation, !, is the product of all integers from 1 to n: n!  1 2  3 
 n
 n
 n
 n
Then the binomial expansion is: (a  b)n    a n    a n 1b    a n 2b 2  ...    b n .
 0
 1
 2
 n
 n  n 
Note:    

r  nr
  n 1  n , and 0!  1.