Download Chapter 10 Review Concepts.

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Chapter 1-10 Review Handout
Factoring
Difference of two squares
a2 - b2 = (a + b)(a - b)
Perfect square trinomial
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
Sum of two cubes
a3 + b3 = (a + b)(a2 - ab + b2)
Difference of two cubes
a3 - b3 = (a - b)(a2 + ab + b2)
Multiplication and Division of Rational Expressions
If P, Q, R, and S are polynomials, then
Multiplication
P
R
PR
--- * --- = ---- (where Q, S cannot equal zero)
Q
S
QS
Division
P
R
P
S
PS
----- = --- * --- = ---(where Q, R, and S cannot equal zero)
Q
S
Q
R
QR
Addition and Subtraction of Rational Expressions
If P, Q, and R are polynomials, then
Addition
P
Q
P+Q
--- + --- = ---- -- (where R cannot equal zero)
R
R
R
Subtraction
P
Q
P-Q
--- - --- = ---- -- (where R cannot equal zero)
R
R
R
General Form of a Linear Equation
Ax + By + C = 0
Where A, B, and C are constants and A and B are not both zero
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Equations of Straight Lines
Vertical line:
x=a
Horizontal line:
y=b
Point Slope form:
y – y1 = m(x – x1)
Slope-intercept form: y = mx + b
The Sum, Difference, Product, and Quotients of Functions
(f + g)(x) = f(x) + g(x)
Sum
(f – g)(x) = f(x) – g(x)
Difference
(fg)(x) = f(x)g(x)
Product
(f /g)(x) = f(x)/g(x) where g(x)  0
Quotient
The Composition of Two Functions
(g  f)(x) = g(f(x))
Exponent Function
f(x) = bx
(b>0, b  1)
Laws of Exponents
Let a and b be positive numbers and let x and y be real numbers. Then,
1. bx  by = bx + y
2. bx/by = bx-y
3. (bx)y = bxy
4. (ab)x = axbx
5. (a/b)x = ax/bx
Logarithm of x to the Base b
y = logbx if and only if x = by (x > 0 and b>0, b  1 )
Laws of Logarithms
If m and n are positive numbers, then
1. logbmn = logbm + logbn
2. logb(m/n) = logbm – logbn
3. logbmn = nlogbm
4. logb1 = 0
5. logbb = 1
note: These properties also hold for the natural log (ln).
 blogbx = x
(x > 0)
x
 logbb = x
(for any real number x)
lnx
 e
=x
(x > 0)
x
 lne = x
(for any real number x)
Exponential Growth Function
Q(t) = Q0ekt
(0  t < )
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Exponential Decay Function
Q(t) = Q0e-kt
(0  t < )
The Gauss-Jordan Elimination Method
1. Write the augmented matrix corresponding to the linear system.
2. Interchange rows, if necessary, to obtain an augmented matrix in which the first entry in the
first row is nonzero. Then pivot the matrix about this entry.
3. Interchange the second row with any row below it, if necessary, to obtain an augmented
matrix in which the second entry in the second row is nonzero. Pivot the matrix about this
entry.
4. Continue until the final matrix is in row-reduced form.
Row Operations
1. Interchange any two rows.
2. Replace any row by a nonzero constant multiple of itself.
3. Replace any row by the sum of that row and a constant multiple of any other row
Limit of a Function
The function f has the limit L as x approaches a, written
lim f(x) = L
x→a
if the value of f(x) can be made as close to the number L as we please by taking sufficiently close
to (but not equal to) a.
Theorem
Let f be a function that is defined for all values of x close to x = a with the possible exception of
a itself. Then
lim f(x) = L
x→a
if and only if
lim f(x) = lim f(x) = L
x→a+
x→a¯
Derivative of a Function
The derivative of a function f with respect to x is the function f ’
f ’(x) = lim
h→0
f(x + h) – f(x)
h
Theorem
a. If f ’(x) > 0 for each value of x in an interval (a,b), then f is increasing on (a,b).
b. If f ’(x) < 0 for each value of x in an interval (a,b), then f is decreasing on (a,b).
c. If f ’(x) = 0 for each value of x in an interval (a,b), then f is constant on (a,b).
Theorem
a. if f ”(x) > 0 for each value of x in (a,b), then f is concave upward on (a,b).
b. if f ”(x) < 0 for each value of x in (a,b), then f is concave downward on (a,b).
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Inflection Point
A point on the graph of a continuous function f where the tangent line exists and where the
concavity changes is called an inflection point.
To find an Inflection Point do the following:
1. Compute f ”(x).
2. Determine the numbers in the domain of f for which f ”(x) = 0 or f ”(x) does not exist.
3. Determine the sign of f ”(x) to the left and right of each number c found in step 2. If there
is a change in the sign of f ”(x) as we move across x = c, then (c,f(c)) is an inflection
point of f.
4.
Guide to Curve Sketching
1. Determine the domain of f
2. Find the x- and y-intercepts of f. (note: if you can’t find the intercepts or it is hard then don’t
worry about it)
3. Determine the behavior of f for large absolute values of x
4. Find all horizontal and vertical asymptotes of f.
5. Determine the intervals where f is increasing and where f is decreasing
6. Find the relative extrema of f
7. Determine the concavity of f
8. Find the inflection points of f
9. Plot a few additional points to help further identify the shape of the graph of f and sketch the
graph.