MATH 1081 Calculus for Social Science and Business

Difference of two squares: x²?y²
x² – y² = (x-y)(x+y)
Perfect square: x²+2xy+y²
x² + 2xy + y² = (x+y)²
Difference of 2 cubes: x³-y³
(x-y)(x²+xy+y²)
Quadratic Formula
ax²+bx+c=0, where a?0

x= -b±?(b²-4ac) / 2a

Sum of 2 cubes: x³+y³
(x+y)(x²-xy+y²)
Properties of Inequality
For all real numbers a, b, and c:
1. If a0, then ac bc
Solve: x²-x < 12
Set equation equal to 0:
x²-x-12 = 0
Factor: (x-4)(x+3) = 0
x = 4 and x = -3
these 2 pts represent changes in the graph – so find whether the factor will be positive or negative in each area.
Calculate for each factor:
(x-4)………..-.|..-..|..+
(x+3)……….-.|..+..|..+
(x-4)(x+3) .+.|..-..|..+
line…..(??..-3….4….+?)
This table represents how the factors change based on numbers in that area.
for x-4 between negative infinity and -3 i chose -5 – so -5-4 equals -9 and all #s in that area will result in x-4 being negative
and for (x+3) use -5, and all numbers between neg infinity and -3 will result in negative result for (x+3)
then for multiplying (x-4)(x+3) => – * – = positive, betwix -3 to 4, -*+=-, etc.
Parallel lines
2 lines are parallel if and only if they have the same slope or if they are both vertical
Perpendicular Lines
2 lines are perpendicular if and only if the product of their slopes is -1, or if one is vertical and the other horizontal
Linear Function
a relationship f defined by
y = f(x) = mx + b,
for real #s m and b, is a linear function
Linear Cost Function equation
C(x) = mx + b, the m is the marginal cost and b the fixed cost.
slope of a line formula given 2 pts
(y?-y?)/(x?-x?)=m
where x?,y? and x?,y? are points on the line
and m is the slope of the line
slope-intercept form
y=mx+b
m is the slope of the line
b is the y-intercept of the line
point-slope form
y-y?=m(x-x?)
m is the slope
(x?,y?) is a point on the line
x=k
vertical line, undefined slope, i.e. 1/0
y=k
horizontal line, slope=0
least squares line: y intercept formula
least squares line: y intercept formula
Y=mx+b, slope is m, b is y-intercept:
b=y-mx
that is to say, given a point on the line, just put in the x, y into the equation and solve to get b – the y-intercept
given a point xsubn,ysubn the equation:
b=(?y – m(?x)/n
least squares line: slope formula
m=(n(?xy)-(?x)(?y))/(n(?x²)-(?x)²)
break-even formula
Revenue = Cost
R(x)=C(x)
Profit P(x)=
R(x)-C(x)
Farenheit conversion to Celcuis
F = 9/5C +32
Celcius conversion to Farenheit
C=5/9(F – 32)
Correlation Coefficient
How well the original data fits a straight line:
r = (n(?xy)-(?x)(?y))/(?(n(?x²)-(?x)²) ×?(n(?y²)-(?y)²))
function
a function is a rule that assigns to each element from one set exactly one element from another set
domain and range
the set of all possible values of the independent variable in a function is called the domain of the function (x), and the resulting set of possible values of the dependent variable is called the range (y)
vertical line test
a graph represents a function if and only if every vertical line intersects the graph in no more than one point
graph of a quadratic function
the graph of the quadfunc f(x) = ax²+bx+c is a parabola with its vertex at (-b/2a,f(-b/2a)). the graph opens upward if a>0 and downward if a<0
collnear points
3 or more points that are in a straight line
About r – the correlation coefficient
* the correlation coefficient measures the strength of the linear relationship between 2 variables
* r is between 1 and -1 or equal to 1 or -1
* if r = 1, the least squares line has a positive slope
* r = -1 gives a negative slope
* if r = 0 there is no linear correlation between the data points
(but some nonlinear function might provide excellent fit for the data)
* also a r=0 may also indicate that the data fit a horizontal line.
* the exact value of the linear relationship depends upon n – the number of data points
Suppose a positive linear correlation is found between 2 quantities. Does this mean that one of the quantities increasing causes the other to increase?
No. a positive correlation means that as one of the quantities increases, the other quantity also increases. It does not mean that one of the quantities increasing causes the other to increase. To prove that one of the quantities increasing causes the other to increase would require further research.
least squares line equations
least squares line equations
Zero and Negative Exponents
a?=1
a??=1/a?
slope of horizontal line
m=0
slope of vertical line
m is undefined , i.e., 1/0
(??a)? =
a
??(a?) =
|a| if n is even
a if n is odd
??a × ??b =
??(ab)
??a/??b =
??(a/b)
b?0
??(??a) =
(?*?)?a
If n is an even natural # and a>0, or n is an odd #, then:
a¹/? =
??a
example 9¹/² = ²?9 = 3
a^(m/n)
(a¹/?)^m
example 27^2/3 = (27^1/3)²=3² = 9
Effective rate for compound interest
If r is the annual stated rate of interest and m is the number of compounding periods per year, the effective rate of interest is:
r subE = ( 1 + r/m)^m -1
Exponential growth and decay function
y=y?e^(kt)

Let y? be the amount or # of some quantity present at time t=0. The quantity is said to grow or decay exponentially if for some constant k, the amt present at time t .

Change of base theorem for exponents
For every positive real number a:
a^x = e ^((ln a)x)
change of base theorem for logs
a & b > 0, a?1, b?1

log sub a x = (log sub b x) / (log sub b a)

y = log sub a x means
a^y = x
continuous compounding
A = Pe^(rt) dollars
If a deposit of P dollars is invested at a rate of interest r compounded continuously for t years, the compound amount is A.
Definition of e
As m becomes larger and larger, (1 + 1/m)^m becomes closer and closer to the number e whose approximate value is 2.718281828
Properties of Polynomial Functions. A polynomial function of degree n can have at most ________ turning points
n-1
If a graph has n turning points it must have a degree of
at least n + 1
In the graph of an even polynomial both ends
go up or down
In the graph of an odd polynomial …
one end goes up and one end goes down
If the graph goes up as x becomes a large positive # , the leading coefficient must be _____
positive
If the graph goes down as x becomes a large positive #, the leading coefficient must be _____
negative
Cummulative properties of real numbers
a + b = b + a
ab = ba
Associative property of real numbers
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Distributative property of real numbers
a(b + c) = ab + ac
Given graph of quadratic function:
f(x) = ax²+bx+c
is a parabola with vertex:
-b/2a is the x
y = f(-b/2a)
(-b/2a,f(-b/2a)) point of vertex
Given f(x) =ax²+bx+c
what can you say about a if the parabola opens upward?
a > 0
a^m * a^n =
a^(m+n)
a^m/a^n =
a^(m-n)
(a^m)^n =
a^(mn)
(ab)^m =
a^m * b^m
(a/b)^m =
a^m/b^m
log_a xy=
log_a x + log_a y
log_a (x/y) =
log_a x – log_a y
log_a x^r =
r * log_a x
log_a a =
1
log_a 1=
log_a a^r =
r
Completing the square: ax²+bx+c=0
a(x+d)²+e=0
d = b/2a
e = c – (b²/4a)
completing the square step-by-step
ax²+bx+c=0 into form
y=a(x-h)²+k
1. First, factor a from 1st and 2nd term
a(x²+(b/a)x) + c
2. Then take half of the coefficient of x inside the parans and square it. Subtract a times this number from c
a(x²+(b/a)x+[(b/2a)²]) + c -(a*(b²/4a²)
3. simplifying:
a(x+b/2a)²+c-(b²/4a)=y
Note: so h = -b/2a and k = c-(b²/4a)
Solve 3x²+9x+1=0 by completing the square into the form
y=a(x-h)²+k
and find the vertex.
1. 3(x²+3x)+1
2. 3(x²+3x+9/4)²+1-(3*9/4)
3. 3(x+3/2)²+1-(27/4)
4. y=3(x+3/2)²-(23/4)

vertex of this formula is (-3/2,-23/4)

What is horizontal asymptote if degree of denominator is greater than degree of numerator? x/x²
horizontal asymptote is y=0
what is horizontal asymptote if degree of numerator is greater than degree of denominator? x²/x
no horizontal asymptote
what is horizontal asymptote if the highest degree of the numerator is equal to highest degree of denominator? 2x²/x²
horizontal asymptote is equal to coefficients of highest degree
for this example: y=2