graphs, lines, polynomials, exponentials and logarithms web viewstretching/squeezing the function....
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Graphs, lines, polynomials, exponentials and logarithms
Vertical transformation
Where downwards by k
Horizontal transformation
Where right by k units
Stretching/squeezing the function
Where rises/falls double as quickly
The Quadratic Function
General form
Intercept form
Vertex form
To find the vertex form of a quadratic function:
take ", and add/subtract it to the formula Find numbers that multiplies to give you and adds to give you b (roots) transformations
Finding asymptotes
Where
Vertical Asymptotes:
1. After cancelling common factors, where , there is a vertical asymptote
Horizontal Asymptotes
1. If the degree of the degree of , is the horizontal asymptote
2. If the degree of the degree of , is the horizontal asymptote:
a. is the leading coefficient of
b. is the leading coefficient of
3. If degree of degree of , there is no horizontal asymptote
A graph has an exponential shape where . Exponent Laws
Where a and b are positive, , x & y are real.
1.
2.
3.
4.
5.
6.
etc.
1.
2. iff
3. where iff
Logarithms iff
that is:
Log properties
Where b, M and N are positive, and p & x are real numbers
1.
2.
3.
4.
5.
6.
7.
8.
9. iff
Changing base of log etc.
Note, calculator has and
Financial applications
simple interest
The compound interest formula
Continuous compound interest
Computing growth time
Since ,.
Annual percentage yield
, compounded continuously,
Future value of an ordinary annuity
Present value of an ordinary annuity
Derivatives
Slope of a secant between two points
Average rate of change (slope of a secant between x and x+h)
The derivative from first principles
basic differentiation properties
1. Constant
2. Just an x
3. A power of x
4. A constant*a function
5. Sum/difference
Derivatives of logarithmic and exponential functions
1. Base e exponential
2. Base e exponential with constant in power
3. Other exponential
4. Natural log
5. Other log
The product rule
The quotient rule
the chain rule
The general derivative rules
Local extrema
Where the first derivative is 0, and the sign of the first derivative changes around it, it is a local extrema:
1. 0 + minimum
2. + 0 - maximum
3. 0 or + 0 + not a local extrema
Note, where , finding can also identify whether it is a local extrema: where , it is a local minimum; where , it is a local maximum. This test is invalid where .
Graph sketching
1. Analyze , find domain and intercepts
2. Analyze , find partition numbers and critical values and construct a sign chart (to find increasing/decreasing segments and local extrema)
3. Analyze , find partition numbers and construct a sign chart (to find concave up and down segments and to find inflexion points)
4. Sketch : locate intercepts, maxima and minima and inflexion points: if still in doubt, sub points into
Optimization
1. Maximize/minimize on the interval I.
2. Find absolute maxima/minima: at a critical value or at endpoint
IntegralsIndefinite integrals of basic functions
1.
2. x to the n
3. e to the x
4. x denominator
Indefinite integrals of a constant multiplied by a function, or, two functions
1.
Integration by substitution
Based on chain rule:
General indefinite integral formulae
General indefinite integral formulae for substitution
1.
Method of integration by substitution
1. Select a substitution to simplify the integrand: one such that u and du (the derivative of u) are present
2. Express the integrand in terms of u and du, completely eliminating x and dx
3. Evaluate the new integral
4. Re-substitute from u to x.
Note, if this is incomplete (i.e. du is not present) you may multiply by the constant factor and divide, outside of the integral, by its fraction.
The definite integralError Bounds
For right and left rectangles, f(x) is above the x-axis:
Properties of a definite integral
1.
2.
3.
4. , where k is a constant
5.
6.
The fundamental theorem of calculus
You do not need to know C
Average value of a continuous function over a period
More than 2 dimensionsFunctions of several variables
Find the shape of the graph by looking at cross sections (e.g. y=0, y=1, x=0, x=1).
Partial derivatives
derived with respect to x ||| derived with respect to x, then y
Maxima and minima
1. Express the function as
2. Find , and simultaneously equate
3. Find (A, B, and C)
4. Find A, and .
a. IF AC-B*B>0 & A0 & A>0, f(a,b) is local minimum
c. IF AC-B*B 2
b
1
>b
2
= 1 2
q=b
1
-b
2
Yi = 0 + ( +2 )X1i +2X2i
Y
i
=b
0
+(q+b
2
)X
1i
+b
2
X
2i
n(x) =
n(x)=
Yi = 0 +X1i +2X1i +2X2i
Y
i
=b
0
+qX
1i
+b
2
X
1i
+b
2
X
2i
X = X1 + X2
X=X
1
+X
2
Yi = 0 +X1i +2X
Y
i
=b
0
+qX
1i
+b
2
X
H0 : = 0,H1 : > 0
H
0
:q=0,H
1
:q>0
> 0 B1 > B2
q>0B
1
>B
2
d(x)
d(x)
F =SSRR SSRUR( ) / qSSRUR / n k 1( )
F=
SSR
R
-SSR
UR
()
/q
SSR
UR
/n-k-1
()
F =RUR2 RR
2( ) / q(1 RUR
2 ) / (n k 1)
F=
R
UR
2
-R
R
2
()
/q
(1-R
UR
2
)/(n-k-1)
Elasticity = dydxxy
Elasticity=
dy
dx
x
y
y = a b
y=
a
b
a
a
n(x)
n(x)
b
b
d(x)
d(x)
n(x)>
n(x)>
d(x)
d(x)
f (x) = bx, x 1, x > 0
f(x)=b
x
,x1,x>0
a 1,b 1
a1,b1
axay = ax+y
a
x
a
y
=a
x+y
ax
ay= axy
a
x
a
y
=a
x-y
(ax )y = axy
(a
x
)
y
=a
xy
(ab)x = axbx
(ab)
x
=a
x
b
x
(ab)x = a
x
bx
(
a
b
)
x
=
a
x
b
x
ax = ay
a
x
=a
y
x = y
x=y
x 0,ax = bx
x0,a
x
=b
x
a = b
a=b
y = loga x
y=log
a
x
x = ay
x=a
y
a = bc : logb a = c
a=b
c
:log
b
a=c
b 1
b1
logb1= 0
log
b
1=0
logb b =1
log
b
b=1
logb bx = x
log
b
b
x
=x
blogb x = x, x > 0
b
log
b
x
=x,x>0
logb MN = logb M + logb N
log
b
MN=log
b
M+log
b
N
logbMN= logb M logb N
log
b
M
N
=log
b
M-log
b
N
logb Mp = p logb M
log
b
M
p
=plog
b
M
logb M = logb N
log
b
M=log
b
N
M = N
M=N
"log"= log10
"log"=log
10
"ln"= loge
"ln"=log
e
ln xlnb
= logb x
lnx
lnb
=log
b
x
ex lnb = bx
e
xlnb
=b
x
I = PR t
I=PRt
A = P(1+ i)n = P(1+ rm)mt
A=P(1+i)
n
=P(1+
r
m
)
mt
A = Pert
A=Pe
rt
A = P(1+ i)n
A=P(1+i)
n
lnA = n ln(P(1+ i))
lnA=nln(P(1+i))
APY = (1+ rm)m 1
APY=(1+
r
m
)
m
-1
APY = er 1
APY=e
r
-1
FV = PMT ((1+ i)n 1i
)
FV=PMT(
(1+i)
n
-1
i
)
PV = PMT (1 (1+ i)n
i)
PV=PMT(
1-(1+i)
-n
i
)
f (a) f (b)a b
f(a)-f(b)
a-b
f (a+ h) f (a)h
,h 0
f(a+h)-f(a)
h
,h0
limh0
f (x + h) f (x)h
lim
h0
f(x+h)-f(x)
h
f (x) = c f '(x) = 0
f(x)=cf'(x)=0
f (x) = ax2 + bx + c
f(x)=ax
2
+bx+c
f (x) = x f '(x) =1
f(x)=xf'(x)=1
f (x) = xn f '(x) =