Pre-Calc Final Exam – Flashcards
Flashcard maker : Collin Foley
Definition of a Limit
We say the limit of f(x), as x approaches a, equals L if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a, but not equal to a.
Point slope equation
y1-y2=m(x1-x2)
slope
rise/run or change in y/change in x
slope-intercept form
y=mx+b
vertical line
slope is undefined
general linear equation
ax+by+c=0
Slopes of parallel lines
are the same
Slopes of perpendicular lines
negative reciprocal
Multiplying bases
add the exponents
something raised to the zero power
equals 1
raised to -n
equal to 1/a^n
a^m/a^n
a^mn
(ab)^n
a^nb^n
(a/b)^n
a^n/b^n
(a/b)^ -n
(b/a)^n
a^-n/b^-m
b^m/a^n
square root (a)/b
b^2=a
nsquareroot(a)/b
b^n=a
Logarithmic basic equation
log10^x
loga^x=y
a^y=x
Natural logarithm
e^x
Exponential Functions
y=ab^x
the a in y=ab^x changes what?
y-intercept
y=-2^x
the negative in the front indicates a reflection over the x-axis.
y=3^(x-2)
the (x-2) indicates a shift 2 to the right
Compound Interest
A(t)=P(1+r/n)^nt
A(t)
amount after t years
P
principal
r
interest rate per year
n
number of times interest is compounded per year
t
the number of years
Continuously Compounded Interest
A(t)=Pe^rt
log25^squareroot of 5=x
1/4
Domain
x-values
Range
y-values
[ ]
if the value is included
( )
if the value is not included
loga^1
equals 0
loga^a
1
loga^a^x
x
a^loga^x
x
Log of a product
sum of the logs
Ex: log3^2 + log3^9
log3(2*9)=log3^18
The division of a log
difference of two logs
Quadratic Equation
-b+- squareroot of (b)2-4(a)(c)/2(a)
Distance Formula
squareroot (x2-x1)^2-(y2-y1)^2
Midpoint Formula
(x2+x1/2, y2+y1/2)
Equation of a circle
x2+y2=1
sin
y-values
cosine
x-values
pi/6
(squareroot(3)/2, 1/2)
pi/4
(squareroot(2)/2, squareroot(2)/2)
pi/3
(1/2, squareroot (3)/2)
pi/2
(0,1)
2pi/3
(-1/2, squareroot(3)/2)
3pi/4
(-squareroot(2)/2, squareroot(2)/2)
5pi/6
(-squareroot(3)/2, 1/2)
pi
(-1,0)
7pi/6
(-squareroot(3)/2, -1/2)
5pi/4
(-squareroot(2)/2, -squareroot(2)/2)
4pi/3
(-1/2, -squareroot(3)/2)
3pi/2
(0, -1)
5pi/3
(1/2, -squareroot(3)/2)
7pi/4
(squareroot(2)/2, -squareroot(2)/2)
11pi/6
(squareroot(3)/2, -1/2)
2pi
(1,0)
radius of the Unit Circle
1
Proving a point lies on the Unit Circle
x2+y2=1
Initial point
point where it starts
Terminal point
where it stops
Coterminal point
going around the unit circle once before ending
Circumference of a circle
2pir
reference angle
how far away a point is from the x-axis, always positive
Pythagorean theorem
a^2+b^2=c^2
sin
opp/hyp
cos
adj/hyp
tan
sin/cos
Pythagorean identity
(sin(t))^2 + (cos(t))^2=1
csc
1/sin(x)
sec
1/cos(x)
cos(-t)
cos(t)
sin(-t)
-sin(t)
tan(-t)
-tan(t)
Amplitude
height of the wave, the y-values
y=sin(t) +2
moves the graph up 2
period
how long it takes to complete one cycle
period changes
period * 2pi
reciporcal of sine
larger
1 radian
180/pi
1 degrees
piR/180
Arc length
s=thetha*r
Area of a sector
A=1/2thethar^2
Arc length measures
always in radians
Angular velocity/speed
w=thetha/t
Linear speed velocity
y=s/t
90-45-45 degree triangle
squareroot(2)/2 +squareroot(2)/2=1
Short side of the triangle
is half of the hypotenuse
socahtoa
calculator must be set in degree mode
Law of sines
sin(A)/a = sin(B)/b = sin(C)/c
Law of cosines
c^2=a^2+b^2-2abcos(c)
Sin sum formula
sin(s+a)=sin(a)cos(a)+sin(a)cos(s)
Sine difference formula
sin(s-a)=sin(s)cos(a)-sin(a)cos(s)
cos sum formula
cos(s+a)=cos(s)cos(a)-sin(s)sin(a)
cos difference formula
cos(s-a)=cos(s)cos(a)+sin(s)sin(a)
conjugate
opposite sines
tan sum formula
tan(s+a)=tan(s)+_ tan(a)/1_+tan(s)tan(a)
Cos (x/2) Half Angle formula
+- squareroot (cos(x) +1)/2
Sin (x/2) Half angle formula
+-squareroot(1-cos(x)/2
tan (x/2) half angle formula
+-squareroot ((1-cos(x)/(cos(x)+1))
Sin double angle
sin(2x)=2sin(x)cos(x)
cos double angle
cos(2x)=1-2Sin^2(x)
tan double angle
2tan(x)/1-tan^2(x)
one-one function
f(a)=f(b)
If an equation is one-to-one
there is an inverse
Solving for inverse algebraically
domain of the function is the range of the inverse
Inverse function of f-1(f(x))=x
f(f-1(x))=x
Inverse sin graph
x values are pi/2- -pi/2, y-values are -1 to 1
Inverse cos graph
x values are -pi to pi, y-values are -1 to 1
Inverse tan graph
x values are all real numbers, y values are -pi/2 to pi/2
Reference Triangle
used to solve inverse trig functions
Solve the following 2sin(3x)-1=0
sin(3x)=1/2
General solution format
y=x+2K/3
Substitution methods
solve for 1 equation
Elimination method
elimination one variable to solve for the other
Addition/Subtraction method
part of elimination method
Solution set
{x,y}
parallel lines
no solution
Intersecting lines
exactly one solution
Dependent system
L1=L2, single line
Back substitution
substituting a value in for a variable
Three methods to solve limits
1. Numerical
2. Graphically
3. Algebraically
2. Graphically
3. Algebraically
open bubble
Limit exists but f(1) doesn’t
closed bubble
limit DNE but f(1) does
Right hand limit
lim a–+
Left hand limit
lim a_-
Limit Rule #1
The limit as x approaches any number f(x) is equal to it’s constant. lim (C)/x–a = C
numerator equals 0
quotient is 0
denominator equals 0
undefined
both numerator and denominator are 0
indeterminate
Non-zero factors
cancel out
slope of sec
change in y/change in x
Mtan
f(a+h) – f(a)/h
Msec
f(x) – f(a)/x-a
Rationalizing
multiply by conjugate
Polynomials in Limits
solve
