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101 Cards in this Set
- Front
- Back
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Double Angle Formulas |
sin2x
cos2x tan2x |
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sin2x= |
2sinxcosx |
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cos2x= |
cos^2x-sin^2x = 1-2sin^2x = 2cos^2x-1 |
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tan2x= |
2tanu/(1-tan^2u)
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Triple Angle Formula ex. cos3x= |
cos(x+2x) then use sum formula for cos |
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sin^2x= |
(1-cos2x)/2
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cos^2x= |
(1+cos2x)/2 |
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tan^2x= |
(1-cos2x)/(1+cos2x) |
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sin(u+v)= |
sinucosv+cosusinv |
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sin(u-v)= |
sinucosv-cosusinv |
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cos(u+v)= |
cosucosv-sinusinv |
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cos(u-v)= |
cosu-cosv+sinusinv |
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tan(u+v)= |
(tanu+tanv)/(1-tanutanv) |
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tan(u-v)= |
(tanu-tanv)/(1+tanutanv) |
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sin^2x+cos^2x= |
1 |
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1+tan^2x= |
sec^2x |
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1+cot^2x= |
csc^2x |
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Which trig functions are even? |
cos(-x)=cosx sec(-x)=secx |
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inverse sine function: domain |
[-1,1] |
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inverse sine function: range |
[-pi/2, pi/2] |
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inverse cosine function: domain |
[-1,1] |
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inverse cosine function: range |
[0, pi] |
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inverse tangent function: domain |
(- infinity, + infinity) |
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inverse tangent function: range |
(-pi/2, pi/2) |
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What are natural numbers? |
Counting #s 1, 2, 3, 4... |
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What are integers? |
Negative and positive counting #s -2, -1, 0, 1, 2, 3... |
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What are whole numbers? |
Counting #s and ZERO 0, 1, 2, 3...
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What are rational numbers? |
fractions, integers finite decimals infinitely REPEATING decimals 0.15, -2.18 |
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What are irrational numbers? |
infinite nonrepeating decimals sqrt(2) pi e 1.0384087049.... |
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What are real numbers? |
Rational + Irrational anything that can be drawn on a number line
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How to solve x^3-1 |
(x-1)(x^2+x+1) |
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How to solve x^3+1 |
(x+1)(x^2-x+1) |
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How to solve a^3-b^3 |
(a-b)(a^2+ab+b^2) |
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How to solve a^3+b^3 |
(a+b)(a^2-ab+b^2) |
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x^(5/2) is the same thing as... |
sqrt(x^5) |
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Zeroes |
when the SIMPLIFIED numerator= 0 these are included in the domain |
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Domain |
when the denominator cannot equal 0 |
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Vertical Asymptotes |
when the SIMPLIFIED denominator equals 0 automatically excluded from domain |
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Horizontal Asymptotes |
numerator > denominator= NO HA (limit goes to positive or negative infinity) numerator < denominator= y=0
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X-intercepts |
where numerator= 0 |
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how to find possible rational zeroes |
all multiples of last number over all multiples of leading number |
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how to find coordinates of hole |
plug in "hole" to SIMPLIFIED expression |
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the nth root of a^n if n is even |
|a| |
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the nth root of a^n if n is odd |
a |
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Complete the square ax^2+bx+c=0 |
1. make sure a=1 2. move constant to other side 3. add the square of half x coefficient |
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Quadratic formula |
-b + or - sqrt(b^2-4ac) -------------------------------- 2a |
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disriminant |
b^2-4ac |
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distance between two points |
sqrt((x2-x1)^2+(y2-y1)^2) |
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midpoint formula |
((x2+x1)/2, (y2+y1)/2) |
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vertical line test |
to test whether a function or not each x value gets ONE y-value |
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average rate of change |
change in y/ change in x |
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area of a circle |
A= pir^2 |
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horizontal line test |
to determine whether one-to-one, therefore an inverse function |
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standard form for quadratic function (also the equation for a parabola) |
f(x)= a(x-h)^2 + k to get this, complete the square of a quad. fxn |
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-b/2a is the |
axis of symmetry or where "ball reaches max height" |
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ln1= |
0 |
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lne= |
1 |
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what reflection is ln(-x) |
y-axis |
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y= lnx if and only if |
e^y = x |
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exponential growth model |
y= ae^bt |
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if doubling time T is given, then |
b= ln2/T |
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indeterminants |
0/0 infinity/infinity infinity - infinity infinity x zero |
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evaluating 0/0 absolute value limits |
plug in number close to correct side if positive: remove absolute value signs if negative: negative sign in front of parenthesis |
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#/0 |
answer will be +infinity or -infinity vertical asymptote |
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removable discontinuities |
holes
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Nonremovable discontinuities |
infinites 1/x jump |x|/x |
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Intermediate Value Theorem: how to solve |
plug in numbers if sign change, YES *if it is not continuous at point in between, IVT does NOT work* |
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limx-->0 sinx/x |
1 |
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limx-->0 tanx/x |
1 |
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limx-->0 (cosx-1)/x |
0 |
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how to show work to find HA (limx--> infinity) |
divide by highest power |
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horizontal asymptotes... what limit? |
as x --> BOTH + and - infinitiy |
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#/infinity |
0 |
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sqrt(x^2) |
|x| |
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e^infinity |
infinity |
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e^-infinity |
0 |
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inside of ln or log fxn must be |
> 0 |
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inside of sqrt fxn must be |
> or = 0 |
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multiplicity rules |
even powers= same sign odd powers= change signs |
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when you see inequality WHAT comes to mind |
NUMBER LINE |
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finding domain of composite function |
domain of INNER fxn domain of end result |
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y- axis symmetry (even) |
f(x) = f(-x) |
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origin symmetry (odd) |
f(-x) = -f(x) |
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what points on f(x) = e^x |
(0,1) above x-axis shoots up |
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what points on f(x) = lnx and f(x) = logx |
(1,0) to the right of y-axis down to up |
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exponential growth/decay |
f(t)= initial(factor)^(t/time) |
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area of circle |
pir^2 |
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volume of cylinder |
V= pir^2h |
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surface area of cylinder |
2pir^2 + 2pirh |
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volume of cone |
1/3pir^2h |
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surface area of cone |
pirsqrt(r^2 + h^2) |
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change of base formula |
logb(x)=
logax -------- logab |
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what to do with constant in front of ln |
move to exponent
ex. 2ln(x) = ln(x^2) |
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when looking for instantaneous velocity, |
let h--> 0 |
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0/infinity |
0 |
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lim x--> infinity tan^-1(x) |
pi/2 |
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lim x--> - infinity tan^-1(x) |
-pi/2 |
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ln(0+) |
- infinity |
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ln(infinity) |
infinity |
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infinity x infinity |
infinity |
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instantaneous velocity (limit of slope) equation |
limx-->a of (f(x)-f(a))/(x-a)
limh-->0 of (f(a+h)- f(a))/h |
