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| ORDER OF SOLVING A LONG GRAPH EQUATION |
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Definition
Intercepts
Symmetry
Domain + Range
Vertical Asymptote
Derivatives
Concavity
Relative Extrema
Points of Inflection
Horizontal Asymptotes
Limit at Infinity |
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| You can easily check these on your calculator by just graphing and using table function. |
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| Basically just substitute the x. |
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x--> 0 !x! / x (absolute value)
x--> 0 1/x^2 (unbounded)
x--> 0 sin 1/x (oscillating) |
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Definition
Trig Identity of x --> c
lim sinx = sin (c)
x--> c |
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| Limits that Aren't Direct Substitution |
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Definition
Multiply by conjugate
Factor and then cancel |
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| You can follow it with a pen. |
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f(c) is defined
lim f(x) x--> c exists
lim f(x) x--> c = f(c) |
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| Intermediate Value Theorem |
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| If f is continuous [a,b] and k is between f(a) and f(b) then there is a number c within [a,b] where f(c) = k |
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| Application of Intermediate Value Theorem |
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Definition
x^3 + 2x -1 [0,1]
f(0) = -1 a
f(1) = 1 b
f(0) = 0 f(c)=k |
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denominator = 0
x=c for the graph of f(x) if f(x) approaches infinity
(factor factorable equations more than one possible asymptote) |
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| Derivatives: Constant Rule |
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Constant turns into zero
d/dx 7 = 0 |
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d/dx [x^n] = nx^n-1
d/dx x^4 = 4x^3 |
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| Derivatives: The Constant Multiple Rule |
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Definition
Constant isn't derived.
d/dx [cf(x)] = cf'(x)
d/dx 3x^2 = 6x |
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| Derivatives: Sum and Difference Rule |
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Definition
d/dx (f(x) + g(x))= f'x + g'x
d/dx (f(x) - g(x))= f'x - g'x |
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fx Acceleration
f'x Velocity
f''x Position |
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| Derivatives: Product Rule |
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Definition
d/dx (f(x)*g(x))= f(x)g'(x) + f'(x)g(x)
Example: (3x-2x^2)(5+4x)
(3x-2x^2)(4)+(5+4x)(3-4x)
=-24x^2 + 4x + 15 |
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| Derivatives: QUOTIENT RULE! |
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Definition
d/dx = [fx/gx] =
(g(x)f'(x) - f(x)g'(x))/[(g(x))^2]
Example: (5x-2)/(x^2 + 1)
(x^2 + 1)(5) - (2x)(5x-2)
((x^2 + 1)^2) |
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| Derivatives: Trig Identities |
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Definition
Sinx = Cosx
Cosx = -Sinx
Tanx = Sec^2x
Secx = TanxSecx
Cotx = -csc^2x
Cscx = -cscxcotx |
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Definition
d/dx = [f(g(x)] = f'(g(x))g'(x)
or y'=f'(u)u'
Example: 3(2x^4-3x)^2
6x(2x^4 - 3x) (8x^3 - 3) |
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| Implicit Differentiation (DY/DX) |
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Definition
1. Derivative with Respect to X
2. Collect all terms with dy/dx on left
3. Factor dy/dx out of left side
4. Solve for dy/dx by dividing the leftside factor that doesnt have dy/dx |
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| Guidlines For Solving Related Rate Problems |
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Definition
1. Find all given quantites and ones to be found. Make a sketch and label.
2. Write the equation with rates of change.
3. Use chain rule to implicitly differentiate with respect to time.
4. Solve after finding values. |
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| Critical Numbers and Relative Extrema |
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Definition
| Use first derivative and factor for "critical numbers" |
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| Extrema for Closed Interval |
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Definition
1.find critical numbers (a,b)
2. find f at (a,b)
3. find f at [a,b]
4. smaller is minimum vice versa |
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| If f is continuous [a,b] and differentiable (a,b) and if f(a)=f(b) then there is one number c in (a,b) such that f'(c)=0 |
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| f'(c)= ((f(b)-f(a))/(b-a)) |
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| Increasing and Decreasing |
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| Find critical numbers using first derivative... Plug into f'(x)... Set up intervals with infinity and negatives are decreasing and positives are increasing. |
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| 2nd derivative... make intervals with points of inflection... find positive and negative. positive is concave up vice versa. |
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| Find second derivative. Factor it out and find the critical numbers of the second derivative. Look at calc if you get scurred. |
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| Look at your calculator as numbers get bigger. They approach a number even as they get supa big. |
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| Guidelines for Limits at Infinity |
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Definition
1. If the degree of the numerator is less than the degree of the denominator than it is 0
2. If the numerator is equal to the degree of the denominator than it the limit is the fraction of the leading coefficients
3. IF the degree of the numerator is greater than the denominator than the limit does not exist. |
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1. Identify all Given Quantities and Ones that need to be found.
2. Write the primary equation
3. Reduce primary to secondary with only one variable
4. Find value
5. Solve for others |
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Primary is what you want to find.
Secondary is the guidelines.
V=x^2 h
S=x^2 + 4xh
solve for h then put it into primary |
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| Don't Forget the Fucking C |
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| You have to integrate and then substitute for C |
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Definition
n
E a1 = a1 + a2 + a3 +...an
i=1 |
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Definition
Ec= cn
Ei=n(n+1) / 2
Ei^2= n(n+1)(2n+1) / 6
Ei^3= n^2(n+1)^2 / 4 |
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Definition
right endpoint= a+i /! x
/!= (b-a)/n
Set up
E f(right end point) (/!x)
Simplify then sum all constants and i's |
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Definition
cosx=sinx + c
sinx=-cosx + c
sec^2x=tanx + c
secxtanx=secx + c
csc^2x=-cotx + c
cscxcotx=-cscx + c |
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Definition
Make it Bigger
3x^2 --> x^3 + c
x^4 --> 1/5x^5 |
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Definition
4
! V-x
1
x^(1/2)+1 --> 2/3x^ 3/2
f(4)-f(1) = 5 1/3 - 2/3 = 4 2/3 |
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Definition
ln(1)=0
ln(ab)= lna +lnb
ln(a^n)=n*ln(a)
ln(a/b)=lna - lnb |
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Definition
| b-a/2n [fx+2(fx1)+2(fx2)... (fxn)] |
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