Stuff to memorize for the AP Calculus Test

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64 Terms

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Y=f(x) must be continuous at each:
\-Critical Point or undefined and endpoints
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Local Minimum
Goes (-,0,+) or (-, und, +)
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Local Maximum
Goes (+,0,-) or (+, und, -)
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Point of Inflection
* Concavity Changes


* (+,0,-) or (-,0,+)
* (+,und,-) or (-,und,+)
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D/dx(sinx)
Cosx
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D/dx(cosx)
\-sinx
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D/dx(tanx)
SecĀ²x
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D/dx(cotx)
\-cscĀ²x
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D/dx(secx)
Secxtanx
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D/dx(cscx)
\-cscxcotx
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D/dx(lnx)
1/x
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D/dx(ln(n))
1/n
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D/dx(eāæ)
Eāæ
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āˆ«Cosx
Sinx
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āˆ«-sinx
Cosx
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āˆ«SecĀ²x
Tanx
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āˆ«-cscĀ²x
Cotx
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āˆ«Secxtanx
Secx
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āˆ«-cscxcotx
Cscx
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āˆ«1/n
Ln(n)
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āˆ«Eāæ
Eāæ
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When doing integrals never forget
Constant (+c)
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āˆ«Axāæ
A/n+1(xāæāŗĀ¹)+C
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āˆ«Tanx
* Ln|secx|+c
* Ln|cosx|+c
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āˆ«Secx
Ln|secx+tanx|+c
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D/dx(sinā»Ā¹u)
1/āˆš1-uĀ²
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D/dx(cosā»Ā¹x)
\-1/āˆš1-xĀ²
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D/dx(tanā»Ā¹x)
1/1+xĀ²
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D/dx(cotā»Ā¹x)
\-1/1+xĀ²
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With derivative inverses
You plug in the number of the trigonometric function into x
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D/dx(secā»Ā¹x)
1/|x|āˆšxĀ²-1
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D/dx(cscā»Ā¹x)
\-1/|x|āˆšxĀ²-1
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D/dx(aāæ)
Aāæln(a)
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D/dx(Logā‚™x)
1/xln(a)
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Chain Rule
* Take derivative of outside of parenthesis
* Take derivative of inside parenthesis and keep the original of what was in the parenthesis
* For example, sin(xĀ²+1)ā†’ 2xcos(xĀ²+1)
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Product Rule
d/dx first times second + first times d/dx second
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Quotient Rule
LoDHi-HiDLo/LoLo
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Fundamental Theorem of Calculus
* āˆ«(a to b) f(x) dx = F(b) - F(a)
* Basically saying that Fā€™(x)=f(x)
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*f*Ā relativeĀ maxā†’*f* ā€˜Ā goesĀ from
Positive to negative
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*f*Ā relativeĀ minā†’*f* ā€˜Ā goesĀ from
Negative to positive
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Intermediate Value Theorem
If I pick an X value that is included on a continuous function, I will get a Y value, within a certain range, to go with it.
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Mean Value Theorem
If function is continuous on \[a,b\] and first derivative exist on interval (a,b), then there is at least one number (c) such that fā€™(c)=f(b)-f(a)/b-a
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Rolleā€™s Theorem with Mean Value Theorem
\
\
If function is continuous on \[a,b\] and first derivative exist on interval (a,b), and f(a)=f(b) then there is at least one number x=c such that fā€™(c)=0
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Trapezoidal Rule
The average of left hand point method and right hand point method
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Theorem of mean value and average value
1/b-a āˆ«f(x)dx which is the average value
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Disk Method
V=Ļ€āˆ«\[R(x)\]Ā²dx
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Washer Method
Ļ€āˆ«(R(x))Ā²-(r(x))Ā²dx
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General volume equation
V=āˆ«area(x)dx

\-If no shape is given, then you just take the integral of the function
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Distance, Velocity, and Acceleration
Velocity is d/dx of position

Acceleration is d/dx of velocity
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Displacement
āˆ«(Velocity)dt
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Distance
āˆ«|Velocity|dt
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Speed
The absolute value of velocity
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Average Velocity
Final Position-Initial Position/Total time
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cos(0)
1
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sin(0)
0
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Pie circle starts with
Ļ€/6
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tan(Ļ€/3)
āˆš3
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tan(Ļ€/6)
āˆš3/3
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tan(90ā°)
Undefined
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Double Argument
sin2x=2sinxcosx

cos2x=cosĀ²x-sinĀ²x=1-2sinĀ²x

cosĀ²x=1/2(1+cos2x)

sinĀ²x=1/2(1-cos2x)
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Pythagorean Identity
sinĀ²x+cosĀ²x=1
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Lā€™Hopital Rule
If limit equals 0/0 or āˆž/āˆž, then you can take the derivative