You plug in the number of the trigonometric function into x
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D/dx(secā»Ā¹x)
1/|x|āxĀ²-1
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D/dx(cscā»Ā¹x)
\-1/|x|āxĀ²-1
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D/dx(aāæ)
Aāæln(a)
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D/dx(Logāx)
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
* ā«(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* āĀ goesĀ from
Positive to negative
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*f*Ā relativeĀ minā*f* āĀ 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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Rolleās 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 ā«f(x)dx which is the average value
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Disk Method
V=Ļā«\[R(x)\]Ā²dx
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Washer Method
Ļā«(R(x))Ā²-(r(x))Ā²dx
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General volume equation
V=ā«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
ā«(Velocity)dt
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Distance
ā«|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)
ā3
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tan(Ļ/6)
ā3/3
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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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LāHopital Rule
If limit equals 0/0 or ā/ā, then you can take the derivative