You plug in the number of the trigonometric function into x
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D/dx(sec⁻¹x)
1/|x|√x²-1
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D/dx(csc⁻¹x)
\-1/|x|√x²-1
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D/dx(aⁿ)
Aⁿln(a)
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D/dx(Logₙx)
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(x²+1)→ 2xcos(x²+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
* ∫(a to b) f(x) dx = F(b) - F(a) * Basically saying that F’(x)=f(x)
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*f* relative max→*f* ‘ goes from
Positive to negative
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*f* relative min→*f* ‘ goes 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 f’(c)=f(b)-f(a)/b-a
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Rolle’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 f’(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 ∫f(x)dx which is the average value
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Disk Method
V=π∫\[R(x)\]²dx
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Washer Method
π∫(R(x))²-(r(x))²dx
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General volume equation
V=∫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
∫(Velocity)dt
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Distance
∫|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
π/6
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tan(π/3)
√3
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tan(π/6)
√3/3
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tan(90⁰)
Undefined
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Double Argument
sin2x=2sinxcosx
cos2x=cos²x-sin²x=1-2sin²x
cos²x=1/2(1+cos2x)
sin²x=1/2(1-cos2x)
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Pythagorean Identity
sin²x+cos²x=1
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L’Hopital Rule
If limit equals 0/0 or ∞/∞, then you can take the derivative