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)
Sec²x
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D/dx(cotx)
\-csc²x
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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(eⁿ)
Eⁿ
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∫Cosx
Sinx
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∫-sinx
Cosx
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∫Sec²x
Tanx
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∫-csc²x
Cotx
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∫Secxtanx
Secx
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∫-cscxcotx
Cscx
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∫1/n
Ln(n)
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∫Eⁿ
Eⁿ
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When doing integrals never forget
Constant (+c)
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∫Axⁿ
A/n+1(xⁿ⁺¹)+C
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∫Tanx
* Ln|secx|+c
* Ln|cosx|+c
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∫Secx
Ln|secx+tanx|+c
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D/dx(sin⁻¹u)
1/√1-u²
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D/dx(cos⁻¹x)
\-1/√1-x²
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D/dx(tan⁻¹x)
1/1+x²
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D/dx(cot⁻¹x)
\-1/1+x²
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With derivative inverses
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
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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