AP Calculus Exam
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| Normal Line | line perpendicular to the tangent line at the point of tangency
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| Derivative Of Sine | cosine
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| Derivative Of Cosine | -sine
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| Derivative Of Tangent | sec^2
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| Derivative Of Cosecant | -csc cot
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| Derivative Of Secant | sec tan
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| Derivative Of Cotangent | -csc^2
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| When F'(x)=0, F(x)... | has a CV
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| When F(x) Has A Horizontal Tangent, F'(x)... | is equal to 0
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| Equation Of A Tangent Line | y2-y1=slope(x2-x1)
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| Product Rule | [f'(x) ⋅ g(x)] + [f(x) ⋅ g'(x)]
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| Slope Formula | (y2-y1) / (x2-x1)
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| Quotient Rule | { [f'(x) ⋅ g(x)] + [f(x) ⋅ g'(x)] } / [g(x)^2]
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| Chain Rule | f'[g(x)] ⋅ g'(x) = derive outside, leave inside, derive inside
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| Derivative Of e^any | e^any ⋅ derivative(any)
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| Derivative Of ln(any) | 1/any ⋅ derivative(any)
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| Derivative Of a^(any) | a^any ⋅ ln(any) ⋅ derivative(any)
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| Derivative Of log_a_(any) | 1/any ⋅ 1/lna ⋅ derivative(any)
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| In Order For A Function To Be Differentiable... | the function must be continuous & must have no cusps
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| When F''(x)=0 or DNE, F(x)... | has a POSSIBLE point of inflection
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| When F''(x)=0 or DNE, F'(x)... | has a CV
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| When F''(x)>0, F(x)... | is concave up
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| When F''(x) Goes From Positive To Negative, F(x)... | has a POI
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| When F''(x)<0, F(x)... | is concave down
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| When F''(x) Goes From Negative To Positive, F(x)... | has a POI
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| When F''(x) Goes From Positive To Negative, F'(x)... | has a relative max
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| When F''(x) Goes From Negative To Positive, F'(x)... | has a relative min
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| Extreme Value Theorem | if f(x) is continuous on [a,b], than f(x) is guaranteed to have an absolute maximum & an absolute minimum on [a,b]
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| Absolute Extrema Can Occur... | at endpoints & at CVs
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| Relative Extrema Can Occur... | only at CVs
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| To Find The Absolute Extrema From An Analytical Function... | check if it's continuous, & then use a candidates test
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| Average Rate Of Change | slope of secant --> (y2-y1)/(x2-x1)
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| Instantaneous Rate Of Change | slope of tangent line --> f'(x)
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| Mean Value Theorem | if f(x) is continuous & differentiable on [a.b], then it is guaranteed to exist a value of c where f'(c)= [f(b)-f(a)] / [b-a]
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| Intermediate Value Theorem | if f(x) is continuous on [a,b], & f(a) ≤ k ≤ f(b), then there exists at least one value, x=c, on [a,b] such that=at f(c)=k
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| When You Derive "y"... | put a y' behind it
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| Steps Of Deriving | 1) multiply 2) subtract one from the exponent
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| Steps Of Integration | 1) add one to exponent 2) divide by new exponent
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| Integral Of Cosine | sinx
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| Integral Of Sine | -cosx
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| Steps Of U-Substitution | 1) set "u" equal to inside function # 2) find the derivative of "u" # 3) move coefficients to the "du" side # 4) plug "u" & "du" into the integral # 5) plug original boundaries into "u" to get "u" boundaries # 6) solve # 7) plug original function back in
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| When Left Hand Riemann Sum Is Increasing... | it is an under approximation
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| When Left Hand Riemann Sum Is Decreasing... | it is an over approximation
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| When Right Hand Riemann Sum Is Increasing... | it is an over approximation
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| When Right Hand Riemann Sum Is Decreasing... | it is an under approximation
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| Trapezoidal Riemann Sums | (left hand + right hand) / 2
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| Second Fundamental Theorem Of Calculus | take the "x" boundary, plug it in for t, and multiply it by the derivative of the "x" boundary
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| Average Velocity Equation | p(b)-p(a) / b-a = slope of position OR (1/b-a) ∫a(t)dt
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| Average Acceleration Equation | v(b)-v(a) / b-a = slope of velocity
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| When Velocity Is>0, Position... | is x to the right & y up
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| When Velocity Is<0, Position... | is to the left & down
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| When Velocity Goes From Positive To Negative Or Vice Versa, Position... | changes direction
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| When Acceleration>0, Velocity... | is increasing
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| When Acceleration<0, Velocity... | is decreasing
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| When Acceleration Changes From Positive To Negative, Velocity... | slows down
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| When Acceleration Changes From Negative To Positive, Velocity... | speeds up
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| Speed Is The Absolute Value Of... | velocity
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| Velocity & Acceleration In Relation To Speed | same signs = increasing speed // different signs = decreasing speed
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| Net Distance Equation | ∫v(t)dt
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| Total Distance Equation | ∫|v(t)|dt
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| Area Equation | ∫(TOP-BOTTOM)dx
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| Volume Equation | π∫(OUTER-AXIS)^2-(INNER-AXIS)^2dx
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| Cross Section Volume | value ⋅ ∫(TOP-BOTTOM)^2
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| Cross Section Value Of A Square | 1
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| Cross Section Value Of Semi-Circle | π/8
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| Cross Section Value Of Isosceles Triangle | 1/2
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| Cross Section Value Of Equilateral Triangle | √3/4
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| Area In Respect To "y" | ∫(RIGHT-LEFT)dy & use y-limits
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| Steps For Solving Differential Equations | 1) separate the variables -- 2) integrate both sides -- 3) find c -- 4) use c to solve for y
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