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# AP Calculus Exam

Question | Answer |
---|---|

Normal Line | line perpendicular to the tangent line at the point of tangency |

Derivative Of Sine | cosine |

Derivative Of Cosine | -sine |

Derivative Of Tangent | sec^2 |

Derivative Of Cosecant | -csc cot |

Derivative Of Secant | sec tan |

Derivative Of Cotangent | -csc^2 |

When F'(x)=0, F(x)... | has a CV |

When F(x) Has A Horizontal Tangent, F'(x)... | is equal to 0 |

Equation Of A Tangent Line | y2-y1=slope(x2-x1) |

Product Rule | [f'(x) ⋅ g(x)] + [f(x) ⋅ g'(x)] |

Slope Formula | (y2-y1) / (x2-x1) |

Quotient Rule | { [f'(x) ⋅ g(x)] + [f(x) ⋅ g'(x)] } / [g(x)^2] |

Chain Rule | f'[g(x)] ⋅ g'(x) = derive outside, leave inside, derive inside |

Derivative Of e^any | e^any ⋅ derivative(any) |

Derivative Of ln(any) | 1/any ⋅ derivative(any) |

Derivative Of a^(any) | a^any ⋅ ln(any) ⋅ derivative(any) |

Derivative Of log_a_(any) | 1/any ⋅ 1/lna ⋅ derivative(any) |

In Order For A Function To Be Differentiable... | the function must be continuous & must have no cusps |

When F''(x)=0 or DNE, F(x)... | has a POSSIBLE point of inflection |

When F''(x)=0 or DNE, F'(x)... | has a CV |

When F''(x)>0, F(x)... | is concave up |

When F''(x) Goes From Positive To Negative, F(x)... | has a POI |

When F''(x)<0, F(x)... | is concave down |

When F''(x) Goes From Negative To Positive, F(x)... | has a POI |

When F''(x) Goes From Positive To Negative, F'(x)... | has a relative max |

When F''(x) Goes From Negative To Positive, F'(x)... | has a relative min |

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] |

Absolute Extrema Can Occur... | at endpoints & at CVs |

Relative Extrema Can Occur... | only at CVs |

To Find The Absolute Extrema From An Analytical Function... | check if it's continuous, & then use a candidates test |

Average Rate Of Change | slope of secant --> (y2-y1)/(x2-x1) |

Instantaneous Rate Of Change | slope of tangent line --> f'(x) |

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] |

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 |

When You Derive "y"... | put a y' behind it |

Steps Of Deriving | 1) multiply 2) subtract one from the exponent |

Steps Of Integration | 1) add one to exponent 2) divide by new exponent |

Integral Of Cosine | sinx |

Integral Of Sine | -cosx |

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 |

When Left Hand Riemann Sum Is Increasing... | it is an under approximation |

When Left Hand Riemann Sum Is Decreasing... | it is an over approximation |

When Right Hand Riemann Sum Is Increasing... | it is an over approximation |

When Right Hand Riemann Sum Is Decreasing... | it is an under approximation |

Trapezoidal Riemann Sums | (left hand + right hand) / 2 |

Second Fundamental Theorem Of Calculus | take the "x" boundary, plug it in for t, and multiply it by the derivative of the "x" boundary |

Average Velocity Equation | p(b)-p(a) / b-a = slope of position OR (1/b-a) ∫a(t)dt |

Average Acceleration Equation | v(b)-v(a) / b-a = slope of velocity |

When Velocity Is>0, Position... | is x to the right & y up |

When Velocity Is<0, Position... | is to the left & down |

When Velocity Goes From Positive To Negative Or Vice Versa, Position... | changes direction |

When Acceleration>0, Velocity... | is increasing |

When Acceleration<0, Velocity... | is decreasing |

When Acceleration Changes From Positive To Negative, Velocity... | slows down |

When Acceleration Changes From Negative To Positive, Velocity... | speeds up |

Speed Is The Absolute Value Of... | velocity |

Velocity & Acceleration In Relation To Speed | same signs = increasing speed // different signs = decreasing speed |

Net Distance Equation | ∫v(t)dt |

Total Distance Equation | ∫|v(t)|dt |

Area Equation | ∫(TOP-BOTTOM)dx |

Volume Equation | π∫(OUTER-AXIS)^2-(INNER-AXIS)^2dx |

Cross Section Volume | value ⋅ ∫(TOP-BOTTOM)^2 |

Cross Section Value Of A Square | 1 |

Cross Section Value Of Semi-Circle | π/8 |

Cross Section Value Of Isosceles Triangle | 1/2 |

Cross Section Value Of Equilateral Triangle | √3/4 |

Area In Respect To "y" | ∫(RIGHT-LEFT)dy & use y-limits |

Steps For Solving Differential Equations | 1) separate the variables -- 2) integrate both sides -- 3) find c -- 4) use c to solve for y |