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 (in terms of sine and/or cosine) |
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 (in terms of sine and/or cosine) |
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 (in terms of sine and/or cosine) |
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 (in terms of sine and/or cosine) |
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Graph of 
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Graph of 
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Graph of 
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Graph of 
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Graph of 
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Graph of 
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Graph of 
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Graph of 
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Graph of 
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Graph of 
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Definition: An even function is... |
...symmetric with respect to the  -axis, like  , , or .
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Definition: An odd function is... |
...symmetric with respect to the origin, like  , , or .
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| Two formulas for the area of a triangle |
 
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| Formula for the area of a circle |
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| Formula for the circumference of a circle |
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| Formula for the volume of a cylinder |
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| Formula for the volume of a cone |
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| Formula for the volume of a sphere |
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| Formula for the surface area of a sphere |
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| Point-slope form of a linear equation |
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Definition: A tangent line is... |
| ...the line through a point on a curve with slope equal to the slope of the curve at that point. |
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Definition: A secant line is... |
| ...the line connecting two points on a curve. |
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Definition: A normal line is... |
| ...the line perpendicular to the the tangent line at the point of tangency. |
Definition: is continuous at  when...
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1.  exists;2.  exists; and3.  .
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Limit definition of the derivative of :
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Alternate definition of derivative of at : =
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What  tells you about a function
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• slope of a curve at a point • slope of tangent line • instantaneous rate of change |
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Definition: Average rate of change is... |
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Power rule for derivatives:
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Product rule for derivatives: =
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Quotient rule for derivatives: 
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Chain rule for derivatives: =
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 , orderivative of the outside function times derivative of the inside function |
 =
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 =
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 =
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Derivative of natural log:
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Derivative of log base  :
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Derivative of natural exponential function: =
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Derivative of exponential function of any base:
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Derivative of an inverse function:
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 The derivatives of inverse functions are reciprocals. |
Rolle's Theorem:
If is continuous on  , differentiable on  , and...
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... , then there exists a value of  such that  .
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Mean Value Theorem for Derivatives: If is continuous on and differentiable on , then...
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...there exists a value of such that  .
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Extreme Value Theorem: If is continuous on a closed interval, then...
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... must have both an absolute maximum and an absolute minimum on the interval.
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Intermediate Value Theorem: If is continuous on , then...
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... must take on every -value between  and  .
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| If a function is differentiable at a point, then... |
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...it must be continuous at that point. (Differentiability implies continuity.) |
| Four ways in which a function can fail to be differentiable at a point |
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•Discontinuity •Corner •Cusp •Vertical tangent line |
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Definition: A critical number (a.k.a. critical point or critical value) of is...
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...a value of  in the domain of at which either  or does not exist.
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If  , then...
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... is increasing.
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If  , then...
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... is decreasing,
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If , then...
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... has a horizontal tangent.
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Definition: is concave up when...
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... is increasing.
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Definition: is concave down when...
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... is decreasing.
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 means that is...
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concave up (like a cup). |
 means that is...
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concave down (like a frown). |
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Definition: A point of inflection is a point on the curve where... |
| ...concavity changes. |
| To find a point of inflection,… |
… look for where  changes signs, or, equivalently, where  changes direction.
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| To find extreme values of a function, look for where… |
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… is zero or undefined (critical numbers).
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| At a maximum, the value of the derivative… |
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… changes from positive to negative. (First Derivative Test) |
| At a minimum, the value of the derivative… |
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… changes from negative to positive. (First Derivative Test) |
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The Second Derivative Test: If …
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…and , then  has a maximum; if , then has a minimum.
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Position function
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 , the antiderivative of velocity
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Velocity function
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 , the derivative of position, as well as  , the antiderivative of acceleration
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Acceleration function 
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 , the derivative of velocity, as well as  , the second derivative of position
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| A particle is moving to the left when… |
… .
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| A particle is moving to the right when… |
… .
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| A particle is not moving (at rest) when… |
… .
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| A particle changes direction when… |
… changes signs.
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To find displacement of a particle with velocity from  to  , calculate this:
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To find total distance traveled by a particle with velocity from rom to , calculate this:
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| Area between curves |
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| Volume of a solid with cross-sections of a specified shape |
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| Volume using discs |
 "perpendiscular" |
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Volume using washers (discs with holes) |
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| Volume using shells |
 "parashell" |
| Area of a trapezoid |
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Trapezoidal rule for approximating 
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Average value of on
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Power rule for antiderivatives:
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 (for  )
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Constant multiple rule for antiderivatives:
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 (A constant coefficient can be brought outside.) |
 =
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L'Hôpital's rule for indeterminate limits If  or  ,
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then  , if the new limit exists.
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Mean Value Theorem for Integration: If is continuous on , then...
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...there exists a value of  such that  .
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Fundamental Theorem of Calculus (part 1)
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Fundamental Theorem of Calculus (part 2) =
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 , where  is an antiderivative of
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| A differential equation is... |
| …an equation containing one or more derivatives. |
| To solve a differential equation,... |
| ...first separate the variables (if needed) by multiplying or dividing, then integrate both sides. |
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Exponential Growth and Decay: If  , then...
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 , where  is the quantity at  , and  is the constant of proportionality.
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the amount which that quantity has changed from to
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