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[image]
(in terms of sine and/or cosine) |
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[image]
(in terms of sine and/or cosine) |
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[image]
(in terms of sine and/or cosine) |
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[image]
(in terms of sine and/or cosine) |
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Definition:
An even function is... |
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...symmetric with respect to the [image]-axis, like [image], [image], or [image].
[image] |
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Definition:
An odd function is... |
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...symmetric with respect to the origin, like [image], [image], or [image].
[image] |
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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... |
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| ...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... |
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| ...the line connecting two points on a curve. |
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Definition:
A normal line is... |
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| ...the line perpendicular to the the tangent line at the point of tangency. |
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Definition:
[image] is continuous at [image] when... |
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1. [image] exists;
2. [image] exists; and
3. [image]. |
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Limit definition of the derivative of [image]:
[image] |
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Alternate definition of derivative of [image] at [image]:
[image]= |
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| What [image] 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:
[image] |
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Product rule for derivatives:
[image] = |
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| Quotient rule for derivatives: [image] |
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Chain rule for derivatives:
[image] = |
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[image], or
derivative of the outside function times derivative of the inside function |
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Derivative of natural log:
[image] |
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Derivative of log base [image]:
[image] |
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Derivative of natural exponential function:
[image] = |
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Derivative of exponential function of any base:
[image] |
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Derivative of an inverse function:
[image] |
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[image]
The derivatives of inverse functions are reciprocals. |
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| Rolle's Theorem:
If [image] is continuous on [image], differentiable on [image], and... |
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| ...[image], then there exists a value of [image] such that [image]. |
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Mean Value Theorem for Derivatives:
If [image] is continuous on [image] and differentiable on [image], then... |
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| ...there exists a value of [image] such that [image]. |
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Extreme Value Theorem:
If [image] is continuous on a closed interval, then... |
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| ...[image] must have both an absolute maximum and an absolute minimum on the interval. |
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Intermediate Value Theorem:
If [image] is continuous on [image], then... |
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| ...[image] must take on every [image]-value between [image] and [image]. |
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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.) |
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| 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 [image] is... |
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| ...a value of [image] in the domain of [image] at which either [image] or [image] does not exist. |
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| ...[image] is increasing. |
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| ...[image] is decreasing, |
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| ...[image] has a horizontal tangent. |
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Definition:
[image] is concave up when... |
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| ...[image] is increasing. |
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Definition:
[image] is concave down when... |
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| ...[image] is decreasing. |
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| [image] means that [image] is... |
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| [image] means that [image] 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... |
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| To find a point of inflection,… |
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| … look for where [image] changes signs, or, equivalently, where [image] changes direction. |
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| To find extreme values of a function, look for where… |
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| … [image] is zero or undefined (critical numbers). |
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| At a maximum, the value of the derivative… |
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… [image] changes from positive to negative.
(First Derivative Test) |
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| At a minimum, the value of the derivative… |
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… [image] changes from negative to positive.
(First Derivative Test) |
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The Second Derivative Test:
If [image]… |
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| …and [image], then [image] has a maximum; if [image], then [image] has a minimum. |
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| [image], the antiderivative of velocity |
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| [image], the derivative of position, as well as [image], the antiderivative of acceleration |
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Acceleration function
[image] |
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| [image], the derivative of velocity, as well as [image], 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… |
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| To find displacement of a particle with velocity [image] from [image] to [image], calculate this: |
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| To find total distance traveled by a particle with velocity [image] from rom [image] to [image], calculate this: |
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| Volume of a solid with cross-sections of a specified shape |
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Volume using washers
(discs with holes) |
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| Trapezoidal rule for approximating [image] |
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| Average value of [image] on [image] |
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Power rule for antiderivatives:
[image] |
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Constant multiple rule for antiderivatives:
[image] |
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[image]
(A constant coefficient can be brought outside.) |
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L'Hôpital's rule for indeterminate limits
If [image] or [image], |
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| then [image], if the new limit exists. |
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Mean Value Theorem for Integration:
If [image] is continuous on [image], then... |
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Definition
| ...there exists a value of [image] such that [image]. |
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Fundamental Theorem of Calculus (part 1)
[image] |
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Fundamental Theorem of Calculus (part 2)
[image] = |
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| [image], where [image] is an antiderivative of [image] |
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| A differential equation is... |
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| …an equation containing one or more derivatives. |
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| To solve a differential equation,... |
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| ...first separate the variables (if needed) by multiplying or dividing, then integrate both sides. |
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Exponential Growth and Decay:
If [image], then... |
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| [image], where [image] is the quantity at [image], and [image] is the constant of proportionality. |
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| the amount which that quantity has changed from [image] to [image] |
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