Term

Definition


Term

Definition


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Definition: A tangent line is... 

Definition
...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... 

Definition
...the line connecting two points on a curve. 


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Definition: A normal line is... 

Definition
...the line perpendicular to the the tangent line at the point of tangency. 


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Definition: [image] is continuous at [image] when... 

Definition
1. [image] exists; 2. [image] exists; and 3. [image]. 


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Limit definition of the derivative of [image]: [image] 

Definition


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Alternate definition of derivative of [image] at [image]: [image]= 

Definition


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What [image] tells you about a function 

Definition
• 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... 

Definition


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Rolle's Theorem:
If [image] is continuous on [image], differentiable on [image], and... 

Definition
...[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... 

Definition
...there exists a value of [image] such that [image]. 


Term
Extreme Value Theorem: If [image] is continuous on a closed interval, then... 

Definition
...[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... 

Definition
...[image] must take on every [image]value between [image] and [image]. 


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If a function is differentiable at a point, then... 

Definition
...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 

Definition
•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... 

Definition
...a value of [image] in the domain of [image] at which either [image] or [image] does not exist. 


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Definition
...[image] is increasing. 


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Definition
...[image] is decreasing, 


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Definition
...[image] has a horizontal tangent. 


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Definition: [image] is concave up when... 

Definition
...[image] is increasing. 


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Definition: [image] is concave down when... 

Definition
...[image] is decreasing. 


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[image] means that [image] is... 

Definition


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[image] means that [image] is... 

Definition
concave down (like a frown). 


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Definition: A point of inflection is a point on the curve where... 

Definition


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To find a point of inflection,… 

Definition
… 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… 

Definition
… [image] is zero or undefined (critical numbers). 


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At a maximum, the value of the derivative… 

Definition
… [image] changes from positive to negative. (First Derivative Test) 


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At a minimum, the value of the derivative… 

Definition
… [image] changes from negative to positive. (First Derivative Test) 


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The Second Derivative Test: If [image]… 

Definition
…and [image], then [image] has a maximum; if [image], then [image] has a minimum. 


Term

Definition
[image], the antiderivative of velocity 


Term

Definition
[image], the derivative of position, as well as [image], the antiderivative of acceleration 


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Acceleration function [image] 

Definition
[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… 

Definition


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A particle is moving to the right when… 

Definition


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A particle is not moving (at rest) when… 

Definition


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A particle changes direction when… 

Definition


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To find displacement of a particle with velocity [image] from [image] to [image], calculate this: 

Definition


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To find total distance traveled by a particle with velocity [image] from rom [image] to [image], calculate this: 

Definition


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Definition


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Volume of a solid with crosssections of a specified shape 

Definition


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Definition


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Volume using washers (discs with holes) 

Definition


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Trapezoidal rule for approximating [image] 

Definition


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Average value of [image] on [image] 

Definition


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L'Hôpital's rule for indeterminate limits If [image] or [image], 

Definition
then [image], if the new limit exists. 


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Mean Value Theorem for Integration: If [image] is continuous on [image], then... 

Definition
...there exists a value of [image] such that [image]. 


Term
Fundamental Theorem of Calculus (part 1) [image] 

Definition


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Fundamental Theorem of Calculus (part 2) [image] = 

Definition
[image], where [image] is an antiderivative of [image] 


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A differential equation is... 

Definition
…an equation containing one or more derivatives. 


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To solve a differential equation,... 

Definition
...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... 

Definition
[image], where [image] is the quantity at [image], and [image] is the constant of proportionality. 


Term

Definition
the amount which that quantity has changed from [image] to [image] 

