Calculus Flashcards
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Transcript of Calculus Flashcards
Critical points of f
Endpoints of domain Where f’(x) = 0 Where f’(x) is undefined
Relative/Local Maximum
f’(x) = 0 f’(x) goes from positive to
negative
Relative/Local Minimum
f’(x) = 0
f’(x) goes from negative to positive
Global/Absolute Maximum
Largest local maximum value of f(x) including endpoints
Global/Absolute Minimum
Smallest local minimum value of f(x) including endpoints
Point of Inflection
f”(x) = 0 f”(x) is undefined
indicates a change in concavity (when f”(x) changes sign)
Concavity
When f”(x) > 0, concave up When f”(x) < 0, concave
down
Plateau
When f’(x) = 0 but does not change signs
Cusp
A point at which f(x) is continuous but f’(x) is discontinuous
Area between two functions
A=∫a
b
( y1− y2 )dx
where y1 is the curve on the right or top, and y2 is the curve on the left or the bottom
Volume by disks
V=π∫a
b
r2h
where r is the radius and h is dy or dx
Volume by Washers
V=π∫a
b
(R ¿¿2−r2)h¿
Where R is the top or right-most function, and r is the bottom or left-most function; h is dy or dx
Volume by Cross Sections
∫a
b
( Area of the cross section )(dy∨dx)
dy or dx determined by axis perpendicular to cross sections
Note: π should only be present if the cross sections involve circles
Volume by Cylindrical Shells
2π∫a
b
rh( thickness)
where the thickness is dy or dx determined by the axis parallel to the axis of rotation
General Rules of Area and Volume by Definite Integral
When subtracting functions: (top or rightmost function) - (bottom or leftmost function)
Cannot slice from curve to curve (use two distinct functions)
General Rules of Area and Volume by Definite Integral
(Continued)
Limits of integration: points of intersection of the two functions.
When creating shells, slice parallel to axis of rotation
When creating disks and washers, slice perpendicular to the axis of rotation
Increasing Function
If f’(x) > 0, then the function is increasing
Decreasing Function
If f’(x) < 0, then the function is decreasing.
Optimization: Maximizing and Minimizing
1) Rewrite the equation in terms of one variablea. Solve another equation relating the two
variables in terms of one variableb. Substitute this expression into the original
equation to minimize/maximize2) Take the derivative of the equation to
minimize/maximize3) Set the derivative equal to zero.4) Solve the equation for the variable.