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32 Cards in this Set
- Front
- Back
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differentiability
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cannot have a "hole", jump, vertical asymptote, or other discontinuous behavior, nor can the graph of a function have a sharp corner or a cusp at that point
the derivative is undefined when the tangent line is vertical |
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the original function f is strictly increasing if the derivative...
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if the derivative is greater than zero
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a function defined on the interval x1 and x2 is increasing over the interval if whenever x1 < x2....
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f(x1) is less than or equal to f(x2)
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the function is strictly increasing over the interval is whenever x1 < x2 then
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f(x1) is less than f(x2)
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the function is decreasing over the interval is, whenever x1 < x2 then...
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f(x1) is greater than or equal to f(x2)
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the function is strictly decreasing over the interval if, whenever x1 < x2...
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f(x1) is greater than f(x2)
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increasing decreasing functions
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a function is said to be increasing or decreasing depending on whether its outputs increase or decrease as the inputs increase
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the original function is strictly decreasing if the derivative
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is less than zero over the interval
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when the derivative at x0 is neither positive nor negative then...
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f'(x0)= 0 or f'(x0) is not defined.
x0 is a critical value for the function f |
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a critical value for the function f is...
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an input in the domain of f such that either f'(x0)=0 or f'(x0) is undefined
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a critical value of a function on a graph of y=(fx)
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is either flat (slope of 0) or has a sharp corner, cusp, vertical tangent, or other such behavior making the slope undefined. it must be in the domain of the function
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The Derivative test for local extrema
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x(0)=critical value of f
f'(x) changes sign from positive to negative at x(0),, then a local maximum occurs at x(0) f'(x) changes sign from negative to positive at x(0) local minimum occurs at x(0) |
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absolute maximum or minimum
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function
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absolute extrema,
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f'(x)
tells you what the absolute maximum or absolute minimum is for the function, f(x) |
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strategy for finding absolute extrema of f over [a,b]
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step one: find the derivative f'
step two: find the critical values of f that lie in the interval [a,b]--> find each of those values a < xo < b such that f'(x0)= 0 or f'(x0) is undefined step three: identify which critical values are locations of local minima and which are locations of local maxima step four: evaluate f(x0) for each local extremum and evaluate f(a) and f(b) for comparison to identify the absolute minimum and maximum output values |
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f(x)=sec(x) find the derivative
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f'(x)=sec(x)tan(x)
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f(x)= tan(x) find the derivative
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f'(x)=sec^2x
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f(x)= csc(x)
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f'(x)= -csc(x)cot(x)
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f(x)= cot(x)
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f'(x)= -csc^2(x)
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lim f(x)-f(x0)/(x-x0)
x->0 |
another way of saying f'(x0)
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lim
x-> infinity (x^2-4)/(2-x-4x^2) |
x-> infinity is equivalent to (1/x)-> 0
lim (1/x) -> 0 (x-4)*(1/x^2)/(2-x-4x^2)*(1/x^2) 1-(4/x^2)/(2/x^2 - 1/x -4) plug in zero... everything cancels out to equal 1/-4 |
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average rate of change over (a,b)
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f(b)-f(a)/b-a
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sin(0)
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0
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sin∏/6
30∘ |
1/2
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sin pie/4
45 degrees |
√2/2
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sin pie/3
60 degrees |
√3/2
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sin pie/2
90 |
1
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cos(0)
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1
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cos(pie/6)
30 degrees |
√3/2
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cos (pie/2)
45 degrees |
√2/2
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cos (pie/3)
60 degrees |
1/2
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cos (pie/2)
90 degrees |
0
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