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Definition
f(x) <= g(x) <= h(x) when x is near a (except possibly at a) and
limit of f(x) as x approaches a = the limit of h(x) as x approaches a = L
then
the limit of g(x) as x approaches a = L |
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Definition
| a function f is continues at a number a if the limit of f(x) as x approaches a = f(a). |
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| Continuous functions at every number |
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Definition
| polynomial, rational, root, and trigonometric |
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| intermediate value theorem |
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Definition
| suppose that f is continuous on the closed interval a,b and let N be any number between f(a) and f(b), where f(a) does not equal f(b). Then there exists a number c in (a,b) such that f(c) = N. |
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| average velocity to instantaneous velocity |
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Definition
| v(a) = lim x->a f(a+h)-f(a)/h (derivative) |
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Definition
| f'(x) = lim h->0 f(x+h)-f(x)/h |
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Definition
| a function is differentiable at a if f'(a) exists. It is differentiable on an oopen interval (a,b) or (a,oo) or (-oo, oo) if it si differentibale at every number in the interval. |
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| theorem of differentiation + converse |
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Definition
| If is differentiable at a, then f is continuous at a. Converse is false |
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| function fails to be differentiable |
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Definition
| a corner |x| , discontinuity, and vertical tangents are not differentiable |
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Definition
| simplify using chain rule( trig or rational) implicit differentiation |
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| derivatives ( power function) |
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Definition
if n is a positive integer, then
d/dx (x^-n) = -nx^(-n-1) |
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| trigonometric derivatives |
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Definition
sin = cos
cos = -sin
tan = sec^2
csc = -csc cot
sec = sec tan
cot = -csc^2 |
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Definition
if g is differentiable at x and f is differentiable at g(x), then the composite function F = f o g defined by F(x) = f(g(x)) is differentiable at x and F' is given by the product F'(x) = f'(g(x)) * g'(x)
in leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then dy/dx = dy/du du/dx |
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| implicit differentiation` |
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Definition
| differentiate both sides of the equation with respect to x and then solving the resulting equation for y'. |
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Definition
| if f is continuous on a closed interval (a,b), then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d (a,b). |
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Definition
| if f has a local maximum or minimum at c, and if f'(c) exists, then f'(c) = 0. |
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Definition
to find the absolute max and min values of a cont. func. f on a closed interval (a,b)
1 find teh values of f at the critical numbers of f in (a,b)
2. find the values of f at the endpoints of the interval.
3. the largest of the values from steps 1 and 3 is the absolute max value:
the smallest of these values is the absolute min value. |
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Definition
| function f is a number c in the domain of f such that either f'(c) = 0 or f'(c) does not exist. conditions(local min/max at c) |
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Definition
let f be a function that satisfies the following hypothesis:
1. f is continuous on the closed interval [a,b]
2. f is differentiable on the open interval (a,b)
then there is a number c in (a,b) such that
1) f'(c) = f(b) - f(a)/b-a
or
2) f(b) - f(a) = f'(c)(b-a) |
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Definition
`c is a crit number of a cont func f.
a) if f' changes from + to - at c, then f has a local max at c.
b) if f' changes from - to + at c, then fa has a local minimum at c.
c) if f' does not change sign at c (for example, if f' is positive on both sides of c or negative on both sides), then f has no local max or min at c. |
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Definition
suppose f'' is continuous near c.
a) if f'(c) = 0 and f''(c) > 0, then f has a local min at c.
(b) if f'(c) = 0 and f''(c) < 0, then f has local max at c. |
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Definition
f is a func
1. f is cont on the closed interval [a,b].
2. f is differentiable on the open interval (a,b)
3. f(a) = f(b)
then there is a number c in (a,b) such that f'(c) = 0. |
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Definition
a) if f''(x) > 0 for all x in I, then the graph of f is concave upward on I.
b) if f''(x) < 0 for all x in I, then the graph of f is concave downward on I. |
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