dx & lim
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- m tangent =
find equation of secant line
- - absolute value of velocity; does not have direction; magnitude component of velocity
- - rate of change of position
- - distance moved per unit of time
- - s(t) = position fxn
- - contains both magnitude + direction
- - s'(t) = velocity fxn
- - rate of change of velocity
- - s"(t) = acceleration nfxn
define derivative of a function
how is continuity of a function related to limit?
- f(x) is continuous at x = a when:
range of sin: [-1,1]
lim as x goes to 0: 0 <
derive formulas for:
intermed value therem
- function continuous over closed interval [a, b]
- if :
- for any value d between f(a) & f(b), excluding f(a) & f(b), there is a c between (a,b) such that f(c) = d
- r is a positive rational number
- if r > 0 & x > 0, x^r will go to infinity, & 1/x^r will approach 0
principal values of inverse trig fxns
difference between essential, non-essential, & removal discontinuity
essential: can't be patched; graph has a break:(2,3) & (2, 5) breaks at x = 2
- non-essential: removable discontinuity, can be patched,
- if these are = (both 9), can patch by defining f(3) = 9
what must be true to repair a discontinuity?
- the 2 one-sided limits must be the same: the limit of f(x) as x approaches c from the left and right must be equal
- then f(x) can be repaired by defining f(c) = L
methods for finding/evaluating limit of a fxn
- using table of values as x gets closer & closer to a
- algebraically using limit laws
- using trig identities
if f(x) <g(x) in a neighborhood around x = a; what must be true about their limits?
what conditions must be met before taking the limit of a composite function?
- for 1 > 2:
- is in domain of f(x)
- f & g are continuous
- f(x) is continuous on an interval if it is continuous at q # in the interval
- if f is defined on only one side of an endpoint, continuous at the endpoint means continuous from the right or continuous from the left
the line y = L is a horizontal asymptote of the curve y = f(x) if:
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