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Definition: Limit of a Function
- The function f has the limit L as x approaches the number a, written
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- lim f(x) = L
- x->a
if the value of f(x) can be made as close to the number L as we please by taking x sufficiently close to (but not equal to) the number a
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Right-handed limit of a function
Limit as x approaches a from the right
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Left-handed limit of a function
Limit as x approaches a from the left
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a non-zero number
0
= infinity
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the Intermediate value theorm
If f is a continuous function on a closed interval [a,b] and M is any number between f(a) and f(b) then there is at least one number 'c' in [a,b] such that f(c)=M
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Existence of Zeros in a continuous function
If f is a continuous function on a closed interval [a,b], and if f(a) and f(b) have opposite signs, then there is at least 1 solution of the equation f(x)=0 in the interval [a,b]
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Secant line
Passes through at least 2 points on a curve
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Tangent Line
a straight line that "just touches" the curve at a given point
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Determine the equation of the tangent line of f(x)=x2-2x+1 at the point (0,1)
- A. Find Derivative: f'(x)=2x-2
- B. Determine slope of tan line (M) at the point (0,1): f'(0)=-2
- C. determine equation of tan line at the point(0,1): y-y1=M(x-x1) M=-2, x1=0, y1=1 y=-2x+1
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Average Rate of Change
- Slope of the Secant Line
- f(x+h)-f(x)
- h
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Instantaneous Rate of Change
- Slope of the tangent line
- lim f(x+h)-f(x)
- h->0 h
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Product Rule
(fg)'=f · g' + g · f'
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Quotient Rule
- f(x) = g(x) · f'(x) - f(x) · g'(x)
- g(x) [g(x)]2
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