Calculus I exam 1

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ithuestad
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106908
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Calculus I exam 1
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2011-10-07 11:44:29
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Calculus limits tangent secant
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Chapter 1 and sections 2.1 and 2.2
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  1. Deffiniation of a Limit

    -Let f be a function defined on an open interval containing c (exept possibly at c) and let L be a real number
     \lim_{x \to c}f(x) = L
    means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, it can be stated that "the limit of f of x, as x approaches c, is L". Note that this statement can be true even if f(c) ≠ L. Indeed, the function f(x) need not even be defined at c.
  2. commontypes of behaviors associated with the non-exitence of a limit
    • 1. f(x) approaches a different number from the right side of c than it approaches from the left side.
    • 2. f(x) increases or decreases without bound as x approaches c
    • 3. f(x) oscillates between two fixed values as x approaches c
  3. What are three basic limits
    \lim_{x \to c} a = a
    \lim_{x \to c} x = c
    \lim_{x \to c} x^r = c^r \qquad \mbox{ if } r \mbox{ is a positive integer}
  4. Properties of limits: Sum or difference
    \text{If }\lim_{x \to c} f(x) = L_1 \text{ and }\lim_{x \to c} g(x) = L_2 \text{ then:}
    \lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2
  5. Properties of limits: Product
    \text{If }\lim_{x \to c} f(x) = L_1 \text{ and }\lim_{x \to c} g(x) = L_2 \text{ then:}
    \lim_{x \to c} \, [f(x)g(x)] = L_1 \times L_2
  6. Properties of limits: Quotient
    \lim_{x\to c} f(x) =L and \lim_{x\to c} g(x) =M
      •  \lim_{x\to c} \frac{f(x)}{g(x)} = \frac{\lim_{x\to c} f(x)}{\lim_{x\to c} g(x)} = \frac{L}{M} \,\,\, \mbox{ provided } M\neq 0




  7. Properties of limits: Power
    \lim_{x\to c} f(x) =L and \lim_{x\to c} g(x) =M
      •  \lim_{x\to c} [f(x) g(x)] = \lim_{x\to c} f(x) \lim_{x\to c} g(x) = L M


  8. Limit of a composite Function

    If f is continuous at b and lim x→a g(x) = b
    then lim x→a f(g(x)) = f (lim x→a g(x)) = f (b).
  9. Limit of Trigonometric Functions:
    find limx→c for sin(x) ; cos (x) ; sin(x)/x ; (1-cos(x))/(x)
  10. Example : Evaluate
    • Substituting 0 for x yields 5/0, which is meaningless; hence, = DNE. (Remember, infinity is not a real number.)
    • THIS IS AN ASYMPTOTE, NON-REMOVABLE DISCONTINUITY (numerator ≠ 0 , denominator = 0)
  11. A stradegy For finding limits
    • 1. Learn to recognize which limits can be evaluated by direct substituation
    • 2. If the limit of f(x) as x approaches c cannot be evaluated by direct substituaiton, try to find a afunction g that agrees with f for all of x other than x=c
    • 3. apply the theorem for removable discontinuities such that; lim x→c f(x) = lim x→c g(x) = g(c)
    • 4. use a graph or table to reinforce conclusion
  12. Functions that agree at all but one point

    let c be real and let f(x) = g(x) for all x ≠ c in an open interval containing c
    If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists

    lim x→c f(x) = lim x→c g(x)
  13. Definition of Continuity
    • 1. f(c) is defined
    • 2. lim x→c f(x) exists
    • 3. \lim_{x \to c}{f(x)} = f(c).

    • continuity on an open interval: a function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line ( -\infty , +\infty ) is everywhere continuous.
  14. The existence of a limit

    let f be a function and let c and L be real numbers
     \lim_{x \to c}f(x) = L if and only if  \lim_{x \to p^+}f(x) = L and  \lim_{x \to p^-}f(x) = L
  15. Definition of continuity on a closed interval
    A function is said to be continuous on [a,b] if and only if

    • 1. it is continuous on (a,b),
    • 2. it is continuous from the right at a and
    • 3. it is continuous from the left at b.
  16. properties of continuity

    1. scalar multiple:
    2. sum or difference:
    3. Product:
    4. quotient:
    • 1. scalar multiple: bf
    • 2. sum or difference: f (+ - ) g
    • 3. Product: fg
    • 4. quotient: f/g, if g(c) ≠ 0
  17. Continuity of a compsoite function
    If the function g is continuous at a number a, and f is continuous at g(a), the composite function is also continuous at a.
  18. intermediate value theorem
    If a function f is continuous on a closed interval [a,b], then for every value k between f(a) and f(b) there is a value c between a and b such that f(c) = k.
  19. definition of infinite limits
    We say if we can make f(x) arbitrarily large for all x sufficiently close to x=a, from both sides, without actually letting .

    We say if we can make f(x) arbitrarily large and negative for all x sufficiently close to x=a, from both sides, without actually letting .
  20. Definition of a vertical asymptote
    if f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f
  21. finding vertical asymptote
    • let h and g be continuous on an open interval containing c. If g(c) ≠ 0, h(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of the function given by:
    • has a vertical asymptote at x = c
  22. properties of infinite limits: sum or difference
    Given the functions and suppose we have,

    for some real numbers c and L.
  23. Properties of Infinite Limits: Product
    Given the functions and suppose we have,

    for some real numbers c and L. Then,
    then

    then
  24. Properties of infinite limits: Quotient

    Given the functions and suppose we have,

    for some real numbers c and L. Then,
  25. Definition of a tangent line with slope m

    also known as the slope of the graph of f at x = a
    if f is defined on an open interval containing a, and if the limit: \frac{f(a+h)-f(a)}{h}. exists, then the lines passing through (a, f(a)) with slpoe m is the tangent line to the graph of f at the point (a, f(a))
  26. Definition of the derivative of a function
    the derivative of f at x
    m = \frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{h}.
  27. differentialbility implies continuity
    if f is differentiable at x = c, then f is continuous at f = c
  28. The constant rule of the derivative
    the derivative of a constant function is 0. That is, if c is a real number, then
  29. the power rule of the derivative
    \begin{displaymath}\boxed{ \frac {d} {dx} (x^n) = n x^{n-1} }\end{displaymath} if n is a rational number, then the function f(x) = xn is differentialble. For f to be differentiable at x=0 then n must be a number such that xn-1 is defined on the interval containing 0

  30. The sum and differnece rules of the derivative
    • The derivative of the sum of two functions is the sum of the derivatives of the two functions:
    • .
    • Likewise, the derivative of the difference of two functions is the difference of the derivatives of the two functions.
  31. the derivative of the sin and cos function
    • \begin{displaymath}\boxed{ \frac {d} {dx} (\sin x) = \cos x }\end{displaymath}
    • \begin{displaymath}\boxed{ \frac {d} {dx} (\cos x) = -\sin x. }\end{displaymath}
  32. Alternate limit form of the derivative
  33. How to find the equation of the tangent line through a point (x1,y1)
    find the derivative f'(c) = m = slope

    plug m into the point slope formula y-y1=m(x-x1)

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