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Calculus crash course

  1. May 31, 2010 #1
    I'm a college student needing to take calculus 2 this fall. However, I have not taken calc 1, so since I'm good at teaching myself I'm planning on giving myself a crash course in the material covered in calc 1 over this summer. Can anyone recommend a good calculus book for someone who's fairly good at math but has limited experience with calc? Thanks!
     
  2. jcsd
  3. Jun 1, 2010 #2
    Hm.. It's going to be pretty hard for you to excel in calc II without taking calc I, frankly, I'm not sure how you even got enrolled into a calc II class without having calc I under your belt, but it's not my business.

    There's a book called the calculus life saver, which may be of interest to you. Hope this helps.
     
  4. Jun 1, 2010 #3
    PROTIP:
    Rules for differentiation:
    1. derivative of a constant
      In general, if [itex]f(.)[/itex] does not depend explicitly on some variable, say [itex]x[/itex] it's derivative is zero:
      [tex]
      \frac{d}{d x}\left(C\right) = 0
      [/tex]
    2. derivative with respect to the argument:
      [tex]
      \frac{d x}{d x} = 1
      [/tex]
    3. rule of sums
      [tex]
      \frac{d}{d x}\left[ f(x) + g(x) \right] = \frac{d f(x)}{dx} + \frac{d g(x)}{dx}
      [/tex]
    4. product tule
      [tex]
      \frac{d}{d x}\left[ f(x) \cdot g(x) \right] = \frac{d f(x)}{dx} \cdot g(x) + f(x) \cdot \frac{d g(x)}{dx}
      [/tex]
    5. chain rule
      [tex]
      \frac{d}{d x} \left( f[g(x)] \right) = \left. \frac{d f(u)}{du} \right|_{u = g(x)} \cdot \frac{d g(x)}{d x}
      [/tex]
    6. derivative of the exponential function
      [tex]
      \frac{d \exp(x)}{dx} = \exp(x)
      [/tex]

    Using the above, see if you can derive the following:
    1. Quotient rule
      [tex]
      \frac{d}{d x}\left( \frac{f(x)}{g(x)}\right) = \frac{f'(x) \, g(x) - f(x) \, g'(x)}{[g(x)]^{2}}
      [/tex]
    2. Derivative of a power function:
      [tex]
      \frac{d}{d x}\left( x^{\alpha} \right) = \alpha \, x^{\alpha - 1}, \ \alpha \in \mathbf{R}
      [/tex]
    3. Derivative of an inverse function
      [tex]
      y = f(x) \Rightarrow x = f^{-1}(y)
      [/tex]

      [tex]
      f[f^{-1}(x)] = x
      [/tex]

      [tex]
      \frac{d}{d x}\left( f^{-1}(x) \right) = \frac{1}{f'[f^{-1}(x)]}
      [/tex]
    4. Derivative of a logarithm
      [tex]
      (\log_{a} {x})' = \frac{1}{x \, \ln{a}}
      [/tex]
    5. Derivative of trigonometric functions
      Using Euler's identity:
      [tex]
      e^{\textup{i} \, x} = \cos{x} + \textup{i} \, \sin{x}
      [/tex]

      and taking the real and imaginary part of the derivative, prove:
      [tex]
      \begin{array}{l}
      (\cos{x})' = -\sin{x} \\

      (\sin{x})' = \cos{x}
      \end{array}
      [/tex]
    6. Find the derivative of
      [tex]
      x^{x}
      [/tex]
     
    Last edited: Jun 1, 2010
  5. Jun 2, 2010 #4

    Char. Limit

    User Avatar
    Gold Member

    Shouldn't you include the definition of a derivative before introducing the rules for it? Just a thought...

    [tex]\frac{d f\left(x\right)}{dx}= \text{lim}_{h\rightarrow0} \frac{f\left(x+h\right)-f\left(x\right)}{h}[/tex]

    To solve it, you first have to eliminate h from the denominator.
     
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