1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook