Homework 0

Problem 1

Set the equation to 0 and solve for x:

  1. \ln(x^2+5)
  2. x^2+6x+7
  3. 3x^2-5x+2
  4. e^{x^2-4}
  5. \ln(5x) + 3

Problem 2

Complete the following derivatives:

  1. f(x)=e^x

  2. f(x) = e^{x^2}

  3. f(x) = e^x x^2

  4. f(x) = \frac{\ln(x^2)}{x^2+3x}

  5. f(x) = \ln(x)

Problem 3

Complete the following integrals:

  1. \int-\frac{9}{x^4}dx
  2. \int x^2\ln(x) dx
  3. \int x^2\sqrt x dx
  4. \int x^2e^{-x^3}dx
  5. \int2x(x^2+1)^4dx

Problem 4

Evaluate the following identities1:

  1. (x+y)^n
  2. \sum^\infty_{i=1}r^i; |r|<1
  3. \sum^{m}_{i=1}r^i; |r|<1
  4. \sum^\infty_{i=0}\frac{x^i}{i!}
  5. \frac{n!}{(n-1)!}

Problem 5

Evaluate the following integrals:

  1. \int_{-\infty}^{\infty} \frac{x}{\sqrt{2\pi}}e^{-x^2/2}dx
  2. \int_{0}^{\infty} x \lambda e^{-\lambda x}dx

Problem 6

Find the derivative with respect to \mu:

  1. \log{\prod^n_{i=1}\frac{1}{\sqrt{2\pi}}}e^{-\frac{(x_i - \mu)^2}{2}}

Footnotes

  1. Either convert to summation notation or evaluate the summation.↩︎