Calculus BC

• Sharp turns on graphs are not differentiable.
• Common Mistake: When doing a derivative, I forget that it is a product rule/quotient rule.


  Before doing a derivative right down, what properties need to be done (example: product, quotient, chain rule)
• How do I solve this problem: $\frac{dx}{dy}[a^{f(x)} ]$


  $\frac{dx}{dy}[a^{f(x)} ]=$$ln(a) \cdot a^ {f(x)} \cdot f'(x)$

  
• What is the formula for derivative of inverse?

  It is the following :

  $g'(x)=$$\frac{1}{f'(g(x)}$
• Most of the time, we want to start at i = 0 or i = 1. It is more traditional to use i = 0 for left or midpoint sums and i = 1  for right sums
• How do I solve these types of problems (Consider the sum 5+ (-7)+(-27)+(-55). Which expression is equal to the above sum 


  Understand that the number of the top --> # of times you are doing this

  A) correct

  
• Let's say you are asked to take the derivative of this $g(x)=$$\int_{0}^{x^4} cos(x) \,dx$

  how would you do it?

  Since the integral has a variable in the upper bound you plug in the upper bound and multiply it with the derivative of the variable.
• How do you set up a partial decompostion

  function = A/(something)+ B/(something)
• To find the r value in a geometric series, you divide any term by the term before it. A geometric series with ratio r, diverges if r greater than or equal to 1. If r is in between 0 and 1 then it converges to this$s =$$\frac{a}{1-r}$
• The criteria to use the Integral test is that:

  1. The function has to be positive
  2. The function has to be decreasing
  3. The function has to be continuous
• The p-series looks like this : $\sum_{n=1}^{\infty} \frac {1}{n^p}$where p can only be a positive constant. If p is greater than 1 it converges. If p is between 1 and zero it diverges.
• Direct Comparison Test:

  
• What is the limit comparison test?


  An and Bn is greater than 0. If Iimit of the ratio (an/bn) is L a finite and positive number that both series either both converge or diverge