Limits and Derivatives Review

Limits and Derivatives Review

Limits

• Limits measure what a function looks like its doing.

• Follow the path on the graph. We do not care what happens at the actual point.

Limits

Evaluate the following:

 $\lim_{x\rightarrow0}\ f\left(x\right)$ 

 $\lim_{x\rightarrow1}\ f\left(x\right)$  

 $\lim_{x\rightarrow3}\ f\left(x\right)$  

Limits

• When looking at limits algebraically, your first step is to always plug the limit in and see what happens.

• As long as your answer is not  $\frac{0}{0}$  or

   $\frac{\infty}{\infty}$  you will have your answer.

Multiple Choice

Evaluate

 $\lim_{x\rightarrow4}\ \frac{x^2\ +\ 16}{x\ -\ 4}$  

• 8

• DNE

• 0

• -4

Multiple Choice

Evaluate

 $\lim_{x\rightarrow9}\ \frac{x-9}{\sqrt{x}-3}$ .

• 6

• 1/6

• 0

• DNE

Power Rule of Derivatives

• Let  $f\left(x\right)\ =\ x^n$ .  
  

• Then $f'\left(x\right)\ =\ nx^n-1$ .

Multiple Choice

Differentiate

 $f\left(x\right)\ =\ 12x^3\ -\ \frac{4}{x^4}.$  

•  $f'\left(x\right)\ =\ 36x^2\ +\ 16x^{-5}$  

•  $f'\left(x\right)\ =\ 12x^3\ -\ 16x^{-5}$  

•  $f'\left(x\right)\ =\ 36x^2\ +\ 16x^{-3}$  

•  $f'\left(x\right)\ =\ 36x^2\ -\ 16x^{-3}$  

Product Rule of Derivatives

• Let $f\left(x\right)\ =\ a\left(x\right)\ \cdot b\left(x\right)$  .
  

• Then $f'\left(x\right)\ =\ a\left(x\right)\cdot b'\left(x\right)\ +\ a'\left(x\right)\cdot b\left(x\right)$  .

Multiple Choice

Find $g'\left(x\right)$ if   $g\left(x\right)\ =\ \left(4x^3-\ 17x\right)\left(3x^5\ +\ 9x^2\right)$  

•  $g'\left(x\right)\ =\ \left(4x^3\ -\ 17x\right)\left(15x^4+\ 18x\right)\ +\ \left(12x^{2\ }-\ 17\right)\left(3x^5\ +\ 9x^2\right)$  

•  $g'\left(x\right)\ =\ \left(12x^2\ -\ 17\right)\left(15x^4+\ 18x\right)$  

Quotient Rule for Derivatives

• If $f\left(x\right)\ =\ \frac{a\left(x\right)}{b\left(x\right)}$  ,

• then $f'\left(x\right)\ =\ \frac{\left(b\left(x\right)\cdot a'\left(x\right)\ -\ a\left(x\right)\cdot b'\left(x\right)\right)}{\left(b\left(x\right)\right)^2}$  .

Multiple Choice

Differentiate $f\left(x\right)\ =\ \frac{5x^2}{x^3\ +\ 1}$  .

•  $f'\left(x\right)\ =\ \frac{\left(\left(x^3\ +\ 1\right)\left(10x\right)\ -\ \left(5x^2\right)\left(3x^2\right)\right)}{\left(x^3\ +\ 1\right)^2}$  

•  $f'\left(x\right)\ =\ \frac{\left(\left(5x^2\right)\left(3x^2\right)\ -\ \left(x^3\ +\ 1\right)\left(10x\right)\right)}{\left(x^3\ +\ 1\right)^2}$  

•  $f'\left(x\right)\ =\ \frac{10x}{3x^2}$  

The Chain Rule for Derivatives

• If $f\left(x\right)\ =\ a\left(b\left(x\right)\right)$ ,

• then $f'\left(x\right)\ =\ a'\left(b\left(x\right)\right)\cdot b'\left(x\right)$  .

• Think derivative of the outer function times the derivative of the inner function.

• Leave the candy inside alone until we get to it.

Multiple Choice

Differentiate $f\left(x\right)\ =\ \sqrt{3x^5-\ 11x^3}$  .

•  $f'\left(x\right)\ =\ \frac{1}{2}\left(3x^5\ -\ 11x^3\right)^{-\frac{1}{2}}\left(15x^4\ -\ 33x^2\right)$  

•  $f'\left(x\right)\ =\ \frac{1}{2}\left(15x^4\ -\ 33x^2\right)^{-\frac{1}{2}}$  

•  $f'\left(x\right)\ =\ \frac{1}{2}\left(3x^5\ -\ 11x^3\right)^{\frac{1}{2}}\left(15x^4\ -\ 33x^2\right)$  

Multiple Choice

The average rate of change is the same thing as

• the slope between two points.

• the derivative.

• marginal cost.

• the cost of a stamp.

Multiple Choice

The instantaneous rate of change is the same things as

• the derivative.

• the slope between two points.

• the difference quotient.

• the times of our lives.

Poll

How confident do you feel going into Thursday's test?

• I got this!

• I will feel better after tomorrow's review.

• I am unsure.

• I need help.