Lazy Lulu is looking at this limit: $\lim\limits _ { x \rightarrow 3 } \large\frac { x ^ { 3 } + x - 30 } { x - 3 }$ and does not want to calculate it using algebra. Lulu recognizes this limit as a definition of the derivative at a point. She thinks she can use the Power Rule instead.

1. What variation of the definition of the derivative is this?

   Answer (a):

   This is Ana's Definition of the Derivative:

   $f'(a)=\lim\limits_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$

   $=\lim\limits_{x\rightarrow 3}\frac{(x^{3}+x)-((3)^{3}+3)}{x-3}=\lim\limits_{x\rightarrow 3}\frac{x^{3}+x-30}{x-3}$

2. What is the equation of $f$ ? What is the value of $a$?

   Hint (b):

   Observe that $x^3 + x = 30$ when $x = 3$.

   Answer (b):

   $f(x) = x^3 + 3$

   Answer (b):

   $a = 3$
   Notice that $f(3) = 3^3 + 3 = 30$, which confirms that equation of $f(x)$ is correct.

3. Use the Power Rule to write an equation for $f^\prime$ and then use it to calculate $f^\prime(a)$.

   Hint (c):

   $f(x) = x^3 + x$
   $f^\prime (x) =$_______________________
   $f^\prime (3) =$__________
   You just avoided a lot of Algebra!