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3-77.

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?

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 $a$?

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

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

$a = 3$
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 $f^\prime(a)$.

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