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2-145.

Given the function $f(x) = 2x^2 − x + 3$ , calculate the following values.

$\frac { f ( 3 ) - f ( 2 ) } { 1 }$
Hint (a):
This is the slope of the secant between $x = 3$ and $x = 2$ , otherwise known as Average Rate of Change (AROC).

$\frac { f ( 2.1 ) - f ( 2 ) } { 0.1 }$
Hint (b):
This is the slope of the secant between $x = 2.1$ and $x = 2$ .

$\frac { f ( 2.01 ) - f ( 2 ) } { 0.01 }$
Hint (c):
And this is the slope of the secant between $x =$ ____ and $x =$ ____.

Estimate $\lim\limits _ { x \rightarrow 2 } \frac { f ( x ) - f ( 2 ) } { x - 2 }$ .
Hint (d):
Notice that as you progressed through parts (a), (b) and (c), the secant line becomes smaller and smaller, squeezing in on the actual slope at $x = 2$ . Based on the patterns you see as the secant line decreases, what do you think the slope of the smallest possible secant line will be?

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