### Home > APCALC > Chapter 2 > Lesson 2.4.1 > Problem2-145

2-145.

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

1. $\frac { f ( 3 ) - f ( 2 ) } { 1 }$

This is the slope of the secant between $x = 3$ and $x = 2$, otherwise known as Average Rate of Change (AROC).

1. $\frac { f ( 2.1 ) - f ( 2 ) } { 0.1 }$

This is the slope of the secant between $x = 2.1$ and $x = 2$.

1. $\frac { f ( 2.01 ) - f ( 2 ) } { 0.01 }$

And this is the slope of the secant between $x =$____ and $x =$____.

1. Estimate $\lim\limits _ { x \rightarrow 2 } \frac { f ( x ) - f ( 2 ) } { x - 2 }$.

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?