Make a table of values for the function F(x)=(x+2) /(x-2) at...

Make a table of values for the function $F(x)=(x+2) /(x-2)$ at the points $x=1.2, x=11 / 10, x=101 / 100, x=1001 / 1000$ $x=10001 / 10000,$ and $x=1$ a. Find the average rate of change of $F(x)$ over the intervals $[1, x]$ for each $x \neq 1$ in your table. b. Extending the table if necessary, try to determine the rate of change of $F(x)$ at $x=1$

Make a table of values for the function $F(x)=(x+2) /(x-2)$ at the points $x=1.2, x=11 / 10, x=101 / 100, x=1001 / 1000$ $x=10001 / 10000,$ and $x=1$
a. Find the average rate of change of $F(x)$ over the intervals $[1, x]$ for each $x \neq 1$ in your table.
b. Extending the table if necessary, try to determine the rate of change of $F(x)$ at $x=1$

Key Concepts

Limits

Limits are used to describe the behavior of a function as its input approaches a particular value. They are essential in calculus for defining both derivatives (instantaneous rates of change) and continuity of functions. The concept of a limit provides the rigorous foundation for moving from average to instantaneous rates of change.

Instantaneous Rate of Change (Derivative)

This concept extends the idea of the average rate of change to a single point, defining the exact rate at which a function is changing at that point. It is found by taking the limit of the average rate of change as the interval narrows to zero, effectively giving the slope of the tangent line to the function's graph.

Average Rate of Change

The average rate of change represents how much the function's output changes per unit change in the input over an interval. It is calculated as the difference in the function values at the end points of the interval divided by the difference in the input values. This concept is fundamental in understanding the overall trend of the function between two points.

Function Evaluation and Tabulation

This involves calculating the output values of a function for specific inputs. Tabulating these values helps in visualizing and understanding the behavior of the function over selected intervals, which is a foundational method in mathematical analysis.

Transcript

00:01 First we're going to find some function values for the given f of x is x plus 2 over x minus 2.

00:06 And then we're given x is 1 .2, 11 over 10, 101 over 100, 1 ,000 over 1 ,000, and x is 1 as well.

00:18 So first of all, if i fill in 1 .2, make sure i evaluate it in both numerator and denominator.

00:23 So 1 .2 plus 2 over 1 .2 minus 2.

00:27 And that gives me, negative 4.

00:35 Now if i use 11 over 10, which is really just 1 .1, in the function for f of x, i would have negative 3 .444.

00:47 If i use 1 .01, i have negative 3 .040.

00:53 If i use 1 .001, if i use 1 .001, i get negative 3 .004.

01:00 And then if i use 1 .00001, i get negative 0 .001.

01:05 I get negative.

01:05 Negative 3 .00, rounding that.

01:12 And if i were to fill in 1 as the x, i get negative 3.

01:19 So we can see those values approaching there.

01:21 So now that i have my table filled in, i want to find average rate of change on the interval 1 to x, and each of these x values is going to be one of my end points.

01:34 So for average rate of change, remember i'm going to do f of x, f of 1 over x minus 1.

01:41 So for the first interval, i'm going to look from x is 1 .2.

01:50 And if i were to fill that in to that average rate of change setup, i would have f of 1 .2, which i know is negative 4, minus f of 1, which is negative 3, over 1 .2 minus 1, which is 0 .2, which is 0 .2.

02:08 If i fill that into that average rate of change, i get negative 5.

02:14 So actually, let's just make a table using our x values and that result we get from that average rate of change...

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