Find the average rate of change for `f(x)=x^2+2x` from 4 to 9 and the equation of the secant line containing (4,f(4)) and (9,f(9)).
There are a couple of ways to do this, and they're based on what it means to find the "average rate of change." Without getting to deep into the details, the rate of change for a function is quantifying how much the function changes given a certain change in its input. In other words:
If that looks familiar, it's because it's the slope of a line!
Now, when its asking you to find the average rate of change, it's actually a blessing. It means that if you draw a line between the two points, all you're doing is finding the slope of that line. It's good because it ignores EVERYTHING that is happening between the two points. It only concerns you with those two points and those two points only.
What this question is asking you to do, effectively is to find the slope of the line between x = 4 (and its f(x)) and x = 9 (and its f(x)). So, let's calculate f(4) and f(9) to get our two points:
`f(4) = 4^2 + 2*4 = 24`
`f(9) = 9^2 + 2*9 = 99`
So we are trying to find the slope of the line going between the points `(4,24)` and `(9,99)`. Well, all we have to do is use our slope formula:
`m = (Deltay)/(Deltax) = (y_2-y_1)/(x_2-x_1) = (99-24)/(9-4) = 75/5 = 15`
A few steps later, and we have the answer to the first portion of the problem. Our average rate of change is 15.
There is another way to do this, but I'll do it at the end after we complete the second part of the problem.
Now, we need to find the equation for the secant line (different from the trigonometric secant function! see link), which is just the line between the two points that we already figured out. Recall, the usual equation for a line:
`y = mx+b`
All we need to do is put in one of our two points, plug in the slope we just calculated, and solve for "b" to get our equation. Let's use (4,24) for our point and, remember, our slope is 15:
`24 = 15(4) + b`
`24 = 60 + b`
Now, we just subtract 60 from both sides...
`b = -36`
so, all we need to do is put in our numbers for "m" and "b," and we'll have our secant line equation:
`y = 15x - 36`
And we're done!
Now, like I said before, I'm going to show you another way to calculate average rate of change. If you're in calculus, you know that the rate of change is represented by the derivative `(df(x))/(dx)`. You also would have learned the equation for the average value of a function (let's call A) over a certain interval:
`A = 1/(b-a) int_a^b f(x) dx`
So, when we're trying to find the average (think the equation) rate of change (think derivative) we can combine these two concepts in the following way:
`A = 1/(b-a) int_a^b f'(x) dx`
Notice that all we did was put in the derivative for the function. To do that in our case, we need to calculate the derivative first:
`f'(x) = 2x+2`
Using this as our function to average, and our bounds of x = 4 and x = 9, we can find the average value:
`A = 1/(9-4) int_4^9 2x+2 dx`
Evaluating the integral:
`A = 1/(9-4) (x^2 + 2x)`
Keep in mind, we need to evaluate for our boundary values (f(b)-f(a))
`A = 1/(9-4) ( 9^2 + 2(9) - 4^2 - 2(4))`
`A = (99 - 24)/(9-4) = 15`
In case you didn't notice, when we were evaluating the integral and subtracting based on the function values at the boundaries, that was the same thing as just doing `y_2-y_1`.
I only show this way as a systematic way that can be applied in all situations, because not in all situations will you have the exact function spelled out to you as in this problem! I hope that helped!