How do you find the average slope over the indicated intervals? The question in my book is $g(x)=4x+5$ from $x=-3$ to $x= -1$ what I don't understand is how do find the average slope and is there a easy way of doing math problems like this?
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$\begingroup$The average slope in $[a,b]$ is computed as follows $$\frac{f(b)-f(a)}{b-a}$$ you can see the function $f$ as the distance of a moving object from a certain point, in that case the average slope means the average speed, therefore you just compute the distance of the trajectory divided by the time.
$\endgroup$ $\begingroup$Slope is rise/run, that is how I was taught. So you take the value of the function at the end points and subtract, then divide be the interval length. This is the simplest way I can explain without using a formula
$\endgroup$ $\begingroup$You probably mean the "average rate of change."
Many students get tripped up by this without knowing it's something they already know - this is just the slope of the line that joins two points on a curve.
Remember that the slope of a line joining $(x_1, y_1)$ and $(x_2, y_2)$ is
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Oh, memories from algebra I. Now, in calculus, it is of interest to examine an average rate of change and compare that to the instantaneous rate of change - the derivative.
To find the average slope of $f$ on $[a, b]$, first find $f(a)$ and $f(b)$ and use the above formula. Done.
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