Yeah, need help solving this one…
A function is given below. Determine the average rate of change of the function between x = 1 and x = 7.
g(x)= 3 + 1/2x
Average Rate of Change – Determining the Average Rate of Change?
algebra-precalculus
Related Solutions
We have from the fundamental theorem of calculus that:
$$\int_{x_{1}}^{x_{2}}\frac{\mathrm{d}}{\mathrm{d}x}\left(f(x)\right)\:\mathrm{d}x=f(x_{2})-f(x_{1})$$
We also have the formula for an average of a function $f(t)$ between two values $t_{1}$ and $t_{2}$:
$$f_{\text{avg}}=\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}f(t)\:\mathrm{d}t$$
In your case, we therefore have:
$$g'_{\text{avg}}=\frac{1}{a-1}\int_{1}^{a}\frac{\mathrm{d}g(x)}{\mathrm{d}x}\:\mathrm{d}x=\frac{g(a)-g(1)}{a-1}=\frac{\frac{1}{a}-1}{a-1}=-\frac{1}{a}$$
It appears you are having trouble simplifying the fraction; note that if you multiply top and bottom by $a$ (which is valid because $\frac{a}{a}=1$) you have:
$$\frac{\frac{1}{a}-1}{a-1}=\frac{\frac{a}{a}-a}{a^{2}-a}=\frac{1-a}{a^{2}-a}=\frac{1-a}{a(a-1)}=-\frac{a-1}{a(a-1)}=-\frac{1}{a}$$
Given the graph above we can conclude the following (some values of $x$ don't have an easy to identify $y$ value on the graph because the interval of the y-axis is large).
Best Answer
Suppose that $g(1)=4$ and $g(7)=2$. (Neither of these is true, but I’ll use them to illustrate what’s going on.) Then the function has changed by $2-4=-2$ units between $x=1$ and $x=7$. That’s $-2$ units of change in $g$ while $x$ changed by $7-1=6$ units, so on average $g$ changed by $\frac{-2}6=-\frac13$ of a unit every time $x$ increased by $1$ unit. That is, the average rate of change was $-\frac13$ (units change in $g$ per $1$-unit change in $x$).
Now you try it with the actual values of $g(1)$ and $g(7)$.