# What is the average rate of change of the function y = x^2+1 over the interval [5, 5+a] ?

##### 1 Answer
May 12, 2015

The average rate of change will be the total change in $y$ divided by the total change in $x$, so we just have to look at the values at each end of the given interval:

When $x = 5$

$y = {5}^{2} + 1 = 26$.

When $x = 5 + a$:

$y = {\left(5 + a\right)}^{2} + 1 = 25 + 10 a + {a}^{2} + 1$

$= 26 + 10 a + {a}^{2}$

$\Delta y = \left(26 + 10 a + {a}^{2}\right) - 26 = 10 a + {a}^{2}$

$\Delta x = \left(5 + a\right) - 5 = a$

So $\frac{\Delta y}{\Delta x} = \frac{10 a + {a}^{2}}{a} = 10 + a$