Find the average rate of change for $f(x)=7 x^{2}+1$ over the interval $[x, x+h]$. help (formulas)

Step 1 ：The average rate of change of a function over an interval [a, b] is given by the formula: \(\frac{f(b) - f(a)}{b - a}\)

Step 2 ：In this case, we are asked to find the average rate of change of the function \(f(x) = 7x^2 + 1\) over the interval [x, x+h]. So, we can substitute a = x and b = x + h into the formula: \(\frac{f(x + h) - f(x)}{(x + h) - x}\)

Step 3 ：First, we need to find \(f(x + h)\) and \(f(x)\):

Step 4 ：\(f(x + h) = 7(x + h)^2 + 1 = 7(x^2 + 2xh + h^2) + 1 = 7x^2 + 14xh + 7h^2 + 1\)

Step 5 ：\(f(x) = 7x^2 + 1\)

Step 6 ：Now, we can substitute these into the formula:

Step 7 ：Average rate of change = \(\frac{7x^2 + 14xh + 7h^2 + 1 - (7x^2 + 1)}{h} = \frac{14xh + 7h^2}{h} = 14x + 7h\)

Step 8 ：\(\boxed{14x + 7h}\) is the average rate of change of the function \(f(x) = 7x^2 + 1\) over the interval [x, x+h]