The Average Rate of Change of a function over an interval gives us an idea of how the function values change on average between two points. This concept is analogous to the slope of the secant line that passes through these points on the graph of the function, \(a\) and \(b\).
The Average Rate of Change can be expressed algebraically as:
Calculate the Average Rate of Change of the function \(f(x) = x^2\) where \(a = 1\) and \(b = 4\)
First, we can evaluate \(f(b)\) by substituting \(4\) for \(x\) in the original function:
\(f(4) = (4)^2\)
\(\textcolor{red}{f(4) = 16}\)
Next, we can evaluate \(f(a)\) by substituting \(1\) for \(x\) in the original function:
\(f(1) = (1)^2\)
\(\textcolor{blue}{f(1) = 1}\)
Finally, we can substitute all pertinent values into the secant formula to determine its slope at these points:
Therefore, we can determine that the Average Rate of Change of \(f(x) = x^2\) where \(a = 1\) and \(b = 4\) is \(\boldsymbol{5}\).
First, we can evaluate \(f(b)\) by substituting \(0.5\) for \(x\) in the original function:
\(f(0.5) = -4.9(0.5)^2 + 14(0.5) + 1\)
\(\textcolor{red}{f(0.5) = 6.775}\)
Next, we can evaluate \(f(a)\) by substituting \(0\) for \(x\) in the original function:
\(f(0) = -4.9(0)^2 + 14(0) + 1\)
\(\textcolor{blue}{f(0) = 1}\)
Finally, we can substitute all pertinent values into the secant formula to determine its slope at these points:
Therefore, we can determine that the Average Rate of Change of \(f(x) = -4.9x^2 + 14x + 1\) where \(a = 0\) and \(b = 0.5\) is \(\boldsymbol{11.55}\).