Let g(x)=√(x) for x ≥ 0 a. Find the average rate of change of g(x) with respect to

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Let g(x)=√(x) for x ≥ 0 a. Find the average rate...

Let \begin{equation} g(x)=\sqrt{x} \text { for } x \geq 0 \end{equation} \begin{equation} \begin{array}{l}{\text { a. Find the average rate of change of } g(x) \text { with respect to } x \text { over }} \\ {\text { the intervals }[1,2],[1,1.5] \text { and }[1,1+h] .} \\ {\text { b. Make a table of values of the average rate of change of } g \text { with }} \\ {\text { respect to } x \text { over the interval }[1,1+h] \text { for some values of } h} \\ {\text { approaching zero, say } h=0.1,0.01,0.001,0.0001,0.00001,} \\ {\text { and } 0.000001 .}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { c. What does your table indicate is the rate of change of } g(x)} \\ {\text { with respect to } x \text { at } x=1 ?} \\ {\text { d. Calculate the limit as } h \text { approaches zero of the average rate of }} \\ {\text { change of } g(x) \text { with respect to } x \text { over the interval }[1,1+h] \text { . }}\end{array} \end{equation}

Let \begin{equation}
g(x)=\sqrt{x} \text { for } x \geq 0
\end{equation}
\begin{equation}
\begin{array}{l}{\text { a. Find the average rate of change of } g(x) \text { with respect to } x \text { over }} \\ {\text { the intervals }[1,2],[1,1.5] \text { and }[1,1+h] .} \\ {\text { b. Make a table of values of the average rate of change of } g \text { with }} \\ {\text { respect to } x \text { over the interval }[1,1+h] \text { for some values of } h} \\ {\text { approaching zero, say } h=0.1,0.01,0.001,0.0001,0.00001,} \\ {\text { and } 0.000001 .}\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}{\text { c. What does your table indicate is the rate of change of } g(x)} \\ {\text { with respect to } x \text { at } x=1 ?} \\ {\text { d. Calculate the limit as } h \text { approaches zero of the average rate of }} \\ {\text { change of } g(x) \text { with respect to } x \text { over the interval }[1,1+h] \text { . }}\end{array}
\end{equation}

Key Concepts

Derivative

The derivative at a point is the instantaneous rate of change of a function with respect to its variable at that specific point. It represents the slope of the tangent line to the function's graph at that point and is obtained by taking the limit of the difference quotient as the interval shrinks to zero.

Average Rate of Change

The average rate of change of a function between two points is a measure of how much the function's value changes per unit change in the independent variable over that interval. It is computed as the difference in function values divided by the difference in input values, essentially representing the slope of the secant line connecting the two points on the graph.

Difference Quotient

The difference quotient is the expression used to compute the average rate of change of a function and is given by [f(x + h) - f(x)]/h. This formula forms the basis for determining changes in the function as the interval h becomes very small, and is a precursor to the concept of derivatives.

Limit Process

The limit process involves evaluating the behavior of a function or expression as the input approaches a particular value. In this context, taking the limit of the difference quotient as h approaches zero refines the average rate of change into the instantaneous rate of change, which is essential in defining the derivative.

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