Limits and the Derivative

(Hypothetical) Production Function for Grades

Average Rate of Change

Use \(\Delta\) to denote change: \[\Delta x=x_{1}-x_{0}\]

Change in \(y\) per unit change in \(x\): \[\frac{\Delta y}{\Delta x}=\frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x}\]

Average Rate of Change

\[y=f(x)=x^{2}+20\]

What happens if you go from 6 to 8 hours of studying? \[x_0 = 6, \; x_1 = 8 \; \rightarrow \; \Delta x = x_{1}-x_{0}=2\]

Total change in grades: \[f(x_0+\Delta x) - f(x_0) = \hspace{4em}\]

Per hour change in grade: \[\frac{\Delta y}{\Delta x}=\frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x} = \hspace{4em}\]

The Derivative

Usually we are interested in minuscule changes from \(x_0\).

The derivative of a function is defined as: \[\frac{d y}{d x} = f'(x)= \lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}\]

Note that \[\frac{\Delta y}{\Delta x}=\frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x}\]

The Derivative

For the function \(y=f(x)=x^{2}+20\): \[\begin{aligned} \frac{\Delta y}{\Delta x}&=\frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x} \\ &= \frac{(x_0+\Delta x)^2+20-(x_0^2+20)}{\Delta x} \\ &= \frac{x_0^2+ (\Delta x)^2+2 x_0 \Delta x-x_0^2}{\Delta x} \\ &= 2 x_0 + \Delta x \end{aligned}\]

Then the derivative is \[\frac{d y}{d x} = f'(x_0)= \lim_{\Delta x \rightarrow 0} \, (2 x_0 + \Delta x) = 2 x_0\]

The Derivative

The derivative: \[\frac{d y}{d x} = f'(x_0)= \lim_{\Delta x \rightarrow 0} \frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x}\]

Alternatively, \[\frac{d y}{d x} = f'(x_0)= \lim_{x \rightarrow x_0} \frac{f(x)-f(x_{0})}{x-x_0}\]

The Derivative

For the function \(y=f(x)=x^{2}+20\): \[\begin{aligned} \frac{d y}{d x}&= \lim_{x \rightarrow x_0} \frac{f(x)-f(x_{0})}{x-x_0} \\ &= \lim_{x \rightarrow x_0} \frac{x^2+20-x_0^2-20}{x-x_0} \\ &= \lim_{x \rightarrow x_0} \frac{x^2-x_0^2}{x-x_0} \\ &= \lim_{x \rightarrow x_0} \, (x+x_0) = 2x_0 \end{aligned}\]

Derivative = Slope of the Tangent Line

\[y=f(x)=x^{2}+20 \; \rightarrow \; f'(x) = 2x\]

Concept of a Limit

We say that \(L\) is the limit of \(f(x)\) at \(a\), i.e. \[\lim_{x \rightarrow a} f(x) = L\] if \(f(x)\) approaches \(L\) as \(x\) approaches \(a\) from any direction.

Note that we don’t actually set \(x=a\).

Also, for the limit to exist at a point we need the function to approach the same value from both directions.

Concept of a Limit

Left-side limit: if \(x\) approaches \(a\) from the left side \[\lim_{x \rightarrow a^-} f(x)\]

Right-side limit: if \(x\) approaches \(a\) from the right side \[\lim_{x \rightarrow a^+} f(x)\]

Only when both the left-side and right-side limits have a common finite value do we say that the limit exists.

Example: Limit Doesn’t Exist

Limit: Another Example

\[f(x) = \frac{4x-4}{\left|x-1\right|}\]

\[\lim_{x \rightarrow 1^-} f(x) = \hspace{6em}\]

\[\lim_{x \rightarrow 1^+} f(x) = \hspace{6em}\]

Limit: Another Example

Homework Questions

• Exercise 6.2: 2, 3

Textbook reference: 6.2-6.4