Section3.6CAP 2017, HW 2 due January 31¶ permalink

Subsection3.6.1Exercises

Definition3.6.1The Definition of a Derivative

The computation of the slope of a tangent line, the instantaneous rate of change of a function, at \(x=a\) can be found from the following limit:

\begin{equation}f'(x)=\lim_{a \to x} \frac {f(x) - f(a)} {x-a}\label{men-4}\tag{3.6.1}\end{equation}

With a small adjustment in notation this limit can be rephrased.

The derivative of \(f(x)\) with respect to \(x\) is the function \(f'(x)\) and is defined as:

\begin{equation}f'(x)=\lim_{h \to 0} \frac {f(x+h) - f(x)} {h}\label{men-5}\tag{3.6.2}\end{equation}

Definition3.6.2The Power Rule

If \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a function such that \(f(x) = x^r\), and \(f\) is differentiable at \(x\), then, : \begin{equation}f'(x) = rx^{r-1}\label{men-6}\tag{3.6.3}\end{equation}

Definition3.6.3Linear Approximation

Given a twice continuously differentiable function \(f\) of one real number variable, Taylor's theorem for the case \(n = 1 \) states that: \begin{equation*} f(x) = f(a) + f'(a)(x - a) + R_2\ \end{equation*} where \(R_2\) is the remainder term. The linear approximation is obtained by dropping the remainder: \begin{equation} f(x) \approx f(a) + f'(a)(x - a)\text{.}\label{men-7}\tag{3.6.4}\end{equation}This is a good approximation for \(x\) when it is close enough to \(a\); since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of \(f\) at \((a,f(a))\). For this reason, this process is also called the tangent line approximation.