Linear Approximations and Differentials
Linear Approximations and Differentials
Objectives
- Describe the linear approximation to a function at a point.
- Write the linearization of a given function.
- Draw a graph that illustrates the use of differentials to approximate the change in a quantity.
- Calculate the relative error and percentage error in using a differential approximation.
Summary
The tangent line to a differentiable function \(y = f(x)\) at the point \((a,f(a))\) is given in point-slope form by the equation
\begin{equation*} y - f(a) = f'(a)(x-a)\text{.} \end{equation*}The principle of local linearity tells us that if we zoom in on a point where a function \(y = f(x)\) is differentiable, the function will be indistinguishable from its tangent line. That is, a differentiable function looks linear when viewed up close. We rename the tangent line to be the function \(y = L(x)\text{,}\) where \(L(x) = f(a) + f'(a)(x-a)\text{.}\) Thus, \(f(x) \approx L(x)\) for all \(x\) near \(x = a\text{.}\)
If we know the tangent line approximation \(L(x) = f(a) + f'(a)(x-a)\) to a function \(y=f(x)\text{,}\) then because \(L(a) = f(a)\) and \(L'(a) = f'(a)\text{,}\) we also know the values of both the function and its derivative at the point where \(x = a\text{.}\) In other words, the linear approximation tells us the height and slope of the original function. If, in addition, we know the value of \(f''(a)\text{,}\) we then know whether the tangent line lies above or below the graph of \(y = f(x)\text{,}\) depending on the concavity of \(f\text{.}\)