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Section 3.11 Linearization and Differentials

Subsection 3.11.1 Linearization

Definition 3.11.1. Linearization.

If \(f\) is differentiable at \(x=a\text{,}\) then the function

\begin{equation*} L(x)=f(a)+f'(a)(x-a) \end{equation*}

is the linearization of \(f\) at \(a\text{.}\) The approximation \(f(x) \approx L(x)\) of \(f\) by \(L\) is the standard linear approximation of \(f\) at \(a\text{.}\) The point \(x=a\) is called the center of the approximation.

Example 3.11.2.

Find the linearization of \(y=\sqrt[3]{x}\) at \(x=8\text{.}\)

Answer.

\(L(x) = 2 + \frac{1}{12}(x-8)\)

Example 3.11.3.

Find the linearization of \(f(x)=\sin(x)\) that includes the point \(x_0=\frac{\pi}{7}\text{.}\) Center the linearization at a point \(x=a\) near \(\frac{\pi}{7}\) making the function easy to evaluate.

Answer.

\(L(x) = \frac12 + \frac{\sqrt{3}}{2}\left(x-\frac{\pi}{6}\right)\)

Subsection 3.11.2 Differentials

Definition 3.11.4. Differential.

Let \(y=f(x)\) be a differentiable function. We use the special symbol \(dx\text{,}\) called a differential, as the independent variable and the differential \(dy\) as the dependent variable, to form the equation

\begin{equation*} dy = f'(x)\, dx \end{equation*}

\(dy = f'(x)\, dx\) expresses the infinitesimal change in y as a function of the change in x.

Example 3.11.5.

Let \(f(x)=x^2\text{.}\)

  1. Find \(f'\)

  2. Find the value of \(df=f'(x_0)\, dx\) when \(x_0=1\) and \(dx=0.1\text{.}\)

  3. Estimate the value of \(f\) at \(x_0+dx\) as \(f(x_0)+df\text{.}\)

  4. Find the actual change \(\Delta f = f(x_0+dx)-f(x_0)\text{.}\)

  5. Find the error \(|\Delta f - df|\text{.}\)

Function with \(dx\text{,}\) \(dy\text{,}\) and \(\Delta f\)
Figure 3.11.6. Differential and error
Answer.

  1. \(\displaystyle f'(x) = 2x\)

  2. \(\displaystyle 0.2\)

  3. \(\displaystyle 1.2\)

  4. \(\displaystyle \Delta f = 1.21-1=0.21\)

  5. \(\displaystyle |0.21-0.2|=0.01\)

Example 3.11.7. Estimating related change/sensitivity to change.

The diameter of a tree was 14 inches one year. The next year, the circumference increase 3 in.

  1. Estimate the increase in diameter.

  2. Estimate the increase in cross-sectional area.

  3. Estimate the increase in cross-sectional area as a percentage of the previous area.

Example 3.11.8. Percent error.

A cylindrical storage tank is 10 meters tall. Its radius is \(3 \pm 0.25\) m. Estimate the percent error in calculating volume, \(V=\frac{1}{3}\pi r^2 h\text{.}\)

Solution.

This calls for computing \(\frac{dV}{v}\) when \(r=3\) and \(dr=0.25\text{.}\)

Example 3.11.9.

The volume of a sphere must be calculated to within \(2\%\) of the true value. Find the greatest error that can be tolerated in the measurement of the radius, expressed as a percentage of \(r\text{.}\)