AD - Local Linearity, Linear Approximation, and Differentials Lesson

Local Linearity, Linear Approximation, and Differentials

Local Linearity and Linear Approximation

image of a graph with a tangent line Recall that the tangent to a curve y = f(x) lies very close to the curve at the point of tangency. When the graph of a function appears to coincide with its tangent line near x = a, the curve demonstrates local linearity. For a small interval on either side of the point (a, f(a)), the y-values along the tangent line provide excellent approximations to the y-values on the curve.

The linear approximation of a curve is the equation of the tangent line, f(x) ≈ f(a) + f '(a)(x - a). The linear function whose graph is this tangent line is given by L(x) = f(a) + f '(a)(x - a) and is called the linearization of f at a.

Zooming and Local Linearity

If we zoom in toward the point of tangency, the graph of the differentiable function looks more and more like its tangent line. This notion forms the basis for finding approximate values of functions. View the illustration incorporating the zooming capability of a grapher to investigate local linearity and to find the linearization equation.

Examine the graph of equation image indicator $LaTeX: y=6\sqrt[]{x}-4$ $y=6\sqrt[]{x}-4$ at the point (1,2)

Graph the function in the ZDecimal window, trace to (1, 2), and zoom in repeatedly on the curve.

calculator screenshot: plot calculator screenshot: zoom decimal menu calculator screenshot: one calculator screenshot: two

Notice that the result of zooming in on (1, 2) appears to be a line.

Use your calculator to find the equation of this "line": y = 3x - 1.

calculator screenshot: menu calculator screenshot: tangent equation graph

Differentials

image of differentials on a graph The key ideas of linear approximations are sometimes framed in the terminology and notation of differentials. If y = f(x) is differentiable on an open interval containing x, then the differential dx represents an increment of an independent variable x and is any nonzero real number. The differential (dy) is defined by dy = f '(x) dx. Note that both dx and dy are numerical values.

Geometrically, dx is the number of units of change in the x-direction of the tangent line. dy is the number of units of change in the y-direction of the tangent line, which is the amount that the curve rises or falls when x changes by an amount dx.

Differentials definitions

The Magic Multiplier Effect: Derivatives to Differentials

Every derivative formula has a corresponding differential formula that results from multiplying the derivative formula by du or dx.

Derivative of y = f(u)	Differential of y = f(u)
$LaTeX: \frac{d\left(k\right)}{du}=0$ $\frac{d\left(k\right)}{du}=0$	$LaTeX: d\left(k\right)=0$ $d\left(k\right)=0$
$LaTeX: \frac{d\left(u^n\right)}{du}=nu^{n-1}$ $\frac{d\left(u^n\right)}{du}=nu^{n-1}$	$LaTeX: d\left(u^n\right)=nu^{n-1}du$ $d\left(u^n\right)=nu^{n-1}du$
$LaTeX: \frac{d\left(uv\right)}{dx}=\frac{udv}{dx}+\frac{vdu}{dx}$ $\frac{d\left(uv\right)}{dx}=\frac{udv}{dx}+\frac{vdu}{dx}$	$LaTeX: d\left(uv\right)=udv+vdu$ $d\left(uv\right)=udv+vdu$
$LaTeX: \frac{d\left(ku\right)}{dx}=\frac{kdu}{dx}$ $\frac{d\left(ku\right)}{dx}=\frac{kdu}{dx}$	$LaTeX: d\left(ku\right)=kdu$ $d\left(ku\right)=kdu$
$LaTeX: \frac{d\left(u+v\right)}{dx}=\frac{du}{dx}+\frac{dv}{dx}$ $\frac{d\left(u+v\right)}{dx}=\frac{du}{dx}+\frac{dv}{dx}$	$LaTeX: d\left(u+v\right)=du+dv$ $d\left(u+v\right)=du+dv$
$LaTeX: \frac{d\left(\frac{u}{v}\right)}{dx}=\frac{\left(v\frac{du}{dx}-u\frac{dv}{dx}\right)}{v^2}$ $\frac{d\left(\frac{u}{v}\right)}{dx}=\frac{\left(v\frac{du}{dx}-u\frac{dv}{dx}\right)}{v^2}$	$LaTeX: d\left(\frac{u}{v}\right)=\frac{\left(vdu-udv\right)}{v^2}$ $d\left(\frac{u}{v}\right)=\frac{\left(vdu-udv\right)}{v^2}$
$LaTeX: \frac{d\left(\sin u\right)}{dx}=\cos u\frac{du}{dx}$ $\frac{d\left(\sin u\right)}{dx}=\cos u\frac{du}{dx}$	$LaTeX: d\left(\sin u\right)=\cos udu$ $d\left(\sin u\right)=\cos udu$
$LaTeX: \frac{d\left(\cos u\right)}{dx}=-\sin u\frac{du}{dx}$ $\frac{d\left(\cos u\right)}{dx}=-\sin u\frac{du}{dx}$	$LaTeX: d\left(\cos u\right)=-\sin udu$ $d\left(\cos u\right)=-\sin udu$
$LaTeX: \frac{d\left(\tan u\right)}{dx}=\sec^2u\frac{du}{dx}$ $\frac{d\left(\tan u\right)}{dx}=\sec^2u\frac{du}{dx}$	$LaTeX: d\left(\tan u\right)=\sec^2udu$ $d\left(\tan u\right)=\sec^2udu$
$LaTeX: \frac{d\left(\sec u\right)}{dx}=\sec u\:\tan u\frac{du}{dx}$ $\frac{d\left(\sec u\right)}{dx}=\sec u\:\tan u\frac{du}{dx}$	$LaTeX: d\left(\sec u\right)=\sec u\:\tan udu$ $d\left(\sec u\right)=\sec u\:\tan udu$
$LaTeX: \frac{d\:\csc\:u}{dx}=-\csc u\:\cot u\frac{du}{dx}$ $\frac{d\:\csc\:u}{dx}=-\csc u\:\cot u\frac{du}{dx}$	$LaTeX: d\left(\csc u\right)=-\csc u\:\cot udu$ $d\left(\csc u\right)=-\csc u\:\cot udu$

Differentials Problems

View the presentation below illustrating interpretations and applications of differentials.

**Special NOTE** There's a mistake in the video around 8:10. He gets an answer of 16.08 cubic inches but it should be 0.1608 cubic inches.

The concept and notation of differentials are foundational elements for finding antiderivatives, evaluating integrals, and solving differential equations, topics that will be explored in subsequent modules. The symbol dx is used in the physical sciences more frequently than Δx.

Local Linearity, Linear Approximation, and Differentials Practice

1. Find the linearization of y = sin x at equation image indicator $LaTeX: \frac{\pi}{3}$ $\frac{\pi}{3}$ .

text annotation indicator Solution: $LaTeX: L\left(x\right)=\frac{\sqrt[]{3}}{2}+\frac{1}{2}\left(x-\frac{\pi}{3}\right)$ $L\left(x\right)=\frac{\sqrt[]{3}}{2}+\frac{1}{2}\left(x-\frac{\pi}{3}\right)$

2. The circumference of the equator of a sphere is measured as 20 cm with a possible error of 0.4 cm. The measurement is then used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of (a) the radius, (b) the surface area, and (c) the volume.

Solution: a. 2%; b. 4%; c. 6% text annotation indicator

Local Linearity, Linear Approximation, and Differentials: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of local linearity, linear approximation, and differentials.

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