Linear Approximation
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.
Lesson Objectives
By the end of this lesson, you will be able to:
- use differentials to come up with linear approximations.
- solve differential equations.
Key Terms
- differential: 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
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
at the point (1,2).
Graph the function in the ZDecimal window, trace to (1, 2), and zoom in repeatedly on the curve.
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.
Above images used courtesy of Georgia Virtual Learning.
Differentials
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.
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) |
Differentials Problems
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Interpretations and Applications of Differentials
View the presentation below illustrating interpretations and applications of differentials.
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.