Lesson 6

Linear Approximation

Linear Approximation

image: piecewise linear function artworkRecall 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

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

 

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 '(xdx

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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  

y equals six to the square root of x minus 4

 

 

at the point (1,2).

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

Graphing calculator screen shown. In Plot , it has y 1 is equal to 6 times square root x minus 4. In zoom menu go to 4: Z deximal Then on the graph screen it traces to x = 1 and y = 2 z decimial window example 2
z decimial window example 3 z decimial window example 4

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.

Graphing Calculator in DRAW, selection number 5 tangent. image: line equation example 2

Above images used courtesy of Georgia Virtual Learning.

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Differentials

 

image: Differentials ExampleThe 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 If y equals f of x is differentiable on an open interval containing x, then the differential of x (dx) represents and increment of a variable x and is any nonzero real number. The differential of y (dy) is defined by dy equals f prime of x dx Note that both dx and dy are numerical values. Geometric interpretation 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 Uses of Differentials Linearization of functions (approximating functions with a line) The tangent line approximation of f at a. F of x equals f prime of a times x minus a plus f of a is a linearization of f at a. Approximations- Area, volume, surface area, error, etc.

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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)

D of k over d u is equal to 0

D of k is equal to 0

D of u to the nth power over d u is equal to n time u to the n minus 1 power

D of u to the nth power is equal to n time u to the n minus 1 power times d u

D of (u times v) over d x is equal to u times d v over d x plus v time d u over d x

D of (u times v) is equal to u times d v plus v times d u

D of (k times u) over d x is equal to k times d u over d x.

D of (k times u) is equal to k times d u

D of (u plus v) over d x is equal to d u over d x plus d v over d x

D of (u plus v) is equal to d u plus d v

D of (u divided by v) over d x is equal to (v times d u over d x minus u times d v over d x) divided by v squared

D of (u divided by v) is equal to (v times d u minus u times d v) divided by v squared

D of sine u over d x is equal to cosine u times d u over d x

D of sine u is equal to cosine u times d u

D of cosine u over d x is equal to negative sine u times d u over d x

D of cosine u is equal to negative sine u times d u

D of tangent u over d x is equal to secant squared u times d u over d x

D of tangent u is equal to secant squared u times d u

D of secant u over d x is equal to secant u times tangent u times d u over d x

D of secant u is equal to secant u times tangent u times d u

D of cosecant u over d x is equal to negative cosecant u times cotangent u times d u over d x

D of cosecant u is equal to negative cosecant u times cotangent u times d u


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Differentials Problems

Change slides by using the arrows or drop-down menu. If the slide has sound, the Play button will be active.

Differentials Problems

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Interpretations and Applications of Differentials

 

View the presentation below illustrating interpretations and applications of differentials.

Video: 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.

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