Presentation on theme: "4.5 Linearization & Newton’s Method"— Presentation transcript:
1 4.5 Linearization & Newton’s Method What you’ll learn aboutLinear ApproximationNewton’s MethodDifferentialsEstimating Change with Differentials
2 Linear Approximation Any differentiable curve is “Locally Linear” if you zoom in enough times.Do Exploration 1: Appreciating Local Linearity (p 233)A fancy name for the equation of the tangent line at a is “The linearization of f at ay – f(a) = f’(a)(x – a)
3 Definition - Linearization If f is differentiable at x = a,then the equation of the tangent line.L(x) = f(a) + f’(a) (x - a),defines the linearization of f at a.The approximation f(x) =L(x) isthe standard linear approximation of f at a.The point x = a is the center of the approximation.
4 Just Math Tutoring You Tube What is Linearization?Just math tutoringFinding the Linearization at a pointFollowed by25) Linear Approximation10 minutes total time needed – Watch if you miss class this day!
5 Example 1 Finding a Linearization Find the linearization of at x = 0 (center of approximation) and use it to approximate without a calculator.Then use a calculator to determine the accuracy of the approximation.Point of tangency f ‘(0) =L(x) = Equation of the tangent line:Evaluate L(.02)Calculator approximation?Approximation error:
6 Practice: Find linearization L(x) of f(x) at x = a when and a = 2 Practice: Find linearization L(x) of f(x) at x = a when and a = 2. How accurate is the approximation L(a + 0.1) ≈ f(a + 0.1)Point of tangency f(2) = f ’(2)Tangent Line equation: L(x)Evaluate |L(2.1) – f(2.1)|Approximation error:
7 Point of tangency f (π/2) f ’(π/2) Tangent Line equation: L(x) Example 2: Find the linearization of f(x) = cos x at x = π/2 and use it to approximate cos 1.75 without a calculator. Then use a calculator to determine the accuracy of the approximation.Point of tangency f (π/2) f ’(π/2)Tangent Line equation:L(x)Evaluate |L(1.75) – cos 1.75 by calculator |Approximation error:
8 Homework Page 242 Quick Review 1-10 Exercises 3, 5, 7 SummaryEvery function is “locally linear” about a point x = a. If you evaluate the tangent line at x = a for points close to a, you will have a close approximation to the function’s actual value.HomeworkPage 242Quick Review 1-10Exercises 3, 5, 7
9 Warm UpFind the linearization L(a) of f(a) at x = a for f(x) = ln(x+1), a = 0.How accurate is the approximationL(0.1) ≈ f(0.1)?
10 StepsUsing f(x), find the equation of a tangent line at some point (a, f(a)).Find f(a) by plugging a into f(x).Find the slope from f’(a).L(x) = f(a) + f’(a) (x - a).2) Evaluate L(x) for any x near a to get a close approximation of f(x) for points near a.
11 Example 3: Approximating Binomial Powers using the general formula Use the formula to find polynomials that will approximate the following functions for values of x close to zero.b) c) d)How?Rewrite expression as (1 + x) k,Identify coefficients of x and k.Find L(x) = 1 + kx for each expression.
12 Example 4: Use linearizations to approximate roots. Find a) and b) Identify function: f(x) =Let a be the perfect square closest to Find L(x) at x = a.Use L(x) to estimateError?You try b.
13 Just Math Tutoring – Newton’s Method (7:29 minutes) FYI – not tested Newton’s Method for approximating a zero of a functionApproximate the zero of a function by finding the zeros of linearizations converging to an accurate approximation.Just Math Tutoring – Newton’s Method(7:29 minutes)
14 Differentials Let y = f(x) be a differentiable function. Since dy/dx = f ’(x),the “differential dy” is defined as dy = f ’(x) dx,(With dx as in independent variable and dy a dependent variable that depends on both x and dx.)Although Liebniz did most of his calculus using dy and dx as separable entities, he never quite settled the issue of what they were. To him, they were “infinitesimals” – nonzero numbers, but infinitesimally small. There was much debate about whether such things could exist in mathematics, but luckily for the early development of calculus it did not matter: thanks to the Chain Rule, dy/dx behaved like a quotient whether it was one or not.
15 Example 6 Find the differential dy and evaluate dy for the given values of x and dx. How? Find f ’(x), multiply both sides by dx, evaluate for given values.a) y = x5 + 37x b) y = sin 3x c) x + y = xyx=1, dx = x=π, dx = x=2, dx = 0.05You try:
17 Example 7 Finding Differentials of functions Example 7 Finding Differentials of functions. Find dy/dx and multiply both sides by dx.d (tan (2x)) b)You try: d(e5x + x5)
18 Estimating Change with Differentials Suppose we know the value of a differentiable function f(x) at a point a and we want to predict how much this value will change if we move to a nearby point (a + dx).If dx is small, f and its linearization L at “a” will change by nearly the same amount.Since the values of L are simple to calculate, calculating the change in L offers a practical way to estimate the change in f.
19 Differential Estimate of Change Let f(x) be differentiable at x = a.The approximate change in the value of f when x changes from a to a + dx isdf = f ’ (a) dx
20 Area formula for a circle: A = True change: f(10.1) – f(10) = Example 8 The radius r of a circle increases from a = 10 to 10.1 m. Use dA to estimate the increase in the circle’s area A. Compare this estimate with the true change ∆A, and find the approximation error.Area formula for a circle: A =True change: f(10.1) – f(10) =Estimated change: dA/dr =dA =Approximation error: |∆A – dA| =You try: f(x) = x3 - x, a = 1, dx = 0.1
21 SummaryLinearization: The equation of a tangent line to f at a point a will give a good approximation of the value of a function f at a.The Linearization of (1 + x)k = 1 + kxNewton’s Method is used to find the roots of a function by using successive tangent line approximations, moving closer and closer to the roots of f if you start with a reasonable value of a.Differentials: Differentials simply estimate the change in y as it relates to the change in x for given values of x. We learned how to estimate with linearizations, differentials are simply a more efficient method of finding change.