Local Linear Approximation Math 200 Week 6 - Friday Local Linear Approximation

Math 200 Goals Be able to compute the local linear approximation for a function of two or more variables at a given point. Be able to use a local linear approximation to estimate a given quantity.

Math 200 From Calc 1 In Calc 1 we discussed the fact that differentiable functions are locally linear That is, near the point of tangency, a function is approximately equal to its tangent line Another way of saying all this is that if I keep zooming in on a differentiable function at a point, it will eventually look flat.

Zoom in Say we want to approximate sin(π/15) Math 200 Say we want to approximate sin(π/15) We can find the tangent line to f(x) = sin(x) at x=0, which is close to π/15 f’(x) = cos(x) f’(0) = 1 The tangent to f at x=0 is y=x (call this L(x) = x) sin(π/15) is approximately equal to L(π/15) = π/15 Zoom in

Math 200 Summary of Calc 1 stuff Local Linear Approximation for single variable functions says that a differentiable function can be approximated by its tangent line For a differentiable function f(x), the local linear approximation at x = x0 is given by L(x) = f(x0) + f’(x0)(x - x0) Remember: Don’t think of this a formula to be memorized; this is just the tangent line to f at x0!

Math 200 New stuff If a single-variable, differentiable function can be approximated by its tangent line near the point of tangency, then a multi- variable function, f(x,y), can be approximated by its tangent plane near the point of tangency

Revisiting Tangent planes for functions of two variables Math 200 Revisiting Tangent planes for functions of two variables Consider any function of two variables, f(x,y). To find the tangent plane at (x0,y0), we should treat the surface z = f(x,y) as a level surface of some function of three variables: z = f(x,y) can be written as 0 = f(x,y) - z F(x,y,z) = f(x,y) - z Notice that Fx = fx, Fy = fy, and Fz = -1 So, And this will be the case for any function of two variables!

The only thing we’ll do differently now is rename z Math 200 We can use this to write a general formula for the tangent plane to f(x,y) at (x0,y0): Solve for z: Since z0 = f(x0,y0), The only thing we’ll do differently now is rename z z = L(x,y)

Don’t memorize, understand Math 200 Don’t memorize, understand Now, we have this formula for the local linear approximation of a function f(x,y) at (x0,y0): But, it’s most important to remember that we approximate functions of two variables with tangent planes And we know that the normal vector for a tangent plane comes from the gradient You should be able to derive this formula if you forget it

Math 200 Example 1 Consider the function f(x,y) = eysin(x). Use local linear approximation to approximate the value of f(0.1,0.1) We can evaluate the function f at (0,0), which is close to (0.1,0.1), so we’ll pick that as the point of tangency. Following our newfound formula, we need f(0,0), fx(0,0), and fy(0,0) fx(x,y) = eycos(x); fx(0,0) = 1 fy(x,y) = eysin(x); fy(0,0) = 0 f(0,0) = 0

Putting it all together… fx(x,y) = eycos(x); fx(0,0) = 1 Math 200 Putting it all together… fx(x,y) = eycos(x); fx(0,0) = 1 fy(x,y) = eysin(x); fy(0,0) = 0 f(0,0) = 0 So basically 0.1 So… Wolfram vs. Us 0.110332988730203711 7193358278087139888 318352848859486

So, the z-value at (0.1,0.1) for the surface, is really close to the z-value at (0.1,0.1) on the plane

Math 200 Example 2 Find the local linear approximation, L(x,y), for

Math 200