LOCAL LINEAR APPROXIMATION MATH 200 GOALS Be able to compute the - - PowerPoint PPT Presentation

LOCAL LINEAR APPROXIMATION

MATH 200 WEEK 6 - WEDNESDAY

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.

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

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!

− → ∇F = ⟨fx, fy, −1⟩

MATH 200

▸ We can use this to write a general formula for the tangent

plane to f(x,y) at (x0,y0):

fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0) − (z − z0) = 0 z = fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0) + z0 ▸ Solve for z: ▸ Since z0 = f(x0,y0), z = fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0) + f(x0, y0)

THE ONLY THING WE’LL DO DIFFERENTLY NOW IS RENAME Z Z = L(X,Y)

L(x, y) = fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0) + f(x0, y0)

MATH 200

DON’T MEMORIZE, UNDERSTAND

▸ Now, we have this formula for the local linear approximation

• f a function f(x,y) at (x0,y0):

L(x, y) = fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0) + f(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

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 L(x, y) = fx(x0, y0)(x − x0) + fy(x0, y0)(y − y0) + f(x0, y0) L(x, y) = (1)(x − 0) + (0)(y − 0) + 0 L(x, y) = x ▸ So… f(0.1, 0.1) = L(0.1, 0.1) e0.1 sin(0.1) = 0.1 ▸ Wolfram vs. Us

SO BASICALLY 0.1

0.1103329887302037 117193358278087139 888318352848859486 e0.1 sin(0.1) =

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 f(x, y) = ln(x2 − y2) at P(2, √ 3) fx(x, y) = 2x x2 − y2 = ⇒ fx(2, √ 3) = 4 1 = 4 f(2, √ 3) = ln(1) = 0 fy(x, y) = −2y x2 − y2 = ⇒ fy(2, √ 3) = −2 √ 3 1 = −2 √ 3 L(x, y) = 4(x − 2) − 2 √ 3(y − √ 3) + 0 L(x, y) = 4x − 2 √ 3y − 2

MATH 200