4.1 Linear ApproximationsFri Oct 16

Do NowFind the equation of the tangent

line of each function at 1) Y = sinx

2) Y = cosx

Test Review

• Retakes?

Differentials

• We define the valuesas the difference between 2 values

These are known as differentials, and can also be written as dx and dy

Linear Approximations

• The tangent line at a point of a function can be used to approximate complicated functions

• Note: The further away from the point of tangency, the worse the approximation

Linear Approximation of df

• If we’re interested in the change of f(x) at 2 different points, we want

• If the change in x is small, we can use derivatives so that

Steps

• 1) Identify the function f(x)• 2) Identify the values a and• 3) Use the linear approximation of

Ex 1

• Use Linear Approximation to estimate

Ex 2

• How much larger is the cube root of 8.1 than the cube root of 8?

Ex 3,4

• In the book bc lots to type

You try

• 1) Estimate the change in f(3.02) - f(3) if f(x) = x^3

• 2) Estimate using Linear Approximation

Linearization

• Again, the tangent line is great for approximating near the point of tangency.

• Linearization is the method of using that tangent line to approximate a function

Linearization• The general method of linearization1) Find the tangent line at x = a2) Solve for y or f(x) 3) If necessary, estimate the function by

plugging in for xThe linearization of f(x) at x = a is:

Ex 1

• Compute the linearization ofat a = 1

Ex 2

• Find the linearization of f(x) = sin x, at a = 0

Ex 3

• Find the linear approximation to f(x) = cos x at and approximate cos(1)

Closure

• Journal Entry: Use Linearization to estimate the square root of 37

• HW: p.214 #5 7 11 17 30 33 47 51 62 74