Linear Approximation

Learning Objectives Upon completing this module, you should be able to:  Recognize the linear approximation of a function as the tangent line to the function. Derive a linear approximation for a function at a particular value Interpret linear approximation graphically Use Linear approximation of a function to estimate numerical values

What is the Linear Approximation? Linear approximation is the process of finding the equation of a line that is the closest estimate of a function for a given value of x. It is sometimes called “tangent line approximation”

Applications of Linear Approximation

Linear Approximation Formula

Linear approximation Equation of tangent line to y=f(x) at a is y = f(a) + f′(a) (x - a) y y = f(x) f(a) x a

Linear approximation If x is near a, we have: f(x) ≈ f(a) + f′(a) (x - a) y f(a) + f′(a) (x - a) y = f(x) f(x) f(a) x a x

Linear approximation Function L(x) = f(a) + f′(a) (x - a) is called linear approximation (or linearization) of f(x) at a y y = L(x) L(x) y = f(x) f(x) f(a) x a x

Example Find linearization of f(x) = √x at a Use it to find approximate value of √5

Linearization

Approximation of √5 Find a such that Use linearization at a √a is easy to compute a is close to 5 Use linearization at a Take a = 4 and compute linear approximation

Approximation of √5

Approximation of √5 y = L(x) = 2 + ¼ (x - 4) y y = √x a = 4 5 √5 2.25

Example Find approximate value of sin 10o

Example We measure x in radians So, 10o = 10 (π/180) = π/18 radians Consider f(x) = sin x Find a such that sin(a) is easy to compute a is near π/18 Take a = 0 and compute linear approximation

Solution f(x) ≈ f(a) + f′(a) (x - a) = f(0) + f′(0) (x - 0) f(x) = sin x, f′(x) = (sin x) ′ = cos x Therefore we obtain: sin x ≈ sin(0) + cos(0) (x - 0) = 0 +1(x – 0) = x Thus sin x ≈ x (when x is near 0) For x = π/18 we obtain: sin 10o = sin (π/18) ≈ π/18 ≈ 0.1745 Calculator gives: sin 10o ≈ 0.1736

Exercise If f(3) =8, f’(3) = -4, then f(3.02) =7.92

Try a few simple linearizations near x = 0: The graph would look like this:

Try a few simple linearizations near zero: Now try the other three.

Find the linearization of the function f (x) = at a = 1 and use it to approximate the numbers and .