1. Linear Approximation

2. 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

3. 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”

4. Applications of Linear Approximation

7. Linear Approximation Formula

15. 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

16. 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

17. 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

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

19. Linearization

20. 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

21. Approximation of √5

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

23. Example
Find approximate value of sin 10o

24. 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

25. 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 ≈
Calculator gives: sin 10o ≈

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

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

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

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