1. 4.1 Linear Approximations Thurs Jan 7
Do Now
Find the slope of each function at
1) Y = sinx
2) Y = cosx

2. Quiz Review
Quiz retakes until Fri

3. Differentials We define the values as the difference between 2 values
These are known as differentials, and can also be written as dx and dy

4. 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

5. 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

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

7. Ex 1
Use Linear Approximation to estimate

8. Ex 2
How much larger is the cube root of 8.1 than the cube root of 8?

9. You try 1) Estimate the change in f(3.02) - f(3) if f(x) = x^3
2) Estimate using Linear Approximation

10. Closure
Use Linear Approximate to estimate f(3.02) - f(3) if f(x) = x^4
HW: p.213 #1-13 odds, odds

11. 4.1 Linearization Fri Jan 8 Do Now
Find the equation of the tangent line of at

12. HW Review p.213 #
1) )
3) ) 5) 7) 9)
11) -0.03
13)
17) 0.1

13. 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

14. Linearization The general method of linearization
Find the tangent line at x = a
Solve for y or f(x)
If necessary, estimate the function by plugging in for x
The linearization of f(x) at x = a is:

15. Ex 1
Compute the linearization of
at a = 1

16. Ex 2
Find the linearization of f(x) = sin x, at a = 0

17. Ex 3
Find the linear approximation to f(x) = cos x at and approximate cos(1)

18. Ex 4
Use linearization to approximate cos(1)

19. More examples
Use a linear approximation to approximate

20. Closure
Journal Entry: Use Linear Approximation to estimate the square root of 26
HW: p.214 # odds

21. Linear Approximation Practice Mon Jan 11
Do Now
Use linear approximations to estimate

22. HW Review p.214 #45-51 59-63 45) L(x) = 4x - 3
47) L(x) = x - pi/4 + 1/2
49) L(x) = -1/2 x + 1
51) L(x) = 1
59) L(17) =
61) L(10.03) =
63) L(64.1) =

23. Linearization Review
We can use linear approximation (tangent line equations) for 2 uses:
1) Find the difference between to values of f(x)
2) Estimate the value of f(x) at specific points

24. Practice
(green book) Worksheet p.249 #5-10, 17-22

25. Closure Hand in: Use linear approximation to estimate
HW: Finish worksheet p.249 #

26. HW Review p.249 #5-10 5) 6) 7) 8) 9)
10)

27. HW Review p.249 #17-22
17) .842
18) .788
19)
20)
21) 2.005
22) 1.030