1. 4.1 Linear Approximations Mon Dec 21 Do Now Find the equation of the tangent line of each function at 1) Y = sinx 2) Y = cosx

2. Quiz Review Retakes this week

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. Ex 3,4 In the book bc lots to type

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

11. 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

12. Linearization The general method of linearization 1)Find the tangent line at x = a 2)Solve for y or f(x) 3)If necessary, estimate the function by plugging in for x The linearization of f(x) at x = a is:

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

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

15. Ex 3 Find the linear approximation to f(x) = cos x atand approximate cos(1)

16. Closure Journal Entry: Use Linearization to estimate the square root of 37 HW: p.214 #1, 5, 9, 13, 17-33 odds,37 45 47 49, 53-63 odds

17. Linear Approximation Review Tues Dec 22 Do Now Use linear approximations to estimate

18. HW Review p.214

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

20. Closure Find the linearization of f(x) = 1/x at a = 2 HW: Retry hw problems