Tangent lines, cont d. Linear approxima5on and Newton s Method
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1 Tangent lines, cont d Linear approxima5on and Newton s Method
2 Last 5me: A challenging tangent line problem, because we had to figure out the point of tangency.?? (A) I get it! (B) I think I see how we did it (C) I m not sure (D) I m confused (E) I m totally lost ground
3 Last 5me: A challenging tangent line problem, because we had to figure out the point of tangency.?? (A) I get it! (B) I think I see how we did it (C) I m not sure (D) I m confused (E) I m totally lost See Sec5on 5.6, Try problem 5.10 ground
4 Last 5me Use linear approxima5on to calculate 105 Solu5on: - figure out the func5on - Figure out a point nearby where we know the value of the func5on - Use linear approxima5on
5 Last 5me Use linear approxima5on to calculate 105 Solu5on: - figure out the func5on f(x) = x - Figure out a point nearby where we know the value of the func5on. We know 100 = 10 - Use linear approxima5on
6 Geometry: S5ck a tangent line on the func5on at x 0 =100, read the value on TL nearby (at x=105). x 0
7 Linear Approx: -Advantage: easy to calculate: - One step method = = 10+ (5/20)= Disadvantage: not very accurate further away
8 Another way (Newton s method) Convert the problem to the form f(x)=0 Use intui5on or desmos to get a rough es5mate for the zero of the func5on. Use Newton s method to get be_er and be_er approxima5on. Disadv: Need some intui5on Advantage: Outstanding level of accuracy!
9 1. Convert to right form Find decimal approx to x= 105. To use Newton s method I would convert this to solving the problem f(x) = 0 where A. f(x) = x B. f(x) = x C. f(x)= x D. f(x)= 105
10 Convert to right form Find decimal approx to x= Same as: find x such that x 2 =105 x 2-105=0 - Now the problem is in the form f(x) = 0 for f(x) = x 2-105
11 2. My ini5al guess for the zero of f(x) = x would be A. x 0 =100 B. x 0 = 10 C. x 0 = 105 D. x 0 = 105 E. I m lost.
12 Ini5al guess Find value of x somewhere around zero of f(x) = x That will be the star5ng value for Newton s method. x 0 = 10
13 Newton s Method A way to find decimal approxima5on for the zero of a func5on UBC Math 102
14 Plan What is Newton s Method The geometry behind it A prac5cal example where we need to use it How to use a spreadsheet to implement Newton s Method. UBC Math 102
15 What is Newton s Method? Newton s method Solve f(x)=0 UBC Math 102
16 (3) Which of these is problems is not suitable for Newton s method? (A) Solve f(x)=0 for x. (B) Find the zeros of a func5on f(x). (C) Find where the graph of f(x) crosses the x axis. (D) Approximate the value of a func5on close to a known point x 0. (E) Find roots of an equa5on g(x)=c. UBC Math 102
17 Newton s method can be used to (A) Solve f(x)=0 for x. (B) Find the zeros of a func5on f(x). (C) Find where the graph of f(x) crosses the x axis. (E) It can also find roots of F(x)=g(x)-C=0 UBC Math 102
18 Hammer and nail Newton s method Solve f(x)=0 UBC Math 102
19 UBC Math 102 The geometry behind it
20 Tangent line Equa5on of tangent line at x 0 : Find the point x 1 at which the tangent line intersects the x axis. UBC Math 102
21 (4) The tangent line intersects the x axis at: UBC Math 102
22 Deriving Newton s Formula Eqn of TL: TL intersects x axis when
23 Improving the rough es5mate.. Repeat the process: At the k th step:
24 Put it on a spreadsheet! Let the spreadsheet do those repe55ve calcula5ons for you! Goal: find decimal approx to square root of 105 accurate to 10 decimal places!
25 Spreadsheet set up The func5on is f(x) = x We need its deriva5ve f (x) = 2 x Ini5al guess x 0 = 10 Formula to repeat: x 1 = x 0 f(x 0 )/f (x 0 )
26 Spreadsheet The next few slides show how to solve this problem on a spreadsheet.
27 Setup
28 Transfer x1 to Column A Highlight the cells and drag down It should look like:
29 Drag down the en5re row to repeat
30 Part II: Looking at all this geometrically
31 Recap Let us look again at linear approxima5on and at Newton s method We will use desmos and the func5on To illustrate concepts.
32 PART 1 A demo of linear approxima5on UBC Math 102
33 Desmos Graph the func5on: Use desmos to graph its deriva5ve
34 It should look like: UBC Math 102
35 Desmos cont d Add the value x 0 =5 A slider should appear for x 0 so you can shif the loca5on of that tangent line. Add the point (x 0, f(x 0 )) to your graph. UBC Math 102
36 It should look like: UBC Math 102
37 Desmos cont d Add the equa5on of a generic tangent line at x=x 0 You should be able to animate x 0 and see that line sweep across your curve. UBC Math 102
38 It should look like: UBC Math 102
39 Using Desmos Consider the tangent line at x o =5. Use your graph to find the values below: f(5), f(6) and the linear approx to f(6) based at x o =5 UBC Math 102
40 (5) Using Desmos I got the values. f(5), f(6) and the linear approx to f(6) (A) 4.37, 4, 5 (B) 4.37, 5, 2 (C) 4.37, 2, 2.5 (D) huh?? UBC Math 102
41 PART 2 A demo of Newton s Method on Desmos UBC Math 102
42 On desmos Use Newton s Method to find the largest zero of the func5on Start with rough ini5al guess x 0 =5. UBC Math 102
43 Tangent line at x 0 =5 TL and f(x) UBC Math 102
44 Newton s formula Add the formula Plot the point
45 Look like this: (here is x 1 ) Actual zero of the func5on we want to find an accurate decimal expansion for it
46 Desmos does the work We got Desmos to compute x1 using the Newton s Method formula: (We got this by finding where a generic tangent line crosses the x axis, see previous lectures)
47 Repeat the process To get closer to the zero of f: Construct a tangent line at the point x 1 Find the point where the new tangent line intersects x axis (Find x 2 ). h_ps:// UBC Math 102
48 We use the same idea to find x2: UBC Math 102
49 To No5ce: We are gejng closer to the actual zero of the func5on. UBC Math 102 X0 x2 x1
50 Effect of ini5al guess Go to the slider for x0 and shif it to the value 3. What happens to the tangent lines? What happens to the approxima5on for the zero of the func5on? UBC Math 102
51 Bad ini5al guess! An ini5al guess x0=2.3 leads astray! Does not converge to the zero of f! See below: UBC Math 102
52 Important comments: 1. The Desmos demonstra5on is only to show the GEOMETRY that explains how Newton s Method works. You do not need this (elaborate) construc5on in a prac5cal problem. 2. How do I find an ini5al guess? ß Use Desmos, or sketch the func5on. Ensure there is no max/ min between x0 and the true zero.. We will see how to sketch func5ons in future lectures. UBC Math 102
53 Answers 1. A 2. B 3. D 4. A 5. C
54 Solu5ons to previous problems First try out the problems at the end of the last lecture slides. Only then should you peek at the answers.
55 Differen5ate, using power or quo5ent rule solu5on
56
57 Calcula5on we did in class Find the equa5on of the tangent line to the graph of at x=0. Where does the tangent line intersect the x axis? solu5on
58 Solu5on: (a) Eqn of tangent line
59 Related test ques5on (MT1, 2014) Consider the func5on (Tangent lines) solu5on (a) At which points (a,f(a)) does the graph of this func5on have tangent lines parallel to the line y = x. (b) What is the equa5on of the tangent lines at each of these points.
60 Solu5on: (a) UBC Math 102
61 Solu5on (b) UBC Math 102
62 Solu5on to Related test-type problem from last 5me: Find a linear approxima5on that provides a rough es5mate for the value of (1.1) 8. Explain why the approxima5on is an (over/under)-es5mate. solu5on (Note: a calculator value is ) UBC Math 102
63 Test ques5on (Quo5ent rule, ra5onal func5ons) MT At an all-you-can-eat buffet, the total calories you gain can be represented by the func5on where t 0 is the 5me in minutes you spend at the restaurant and A and b are posi5ve constants. (a) If you stayed for a long 5me, what asymptote would your total caloric gain approach? (b) Afer how much 5me do you gain exactly half of that asympto5c caloric amount? (c) At 5me t, what is the instantaneous rate at which your caloric gain changes? solu5on
64 Solu5on: (a) (b) (c) UBC Math 102
65 Test your skill on these problems
66 Linear Approxima5on Find a linear approxima5on that provides a rough es5mate for the value of (1.1) 8. Explain why the approxima5on is an (over/under)- es5mate. UBC Math 102
67 Find the prey density such that P(x)=G(x) Note: this problem is similar to the aphid ladybug problem from week 1 but the func5on P(x) is not the same. UBC Math 102
68 Newton s Method
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