Tangent Line equa,on. Simple Examtype

Transcription

1 Tangent Line equa,on Find the equa,on of the tangent line to the graph of the func,on y=5x 2 at the point x=1. Where does the tangent line intersect the x axis? Simple Examtype ques,on Finished fast? Find the equa,on of the tangent line to the graph of the func,on y=f(x) at the point x=x 0. Where doe the tangent line intersect the x axis?

2 Office hours this week Tu: 3:30-5PM I am at an Open Book seminar on Friday morning, so which replacement office hours are best for you? (A) Wed 10-11:30 (B) Wedn 3:30-5pm (C) Thurs 10-11:30

3 Midterm sign up Sign up for one of the three op,ons. Check back for your room assignment may not be same as your friend s Declare any conflicts (lecture, another midterm, etc)

4 Start Off Strong Spac.science.ubc.ca Sept 26-27, 5-7pm Learn,ps from profs about how to study for midterm exams

5 Help with homework (OSH 2) d 1 (x) that gives the distance from a point on the x-axis (x,0) to the point (2,3). Use distance formula

6 OSH 2 cont d What is there are 2 points and you want the shortest distance?

7 OSH 2 cont d What is there are 2 points and you want the shortest distance?

8 Help with HW In 1938 A.V. Hill proposed a math model for the rate of shortening of a muscle, v, (cm/sec) when it is working against a load p (gms). His so called forcevelocity curve is given by the rela,onship a, b posi,ve constants. HINT: Rewrite in the form v = f(p). Note: v>0 means muscle contracts. v<0 means muscle expands

9 To work on Prac,ce the Product and Quo,en Rules Prac,ce sketching deriva,ves and an,deriva,ves Start studying for the MidTerm Note: online copy of book just uploaded: now has answers to most of the problems in the book.

10 Tangent lines And linear approxima,ons UBC Math 102

11 Generic Tangent Line equa,on Find the equa,on of the tangent line to the graph of the func,on y=f(x) at the point x=x 0. UBC Math 102

12 Solu,on (in class):

13 Tangent lines can be used to (locally) approximate a func,on Func,on à Near x=0 Tangent Line is close to the func,on. TL (lower than f(x) near x=0) underes,mates the func,on at that point UBC Math 102

14 1. Linear approxima,on Find an approxima,on for f(1.5) What is the slope of the TL? (A) 2 (B) 3 (C) 3/2 (D) -3 (E) 1/2

15 2. Linear approxima,on Find an approxima,on for f(1.5) What is the point of tangency?? (A) (0, -1) (B) (2,1) (C) (1,1) (D) (1,2) (E) None of the above

16 3. Linear approxima,on The eqn of the tangent line is (A) y=2x-1 (B) y=2x +1 (C) y=2x-0.5 (D) y= x-1 (E) y= -1-2x

17 4. Linear approxima,on The LINEAR APPROXIMATION to f(1.5) is (A) f(1.5) 1 (B) f(1.5) 1.5 (C) f(1.5) 2 (D) f(1.5) 2.5 (E) f(1.5) 3

18 Linear approxima,on The LINEAR APPROXIMATION to f(1.5) is (A) f(1.5) 1 (B) f(1.5) 1.5 (C) f(1.5) 2 (D) f(1.5) 2.5 (E) f(1.5)

19 5. Linear approxima,on The LINEAR APPROXIMATION to f(1.5) is (A) An overes,mate (B) An underes,mate 2 1.5

20 Rescue me! Rapunzel, Rapunzel, Let down your hair! UBC Math 102

21 Rescue me! No way! You re not climbing up my hair! Go get a ladder! UBC Math 102

22 Rescue! Examtype ques,on What is the shortest ladder that you can use to rescue Rapunzel from the window at 6 meters height if there is an anthill in the way? anthill See P 111 Featured problem 5.1 UBC Math 102 ground

23 Rescue! What is the shortest ladder that you can use to rescue someone from the window at 6 meters height if there is a hill in the way? Tangent Line UBC Math 102 ground

24 ladder anthill ground

25 Solu,on: We need to find the length of the tangent line shown in the diagram. To do so, we should solve the following (restated mathema,cal) problem: UBC Math 102

26 UBC Math 102 Solu,on, cont d

27 Soln cont d: Use the fact that the TL goes through the point (0,6) Now solve this equa,on to find x 0 UBC Math 102

28 Now get the equa,on of the TL: Slope: TL has y-intercept 6 and slope -1 so its eqn: Is: UBC Math 102

29 Soln cont d Now that we have the equa,on of the tangent line, we can determine its length. (See next slide). The problem has been solved.. And this one was less trivial than the earlier problems, since we had to solve for the point of tangency. UBC Math 102

30 Solu,on: y=6-x The length of the shortest ladder is : 6 sqrt(2) UBC Math 102

31 Tangent line (0,6) y = 6 - x (2,4) (6,0)

32 Linear approxima,on Using a tangent line to approximate the value of a func,on (near some point). UBC Math 102

33 (6) Tangent line equa,on Which of the following is the equa,on of the tangent line to f(x) at the point x 0 UBC Math 102

34 (7) Linear approxima,on Close to the point x 0 we can approximate the func,on f(x) by

35 (8) Linear approxima,on The approxima,on is good provided (A) x is small (B) x 0 is small (C) (x-x 0 ) is large (D) (x 0 -x) is small (E) None of the above UBC Math 102

36 (9) A linear approxima,on.. approximates the true value of the func,on at some point close to x 0 by the value shown in E D A B C A, B, C, D, or E UBC Math 102

37 Geometry Δf f(x 0 ) UBC Math 102

38 Geometry Δf f(x 0 ) f (x 0 )(x-x 0 ) (x-x 0 ) Δf UBC Math 102

39 UBC Math 102 (10) Example:

40 UBC Math 102 (11) Example cont d

41 UBC Math 102 (12) Example cont d

42 (13) Over or under-es,mate? The approxima,on we made will (A) Over-es,mate the true value (B) Under-es,mate UBC Math 102

43 From Last Week More about displacement, velocity, and accelera,on --- since we did not have a chance to prac,ce this on Friday: try out this ques,on UBC Math 102

44 How to outrun a cheetah hvps:// I m puwng my money on the livle guy See P 100 Featured problem 4.3 UBC Math 102 Examtype ques,on hvps://en.wikipedia.org

45 How to outrun a cheetah A cheetah can run at top speed v c but it has to slow down (decelerate) to keep from gewng too hot. (Assume a constant decelera,on a=-a c ). A gazelle runs at a slower speed v g but it can maintain that constant speed for a long,me. If the gazelle is ini,ally a distance d from the cheetah, and running away, when would it be caught? Is there a (far enough) distance d such that it never gets caught? UBC Math 102

46 Answers 1. A 2. C 3. A 4. C 5. B 6. C 7. E 8. D 9. B 10. D 11. B 12. A 13. A UBC Math 102

47 Solu,on: to problem done on the board Tangent line intersects x axis at y=0, so at x=10/5 = 1/2 UBC Math 102

48 Solu,ons to previous problems First try out the problems at the end of the last lecture slides. Only then should you peek at the answers.

49 Solu,on to: deriva,ve of sin(x) On desmos: (note that deriva,ve does not exist at all the cusps!)

50 Test your skills Problems to work on

51 Differen,ate, using power or quo,ent rule

52 Calcula,on and prac,ce Find the equa,on of the tangent line to the graph of at x=0. Where does the tangent line intersect the x axis?

53 Related test ques,on (MT1, 2014) Consider the func,on (Tangent lines) (a) At which points (a,f(a)) does the graph of this func,on have tangent lines parallel to the line y = x. (b) What is the equa,on of the tangent lines at each of these points.

54 Test ques,on (Quo,ent rule, ra,onal func,ons) MT At an all-you-can-eat buffet, the total calories you gain can be represented by the func,on where t 0 is the,me in minutes you spend at the restaurant and A and b are posi,ve constants. (a) If you stayed for a long,me, what asymptote would your total caloric gain approach? (b) A{er how much,me do you gain exactly half of that asympto,c caloric amount? (c) At,me t, what is the instantaneous rate at which your caloric gain changes?