Chain Rule. Con,nued: Related Rates. UBC Math 102

Transcription

1 Chain Rule Con,nued: Related Rates

2 Midterm Test All sec,ons Average = 69%

3 I think the test was A Too hard B On the hard side C Fair D On the easy side E too easy

4 I think I can do beper next,me by A) Studying more or differently B) Going over challenging concepts C) Being more careful about silly errors D) More than one of the above E) I studied as hard as I could and not sure what to do to improve

5 Comments about the Midterm: See our Wiki site: hpps://wiki.math.ubc.ca/mathbook/m102/ Midterm_informa,on/Midterm_2015/ Commentary

6 MC answers (see Q s last lecture) 1 C 2 B 3 A 4 D 5 B 6 B

7 Q7 Using any method, determine the value of the following limit

8 Q 8 Consider the func,on Calculate the following. Enter DNE if a limit does not exist.

9 Q 9 (a) Consider a differen,able func,on f(x) for which we want to es,mate f(2). Suppose that we found out that f(1) = 1, fʹ (1) = 1/2. Es,mate f(2) using this informa,on.

10 Q 9 (b) We subsequently discovered that f(3) = 3 and fʹ (3) = 2, and that f is concave up. Use this new informa,on to get another es,mate for f(2).

11 Q 9 (b) cont d Would that es,mate be beper or worse than the one in the previous part?

12 Q 10 Use Newton s method to approximate the value of x at which What polynomial would you choose as your func,on g(x)?

13 Q 10, cont d For what integer value for x 0 would the itera,ons get close to the solu,on the fastest?

14 Q 10 resources: - How do I set up Newton s Method? See course Notes Sec How do I chose a good star,ng guess? See: Newton s Method - how to chose a good x 0 value: Video link [21]

15 Q 11 It is well known to ecologists that diversity of species is greatest close to the equator (la,tude 0) and much lower near the North and South poles (la,tudes ±90 degrees). Shown below are two graphs of the number of species (N(x)) versus la,tude (x) at two geological ages, 300 and 270 million years ago. On the axes provided, sketch the deriva,ves of each of these two func,ons.

16 Q 11 solu,on: IP

17 Q 11 Resources: How do I know that the inflec,on points of f(x) are cri,cal points of f (x)? See: Assignment5: Problem 14 At what value(s) of x on the curve y=7+160x 3 3x 5 does the tangent line have the largest slope? See Video link [23] on concavity and inflec,on pts.

18 Q 12 There is a circular pool in the centre of a square courtyard. Lucas father (F) stands in the corner of the courtyard. Lucas (L) starts on the edge of the pool at the point closest to his father and walks once around the edge of the pool at a constant speed. It takes Lucas two minutes to go around the pool.

19 Q 12 (a) Sketch the graph of the distance between Lucas and his father as a func,on of,me. Do not try to write down an equa,on for the distance - just make an approximate sketch based on the diagram with a focus on gesng key features (minima, maxima) correct.

20 Q 12 (b) If your graph has any minima, maxima and/or inflec,on points, explain which points on the edge of the pool correspond to each of these special points and label them on your graph and on the diagram with the lepers A, B, C

21 Q 13

22 Q 13 solu,on:

23 Chain rule and related rates

24 Related rates: There is an independent variable (,me) that all quan,,es depend on. t

25 Chain Time t à F(t) G(F(t)) y

26 Chain Rule Time t à F(t) = u G(u) y

27 Growing vine A bean grows up a pole in the form of a helix. If the height of the vine,p increases at a constant rate k (cm/day) at what rate is the length of the vine increasing?

28 My bean plant

29 My bean plant

30

31

32 Growing vine A bean grows up a pole in the form of a helix. If the height of the vine,p increases at a constant rate k (cm/day) at what rate is the length of the vine increasing? Assume that the radius of the pole is r and the pitch of the helix is p, (p>0). (pitch= height increase for each complete turn of the helix)

33 The geometry h(t)

34 Unwrap the helix: 2 π r p

35 Unwrap the helix: The pitch of the helix, p, and the circumference of the pole, 2 π r, are constants. 2 π r p

36 In the small triangle: 2 π r p

37 Similar triangles: 2 π r p

38 Solu,on step 1

39 Chain Rule L(t) is related to h(t) which (we are told) increases at a constant rate. h(t) L(h) Time t à

40 So, according to the Chain Rule:

41 Conical pile of sugar Sugar is poured at a constant flow rate, 1 cm 3 /s to form a conical pile. If the ra,o of the height to radius of the cone is constant, at what rate is the radius of the base of the conical pile increasing? At what rate is the base area increasing? Note: the volume of a cone is

42 The conical pile

43 The conical pile

44 The conical pile

45 Chain Rule Time t à r(t),h(t) V(r,h) Use the fact that h(t) = C r (t)

46 Solu,on step 1 Given: - Volume of cone - Ra,o of height to radius is constant: Hence:

47 Chain Rule Time t à r(t) V(r)

48 Solu,on step 2

49 Related problem: Try this one yourself! Sugar is poured at a constant flow rate, 1 cm 3 /s to form a conical pile. At what rate is the base area increasing?

50 Related test problem

51 Solu,on from last,me: Angles in the op,mal Y- shaped ant- trail What angles does the Y trail form? We found that

52 Angles What angle does the Y trail form?

53 Recall: Ra,os in equilateral triangle Side lengths all equal 1 Height: 30 o 30 o 60 o 60 o

54 Angles What angle does the Y trail form? 60 o The branches form a 120 degree angle with each other, a 60 degree angle with the base of the Y