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Chain Rule
Con,nued: Related Rates
UBC Math 102
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Midterm Test – All sec,ons
UBC Math 102
0
10
20
30
40
50
60
0 6 12 18 24 30 36 42 48
Average = 69%
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I think the test was
A Too hard B On the hard side C Fair D On the easy side E too easy
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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
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Comments about the Midterm:
See our Wiki site: hPps://wiki.math.ubc.ca/mathbook/M102/Midterm_informa,on/Midterm_2015/Commentary
UBC Math 102
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MC answers (see Q’s last lecture)
1 C 2 B 3 A 4 D 5 B 6 B
UBC Math 102
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Q7
Using any method, determine the value of the following limit
UBC Math 102
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Q 8 Consider the func,on Calculate the following. Enter DNE if a limit does not exist.
UBC Math 102
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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.
UBC Math 102
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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).
UBC Math 102
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Q 9 (b) cont’d
Would that es,mate be bePer or worse than the one in the previous part?
UBC Math 102
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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)?
UBC Math 102
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Q 10, cont’d
For what integer value for x0 would the itera,ons get close to the solu,on the fastest?
UBC Math 102
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Q 10 resources:
-‐ How do I set up Newton’s Method? See course Notes Sec 5.4.1 -‐ How do I chose a good star,ng guess? See: Newton’s Method -‐ how to chose a good x0 value: Video link [21]
UBC Math 102
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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.
UBC Math 102
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Q 11 solu,on:
UBC Math 102
IP
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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+160x3−3x5 does the tangent line have the largest slope?” See Video link [23] on concavity and inflec,on pts.
UBC Math 102
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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.
UBC Math 102
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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.
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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
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Q 13
UBC Math 102
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Q 13 solu,on:
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Chain rule and related rates
UBC Math 102
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Related rates:
• There is an independent variable (,me) that all quan,,es depend on.
• t
UBC Math 102
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“Chain”
Time t à y
UBC Math 102
F(t) G(F(t))
![Page 26: Chain&Rule&keshet/M102/2015/Lect8.1.pdfIthink&Ican&do&bePer&next,me&by& • A)&Studying&more&or&differently& • B)&Going&over&challenging&concepts& • C)&Being&more&careful&aboutsilly&errors&](https://reader034.vdocuments.mx/reader034/viewer/2022050502/5f9462d93b809a0d503e8380/html5/thumbnails/26.jpg)
Chain Rule
Time t à y
UBC Math 102
F(t) = u G(u)
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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?
UBC Math 102
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My bean plant
UBC Math 102
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My bean plant
UBC Math 102
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UBC Math 102
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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)
UBC Math 102
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The geometry
h(t)
UBC Math 102
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Unwrap the helix:
UBC Math 102
p
2 π r
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Unwrap the helix:
UBC Math 102
p
2 π r
The pitch of the helix, p, and the circumference of the pole, 2 π r, are constants.
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In the small triangle:
UBC Math 102
p
2 π r
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Similar triangles:
UBC Math 102
p
2 π r
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Solu,on – step 1
UBC Math 102
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Chain Rule
L(t) is related to h(t) which (we are told) increases at a constant rate.
Time t à
UBC Math 102
h(t) L(h)
![Page 40: Chain&Rule&keshet/M102/2015/Lect8.1.pdfIthink&Ican&do&bePer&next,me&by& • A)&Studying&more&or&differently& • B)&Going&over&challenging&concepts& • C)&Being&more&careful&aboutsilly&errors&](https://reader034.vdocuments.mx/reader034/viewer/2022050502/5f9462d93b809a0d503e8380/html5/thumbnails/40.jpg)
So, according to the Chain Rule:
UBC Math 102
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Conical pile of sugar
Sugar is poured at a constant flow rate, 1 cm3/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
UBC Math 102
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The conical pile
UBC Math 102
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The conical pile
UBC Math 102
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The conical pile
UBC Math 102
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Chain Rule
Time t à
Use the fact that h(t) = C r (t) UBC Math 102
r(t),h(t) V(r,h)
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Solu,on – step 1
Given: -‐ Volume of cone -‐ Ra,o of height to radius is constant:
Hence:
UBC Math 102
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Chain Rule
Time t à
UBC Math 102
r(t) V(r)
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Solu,on – step 2
UBC Math 102
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Related problem:
Try this one yourself! Sugar is poured at a constant flow rate, 1 cm3/s to form a conical pile. At what rate is the base area increasing?
UBC Math 102
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Related test problem
UBC Math 102
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Solu,on from last ,me: Angles in the op,mal Y-‐shaped ant-‐trail
What angles does the Y trail form? We found that
UBC Math 102
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Angles
What angle does the Y trail form?
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Recall: Ra,os in equilateral triangle
• Side lengths all equal 1 • Height:
UBC Math 102
60o 60o
30o 30o
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Angles
What angle does the Y trail form? The branches form a 120 degree angle with each other, a 60 degree angle with the base of the Y
60o