op#mizaon* - math.ubc.cakeshet/m102/2015/lect5.2.pdf · op#mizaon* finding*max*or*mins*and*op#mal*...
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Op#miza#on
Finding max or mins and op#mal solu#ons to prac#cal problems
UBC Math 102
Course Calendar: Quiz this Friday
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You are here
Quiz on Friday:
Part 1 – individual – no notes, calculators, no laptops, no books, no devises. Part 2 – Group – Please do use any notes, calculators, laptop, spreadsheet, and other devices. ENSURE ONE GROUP MEMBER HAS SUCH DEVICE(S).
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Announcements Assignment 4 was supposed to have problems on Linear Approxima#on and Newton’s Method (but it did not) You could study by doing the following problems to prepare for the quiz: Extra Prac#ce problems 8, 9, 10, 11, 12, or some of the problems in Chapter 5 of the Course Notes.
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Help with homework
Find the dimensions of the rectangle with largest area that can be inscribed in a right triangle with legs of length 15 cm and 20 cm if two sides of the rectangle lie along the legs (the sides adjacent to the right angle).
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(x,y)
Help with Homework
Where is the func#on Increasing? Decreasing? See solu#on next page. (We did not spend #me on this in class.
UBC Math 102 Solu#on
Solu#on
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From Last #me: Consider the func#on y = f(x) = Use pen & paper to sketch the graph of this func#on, indica#ng the zeros, cri#cal points, inflec#on points of the func#on. Determine the absolute minimum and absolute maximum of the func#on on the interval [0,3]
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Actual graph
• Infl Pts
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Local max
Local mins
Absolute min/max on [0,3]
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Evaluate the original func#on at endpoints and cri#cal points in [0,3]
• y=f(x)=
• So f(0)=5, f( ) =-‐4, f(3)= 32 • Smallest Value; Largest Value • (Absolute Min); (Absolute Max)
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Problem
Find the cylinder of maximal volume that would fit inside a sphere of radius R.
• Sketch a diagram and label some quan##es that might be useful
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(1) Problem
Find the cylinder of maximal volume that would fit inside a sphere of radius R. • My diagram looks like:
(A) (B) (C)
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Label some quan##es
• Add these to your diagram:
• (Draw these in so that some rela#onship can be deduced from your sketch!)
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Geometry
No#ce that the top and bolom edges of the cylinder touch the sphere!
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Relate the variables to one another
• What is the rela#onship between these quan##es?
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(2) The rela#onship is
(A) R = r + (h/2) (B) R = r + h (C) R = r2+ h2
(D) R2= r2+ h2 (E) R2= r2+ (h/2)2
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The rela#onship is
(A) R = r + (h/2) (B) R = r + h (C) R = r2+ h2
(D) R2= r2+ h2 (E) R2= r2+ (h/2)2
PYTHAGOREAN TRIANGLE !!
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(3) What else do we need?
Recall Problem: Find the cylinder of maximal volume that would fit inside a sphere of radius R. (A) The volume of the sphere (B) The surface area of the sphere (C) The volume of the cylinder (D) The surface area of the cylinder (E) More than one of the above.
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(4) The volume of the cylinder is
• (A) V= (4/3) π r3 • (B) V= (4/3) π R3 • (C) V=2π R h • (D) V= π R2 (h/2) • (E) V= π r2 h
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The volume of the cylinder is The volume of any cylinder is V = base area X height = A h
The base area is (area of a circle):
A = π r2
Hence the volume of the cylinder is V= π r2 h
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h A
(5) The goal of the problem is to
• (A) Find smallest sphere that can contain the cylinder
• (B) Maximize the volume of the sphere • (C) Maximize the volume of the cylinder while keeping its radius r constant.
• (D) Maximize the volume of the cylinder while also sa#sfying a constraint
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(6) The constraint
In the constraint
• (A) All three quan##es are variables. • (B) Only R and r can be varied, while h stays fixed
• (C) Only r and h can be varied while R is taken to be a given constant
• (D) Only r is a variable
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Domain of interest Q: Can the variables take on any values? A: NO, they have to stay within some ranges. What is the biggest that r could be? What is the biggest that h could be? fixed
The Mathema#cal Problem
Maximize the volume of the cylinder while also sa#sfying the constraint. Hint: use the constraint to eliminate r.
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The mathema#cal problem reduces to
Maximize the func#on (or wrilen in expanded form)
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Sketch V as a func#on of h
Or NOTE: R and π are constants!!
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How to assemble a sketch:
• Sketch the func#on
• If h=0 then V=0 • If h= 2R then V=0 • If 0 <h<2R then V>0
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Sketch looks like:
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(7)The deriva#ve of is
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(8) Cri#cal points of V(h) are
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Cri#cal points:
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±
We can reject the nega#ve values as irrelevant
(9) What type of cri#cal point?
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Type of cri#cal points:
• The second deriva#ve is
Since V’’(h) <0, v(h) is concave down and the cri#cal point is a local maximum.
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From desmos:
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Actually, only 0<h<2R:
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(10) The corresponding radius of the cylinder is
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Radius:
• We find the radius from the constraint
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(11) The volume of the cylinder is
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Volume:
• The volume of the cylinder is:
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Answers • 1 C • 2 E • 3 C • 4 E • 5 D • 6 C • 7 B • 8 E • 9 A • 10 A • 11 D
Related test problems (2014 MT2)
A farmer is building a rectangular enclosure for a pevng zoo. She wants the enclosure to have an area of 100 m2 and as large a perimeter as possible. For the comfort of the animals, both the width and the length of the enclosure should be no less then to 2 m. What is the maximum possible perimeter?
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Test problem (2008)
UBC Math 102 Solu#on
Solu#on
UBC Math 102
Test problem
UBC Math 102 Solu#on
Solu#on
UBC Math 102
Test problem
UBC Math 102 Solu#on
Solu#on to Test problem
UBC Math 102
Test problem
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UBC Math 102 Solu#on
Solu#on to test problem
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UBC Math 102