Why are soda cans shaped the way they are?
... but other cans vary in shape?
Our second Midterm Exam will be on Tuesday 11/4
Two times (check your time following these instructions)
6-8 PM at AUST 108, or 9-11 PM at TLS 154.
There is a practice exam and solutions in outline in the website:
http://alozano.clas.uconn.edu/math1131f14
Covers Sections 1.1 - 4.7 (including 4.7, optimization). Emphasis
is on 3.4 - 4.7.
There will be a review during class on Tuesday 11/4.
My office hours are as usual: Tuesdays 10:15-11:15 and
Thursdays 11-12, at MSB 312. TA’s office hours also as usual
(see website).
There will be a Q Center review session on Monday 11/3 at the
HBL Lecture Center, from 9am to 10am.
There will be a review session on Monday 11/3 (by Amit Savkar at
LH 101 from 3:30 pm to 5:30 pm; will also be available online).
MATH 1131Q - Calculus 1.
Álvaro Lozano-Robledo
Department of Mathematics
University of Connecticut
Day 20
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
4 / 27
MATH 1131Q...
MATH 1131Q...
Sketching Graphs of
Functions
and l’Hospital’s Rule
Using the First Derivative
Theorem (Increasing/Decreasing Test)
Let f be a differentiable function on an interval (a, b).
1
If f 0 (x) > 0 on (a, b), then f is increasing on (a, b).
2
If f 0 (x) < 0 on (a, b), then f is decreasing on (a, b).
Theorem (The First Derivative Test for Critical Points)
Suppose that c is a critical number of a continuous function f . Then:
1
2
3
If f 0 changes from positive to negative at c, then f has a local
maximum at c.
If f 0 changes from negative to positive at c, then f has a local
minimum at c.
If f 0 does not change sign at c, then f has no local maximum or
minimum at c.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
6 / 27
Using the Second Derivative
Theorem (Concavity Test)
Let f be a twice differentiable function on an interval (a, b).
1
If f 00 (x) > 0 on (a, b), then f is concave upward on (a, b).
2
If f 00 (x) < 0 on (a, b), then f is concave downward on (a, b).
Definition
A point P on a curve y = f (x) is called an inflection point if f is
continuous there and the curve changes concavity at P.
Theorem (The Second Derivative Test for Critical Points)
Suppose that c is a critical number of a continuous function f , and
suppose f 00 is continuous near c. Then:
1
If f 0 (c) = 0 and f 00 (c) < 0, then f has a local minimum at c.
2
If f 0 (c) = 0 and f 00 (c) > 0, then f has a local maximum at c.
3
If f 0 (c) = 0 and f 00 (c) = 0, then the test is inconclusive.
Example
Sketch the graph of f (x) = e1/ x .
3
2
1
−3
−2
−1
0
−1
1
2
3
4
5
6
Example
Sketch the graph of f (x) = e1/ x .
Example
Sketch the graph of f (x) = e1/ x .
3
2
1
−3
−2
−1
0
−1
1
2
3
4
5
6
Inderterminate Powers
Example
Calculate lim+ (1 + sin(4x))cot x .
x →0
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
10 / 27
Inderterminate Powers
Example
Calculate lim+ (1 + sin(4x))cot x (Alternative method!)
x →0
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
11 / 27
Optimization
What’s the “best” shape for a can?
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
13 / 27
Example
If we want to build a can (cylinder) that will hold a volume of V = 21.6
cubic inches (≈ 12 ounces), what dimensions will minimize the surface
area?
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
14 / 27
Example
If we want to build a can (cylinder) that will hold a volume of V = 21.6
cubic inches (≈ 12 ounces), what dimensions will minimize the surface
area?
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
15 / 27
Thus, the “best” can (in terms of minimal surface area) is one such that
h = 2r , or h/ 2r = 1, or equivalently, height/diameter = h/ d = 1.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
16 / 27
Thus, the “best” can (in terms of minimal surface area) is one such that
h = 2r , or h/ 2r = 1, or equivalently, height/diameter = h/ d = 1.
Brand
Soda Can
Swanson Chicken Broth
Campbell’s Soup
Chicken of the Sea
Carnation Cond. Milk
Eagle Brand Cond. Milk
Progresso Chunky Soup
Blue Diamond Almonds
Álvaro Lozano-Robledo (UConn)
Diameter
2.5
2.95
2.55
3.34
2.95
2.91
3.34
3.3
Height
4.8
4.13
3.85
1.57
3.7
3.03
4.21
2.24
MATH 1131Q - Calculus 1
Height/Diameter
1.92
1.4
1.5
0.47
1.25
1.04
1.26
0.67
16 / 27
The Best Can in the Market!
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
17 / 27
Business Idea...
A BETTER CAN!
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
18 / 27
Business Idea...
A BETTER CAN!
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
18 / 27
Business Idea...
A BETTER CAN!
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
A BETTER CAN!
Volume of a can of soda (2.5 inches x 4.8 inches) is 23.56 cubic
inches.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
A BETTER CAN!
Volume of a can of soda (2.5 inches x 4.8 inches) is 23.56 cubic
inches.
Surface of a can of soda is 47.51 square inches.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
A BETTER CAN!
Volume of a can of soda (2.5 inches x 4.8 inches) is 23.56 cubic
inches.
Surface of a can of soda is 47.51 square inches.
Surface of an ideal can (h/ d = 1) with V = 23.56 is 45.49 square
inches. (That’s a reduction of 4.25% in surface area.)
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
A BETTER CAN!
Volume of a can of soda (2.5 inches x 4.8 inches) is 23.56 cubic
inches.
Surface of a can of soda is 47.51 square inches.
Surface of an ideal can (h/ d = 1) with V = 23.56 is 45.49 square
inches. (That’s a reduction of 4.25% in surface area.)
Radius of a sphere with V = 23.56 is 1.778 inches.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
A BETTER CAN!
Volume of a can of soda (2.5 inches x 4.8 inches) is 23.56 cubic
inches.
Surface of a can of soda is 47.51 square inches.
Surface of an ideal can (h/ d = 1) with V = 23.56 is 45.49 square
inches. (That’s a reduction of 4.25% in surface area.)
Radius of a sphere with V = 23.56 is 1.778 inches.
Surface of a sphere with r = 1.778 is 39.73 square inches!
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
A BETTER CAN!
Volume of a can of soda (2.5 inches x 4.8 inches) is 23.56 cubic
inches.
Surface of a can of soda is 47.51 square inches.
Surface of an ideal can (h/ d = 1) with V = 23.56 is 45.49 square
inches. (That’s a reduction of 4.25% in surface area.)
Radius of a sphere with V = 23.56 is 1.778 inches.
Surface of a sphere with r = 1.778 is 39.73 square inches!
That’s a reduction of 16.37% in surface area over traditional
soda cans!
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
A BETTER CAN!
Volume of a can of soda (2.5 inches x 4.8 inches) is 23.56 cubic
inches.
Surface of a can of soda is 47.51 square inches.
Surface of an ideal can (h/ d = 1) with V = 23.56 is 45.49 square
inches. (That’s a reduction of 4.25% in surface area.)
Radius of a sphere with V = 23.56 is 1.778 inches.
Surface of a sphere with r = 1.778 is 39.73 square inches!
That’s a reduction of 16.37% in surface area over traditional
soda cans! And a 12.66% reduction of the best can.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
19 / 27
Business Idea...
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
20 / 27
Business Idea...
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
20 / 27
Business Idea...
Oh well...
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
20 / 27
Optimization Problems
How to approach optimization problems:
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
21 / 27
Optimization Problems
How to approach optimization problems:
1
Understand the problem. What is the unknown? What are the
given variables? What data is given?
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
21 / 27
Optimization Problems
How to approach optimization problems:
1
Understand the problem. What is the unknown? What are the
given variables? What data is given?
2
Draw a diagram relating the variables of the problem.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
21 / 27
Optimization Problems
How to approach optimization problems:
1
Understand the problem. What is the unknown? What are the
given variables? What data is given?
2
Draw a diagram relating the variables of the problem.
3
Assign names to the variables, and find equations among the
variables. Let’s say Q is the variable to be optimized.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
21 / 27
Optimization Problems
How to approach optimization problems:
1
Understand the problem. What is the unknown? What are the
given variables? What data is given?
2
Draw a diagram relating the variables of the problem.
3
Assign names to the variables, and find equations among the
variables. Let’s say Q is the variable to be optimized.
4
Express the optimization variable Q in terms of the other
variables.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
21 / 27
Optimization Problems
How to approach optimization problems:
1
Understand the problem. What is the unknown? What are the
given variables? What data is given?
2
Draw a diagram relating the variables of the problem.
3
Assign names to the variables, and find equations among the
variables. Let’s say Q is the variable to be optimized.
4
Express the optimization variable Q in terms of the other
variables.
5
If the optimization variable Q is expressed in terms of several
variables, use the equations relating the variables to express Q in
terms of one single variable x.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
21 / 27
Optimization Problems
How to approach optimization problems:
1
Understand the problem. What is the unknown? What are the
given variables? What data is given?
2
Draw a diagram relating the variables of the problem.
3
Assign names to the variables, and find equations among the
variables. Let’s say Q is the variable to be optimized.
4
Express the optimization variable Q in terms of the other
variables.
5
If the optimization variable Q is expressed in terms of several
variables, use the equations relating the variables to express Q in
terms of one single variable x.
6
Find the absolute maximum or minimum of the optimization
variable Q(x), using our extreme values methods, in a sensible
interval.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
21 / 27
Example
We have 30 feet of posting to make the front part of a soccer goal.
What are the dimensions of the front frame with the largest possible
area?
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
22 / 27
Example
We have 30 feet of posting to make the front part of a soccer goal.
What are the dimensions of the front frame with the largest possible
area?
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
23 / 27
Example
Find the point on the parabola y 2 = 2x that is closest to the point (1, 4).
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
24 / 27
Example
A box-shaped shipping crate with a square base needs to have a
volume of 80ft3 . The material used to make the base of the crate costs
twice as much (per ft2 ) as the material used for the sides, and the
material used to make the top of the crate costs half as much (per ft2 )
as the material used for the sides. Use calculus to find the dimensions
of the crate that minimize the total cost of the materials.
Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
25 / 27
Happy Halloween!
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Álvaro Lozano-Robledo (UConn)
MATH 1131Q - Calculus 1
27 / 27