Why are soda cans shaped the way they are?
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1 Why are soda cans shaped the way they are?
2 ... but other cans vary in shape?
3 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: Covers Sections (including 4.7, optimization). Emphasis is on 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).
4 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
5 MATH 1131Q...
6 MATH 1131Q... Sketching Graphs of Functions and l Hospital s Rule
7 Using the First Derivative Theorem (Increasing/Decreasing Test) Let f be a differentiable function on an interval (a, b). 1 If f (x) > 0 on (a, b), then f is increasing on (a, b). 2 If f (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 If f changes from positive to negative at c, then f has a local maximum at c. 2 If f changes from negative to positive at c, then f has a local minimum at c. 3 If f 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
8 Using the Second Derivative Theorem (Concavity Test) Let f be a twice differentiable function on an interval (a, b). 1 If f (x) > 0 on (a, b), then f is concave upward on (a, b). 2 If f (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 is continuous near c. Then: 1 If f (c) = 0 and f (c) < 0, then f has a local minimum at c. 2 If f (c) = 0 and f (c) > 0, then f has a local maximum at c. 3 If f (c) = 0 and f (c) = 0, then the test is inconclusive.
9 Example Sketch the graph of f (x) = e 1/x
10 Example Sketch the graph of f (x) = e 1/x.
11 Example Sketch the graph of f (x) = e 1/x
12 Inderterminate Powers Example Calculate lim x 0 +(1 + sin(4x))cot x. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 10 / 27
13 Inderterminate Powers Example Calculate lim x 0 +(1 + sin(4x))cot x (Alternative method!) Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 11 / 27
14 Optimization
15 What s the best shape for a can? Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 13 / 27
16 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
17 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
18 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
19 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 Diameter Height Height/Diameter 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) MATH 1131Q - Calculus 1 16 / 27
20 The Best Can in the Market! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 17 / 27
21 Business Idea... A BETTER CAN! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 18 / 27
22 Business Idea... A BETTER CAN! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 18 / 27
23 Business Idea... A BETTER CAN! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 27
24 Business Idea... A BETTER CAN! Volume of a can of soda (2.5 inches x 4.8 inches) is cubic inches. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 27
25 Business Idea... A BETTER CAN! Volume of a can of soda (2.5 inches x 4.8 inches) is cubic inches. Surface of a can of soda is square inches. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 27
26 Business Idea... A BETTER CAN! Volume of a can of soda (2.5 inches x 4.8 inches) is cubic inches. Surface of a can of soda is square inches. Surface of an ideal can (h/d = 1) with V = is square inches. (That s a reduction of 4.25% in surface area.) Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 27
27 Business Idea... A BETTER CAN! Volume of a can of soda (2.5 inches x 4.8 inches) is cubic inches. Surface of a can of soda is square inches. Surface of an ideal can (h/d = 1) with V = is square inches. (That s a reduction of 4.25% in surface area.) Radius of a sphere with V = is inches. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 27
28 Business Idea... A BETTER CAN! Volume of a can of soda (2.5 inches x 4.8 inches) is cubic inches. Surface of a can of soda is square inches. Surface of an ideal can (h/d = 1) with V = is square inches. (That s a reduction of 4.25% in surface area.) Radius of a sphere with V = is inches. Surface of a sphere with r = is square inches! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 19 / 27
29 Business Idea... A BETTER CAN! Volume of a can of soda (2.5 inches x 4.8 inches) is cubic inches. Surface of a can of soda is square inches. Surface of an ideal can (h/d = 1) with V = is square inches. (That s a reduction of 4.25% in surface area.) Radius of a sphere with V = is inches. Surface of a sphere with r = is 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
30 Business Idea... A BETTER CAN! Volume of a can of soda (2.5 inches x 4.8 inches) is cubic inches. Surface of a can of soda is square inches. Surface of an ideal can (h/d = 1) with V = is square inches. (That s a reduction of 4.25% in surface area.) Radius of a sphere with V = is inches. Surface of a sphere with r = is 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
31 Business Idea... Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 27
32 Business Idea... Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 27
33 Business Idea... Oh well... Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 27
34 Optimization Problems How to approach optimization problems: Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 21 / 27
35 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
36 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
37 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
38 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
39 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
40 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
41 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
42 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
43 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
44 Example A box-shaped shipping crate with a square base needs to have a volume of 80ft 3. The material used to make the base of the crate costs twice as much (per ft 2 ) as the material used for the sides, and the material used to make the top of the crate costs half as much (per ft 2 ) 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
45 Happy Halloween!
46 This slide left intentionally blank Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 27 / 27
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