Applied Calculus
7th Edition
ISBN: 9781337291248
Author: Waner, Stefan.
Publisher: Cengage Learning,
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Textbook Question
Chapter 5.2, Problem 12E
Advertising Costs The cost, in billions of dollars, of airing x five-second hologram commercials during a Galactic Chess game can be approximated by
How many hologram commercials should your company air to minimize average costs? What is the resulting average cost of a five-second ad? (Round your answer to the nearest billion dollars.) [HINT: See Example 1.]
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(7) (12 points) Let F(x, y, z) = (y, x+z cos yz, y cos yz).
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(a) (4 points) Show that V x F = 0.
(b) (4 points) Find a potential f for the vector field F.
(c) (4 points) Let S be a surface in R3 for which the Stokes' Theorem is valid. Use
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Jos
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as denotes the boundary of S. Explain your answer.
(3) (16 points) Consider
z = uv,
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(x, y) = R².
Give a parametric description of the tangent plane to S at the point p = T(x, y).
(d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic
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Q(x,y) F(a+x,b+y).
=
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(z +4ry)drdy .
So S√ ²
0
Chapter 5 Solutions
Applied Calculus
Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - Prob. 6ECh. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - Prob. 10E
Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 112, locate and classify all extrema...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - Prob. 16ECh. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - Prob. 22ECh. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - Prob. 30ECh. 5.1 - Prob. 31ECh. 5.1 - Prob. 32ECh. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.1 - Prob. 35ECh. 5.1 - Prob. 36ECh. 5.1 - Prob. 37ECh. 5.1 - Prob. 38ECh. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - Prob. 40ECh. 5.1 - Prob. 41ECh. 5.1 - Prob. 42ECh. 5.1 - In Exercises 13-44, find the exact location of all...Ch. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 5.1 - In Exercises 4956 the graph of the derivative of a...Ch. 5.1 - Prob. 50ECh. 5.1 - Prob. 51ECh. 5.1 - Prob. 52ECh. 5.1 - In Exercises 4956 the graph of the derivative of a...Ch. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - In Exercises 4956 the graph of the derivative of a...Ch. 5.1 - Prob. 57ECh. 5.1 - Prob. 58ECh. 5.1 - Draw the graph of a function that has stationary...Ch. 5.1 - Draw the graph of a function that has relative,...Ch. 5.1 - If a stationary point is not a relative maximum,...Ch. 5.1 - If one endpoint is a relative maximum, must the...Ch. 5.1 - We said that if f is continuous on a closed...Ch. 5.1 - Prob. 64ECh. 5.1 - Prob. 65ECh. 5.1 - Prob. 66ECh. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - In Exercises 1-8, solve the given optimization...Ch. 5.2 - For a rectangle with perimeter 20 to have the...Ch. 5.2 - For a rectangle with area 100 to have the smallest...Ch. 5.2 - Advertising Costs The cost, in thousands of...Ch. 5.2 - Advertising Costs The cost, in billions of...Ch. 5.2 - Average Cost: iPhones Assume that it costs Apple...Ch. 5.2 - Prob. 14ECh. 5.2 - Pollution Control The cost of controlling...Ch. 5.2 - Prob. 16ECh. 5.2 - Pollution Control (Compare Exercise 15.) The cost...Ch. 5.2 - Pollution Control (Compare Exercise 16.) Repeat...Ch. 5.2 - Fences I would like to create a rectangular...Ch. 5.2 - Fences I would like to create a rectangular orchid...Ch. 5.2 - Prob. 21ECh. 5.2 - Prob. 22ECh. 5.2 - Fences (Compare Exercise 19.) For tax reasons I...Ch. 5.2 - Fences (Compare Exercise 20.) For reasons too...Ch. 5.2 - Revenue Hercules Films is deciding on the price of...Ch. 5.2 - Profit Hercules Films is also deciding on the...Ch. 5.2 - Revenue: Smartphones Worldwide annual sales of...Ch. 5.2 - Projected Revenue: Smartphones Worldwide annual...Ch. 5.2 - Revenue: Monorail Service The demand for monorail...Ch. 5.2 - Revenue: Mars Monorail The demand for monorail...Ch. 5.2 - Revenue Assume that the demand for tuna in a small...Ch. 5.2 - Revenue In the 1930s the economist Henry Schultz...Ch. 5.2 - Revenue During the 1950s the wholesale price for...Ch. 5.2 - Revenue Your underground used-book business is...Ch. 5.2 - Profit: Smartphones (Compare Exercise 27.)...Ch. 5.2 - Projected Profit: Smartphones (Compare Exercise...Ch. 5.2 - Profit The demand equation for your companys...Ch. 5.2 - Profit Because of sales by a competing company,...Ch. 5.2 - Paint Cans A company manufactures cylindrical...Ch. 5.2 - Metal Drums A company manufactures cylindrical...Ch. 5.2 - Tin Cans A company manufactures cylindrical tin...Ch. 5.2 - Metal Drums A company manufactures cylindrical...Ch. 5.2 - Box Design Chocolate Box Company is going to make...Ch. 5.2 - Box Design Vanilla Box Company is going to make...Ch. 5.2 - Box Design A packaging company is going to make...Ch. 5.2 - Prob. 46ECh. 5.2 - Luggage Dimensions American Airlines requires that...Ch. 5.2 - Carry-on Dimensions American Airlines requires...Ch. 5.2 - Luggage Dimensions Fly-by-Night Airlines has a...Ch. 5.2 - Luggage Dimensions Fair Weather Airlines has a...Ch. 5.2 - Package Dimensions The U.S. Postal Service (USPS)...Ch. 5.2 - Prob. 52ECh. 5.2 - Prob. 53ECh. 5.2 - Cellphone Revenues (Refer to Exercise 53.) If we...Ch. 5.2 - Research and Development Spending on research and...Ch. 5.2 - Prob. 56ECh. 5.2 - Asset Appreciation As the financial consultant to...Ch. 5.2 - Plantation Management The value of a fir tree in...Ch. 5.2 - Prob. 59ECh. 5.2 - Prob. 60ECh. 5.2 - Average Profit The FeatureRich Software Company...Ch. 5.2 - Prob. 62ECh. 5.2 - Resource Allocation Your company manufactures...Ch. 5.2 - Prob. 64ECh. 5.2 - Resource Allocation Your automobile assembly plant...Ch. 5.2 - Prob. 66ECh. 5.2 - Prob. 67ECh. 5.2 - Prob. 68ECh. 5.2 - Prob. 69ECh. 5.2 - Prob. 70ECh. 5.2 - Prob. 71ECh. 5.2 - Prob. 72ECh. 5.2 - Prob. 73ECh. 5.2 - Harvesting Forests The following equation models...Ch. 5.2 - Agriculture The fruit yield per tree in an orchard...Ch. 5.2 - Agriculture Two years ago, your orange orchard...Ch. 5.2 - Prob. 77ECh. 5.2 - Prob. 78ECh. 5.2 - Prob. 79ECh. 5.2 - Prob. 80ECh. 5.2 - Prob. 81ECh. 5.2 - Explain why finding the production level that...Ch. 5.2 - Prob. 83ECh. 5.2 - Prob. 84ECh. 5.2 - Prob. 85ECh. 5.2 - Prob. 86ECh. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - Prob. 2ECh. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1-10, calculate d2ydx2. [HINT: See...Ch. 5.3 - In Exercises 1116 the position s of a point (in...Ch. 5.3 - In Exercises 1116 the position s of a point (in...Ch. 5.3 - In Exercises 1116 the position s of a point (in...Ch. 5.3 - In Exercises 1116 the position s of a point (in...Ch. 5.3 - In Exercises 1116 the position s of a point (in...Ch. 5.3 - In Exercises 1116 the position s of a point (in...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 1724 the graph of a function is...Ch. 5.3 - In Exercises 2528 the graph of the derivative,...Ch. 5.3 - In Exercises 2528 the graph of the derivative,...Ch. 5.3 - In Exercises 2528 the graph of the derivative,...Ch. 5.3 - In Exercises 2528 the graph of the derivative,...Ch. 5.3 - In Exercises 2932 the graph of the second...Ch. 5.3 - In Exercises 2932 the graph of the second...Ch. 5.3 - In Exercises 2932 the graph of the second...Ch. 5.3 - Prob. 32ECh. 5.3 - In Exercises 3344, find the x-coordinates of all...Ch. 5.3 - In Exercises 3344, find the x-coordinates of all...Ch. 5.3 - In Exercises 3344, find the x-coordinates of all...Ch. 5.3 - Prob. 36ECh. 5.3 - Prob. 37ECh. 5.3 - In Exercises 3344, find the x-coordinates of all...Ch. 5.3 - Prob. 39ECh. 5.3 - Prob. 40ECh. 5.3 - Prob. 41ECh. 5.3 - Prob. 42ECh. 5.3 - Prob. 43ECh. 5.3 - Prob. 44ECh. 5.3 - Prob. 45ECh. 5.3 - Prob. 46ECh. 5.3 - Prob. 47ECh. 5.3 - Prob. 48ECh. 5.3 - Prob. 49ECh. 5.3 - Prob. 50ECh. 5.3 - Prob. 51ECh. 5.3 - Prob. 52ECh. 5.3 - Prob. 53ECh. 5.3 - Prob. 54ECh. 5.3 - Acceleration on Mars If a stone is dropped from a...Ch. 5.3 - Acceleration on the Moon If a stone is thrown up...Ch. 5.3 - Motion in a Straight Line The position of a...Ch. 5.3 - Motion in a Straight Line The position of a...Ch. 5.3 - Bottled Water Sales Annual sales of bottled water...Ch. 5.3 - Bottled Water Sales Annual U.S. per capita sales...Ch. 5.3 - Prob. 61ECh. 5.3 - Embryo Development The daily oxygen consumption of...Ch. 5.3 - Inflation The following graph shows the...Ch. 5.3 - Inflation The following graph shows the...Ch. 5.3 - Inflation The following graph shows the...Ch. 5.3 - Inflation The following graph shows the...Ch. 5.3 - Scientific Research: 19832003 The percentage of...Ch. 5.3 - Scientific Research: 19832003 The percentage of...Ch. 5.3 - Prob. 69ECh. 5.3 - Prob. 70ECh. 5.3 - Prob. 71ECh. 5.3 - Prob. 72ECh. 5.3 - Epidemics The following graph shows the total...Ch. 5.3 - Prob. 74ECh. 5.3 - Prob. 75ECh. 5.3 - Prob. 76ECh. 5.3 - Prob. 77ECh. 5.3 - Prob. 78ECh. 5.3 - Prob. 79ECh. 5.3 - Prob. 80ECh. 5.3 - Prob. 81ECh. 5.3 - Oil Imports from Mexico Repeat Exercises 81 using...Ch. 5.3 - Prob. 83ECh. 5.3 - Prob. 84ECh. 5.3 - Prob. 85ECh. 5.3 - Prob. 86ECh. 5.3 - Prob. 87ECh. 5.3 - Prob. 88ECh. 5.3 - Prob. 89ECh. 5.3 - Prob. 90ECh. 5.3 - Complete the following: If the graph of a function...Ch. 5.3 - Prob. 92ECh. 5.3 - Prob. 93ECh. 5.3 - The graph of daily sales of Luddingtons Wellington...Ch. 5.3 - Prob. 95ECh. 5.3 - Company Cs profits satisfy P(0)=1 million, P;(0)=1...Ch. 5.3 - Prob. 97ECh. 5.3 - Prob. 98ECh. 5.4 - In Exercises 126, sketch the graph of the given...Ch. 5.4 - Prob. 2ECh. 5.4 - Prob. 3ECh. 5.4 - Prob. 4ECh. 5.4 - Prob. 5ECh. 5.4 - In Exercises 126, sketch the graph of the given...Ch. 5.4 - In Exercises 126, sketch the graph of the given...Ch. 5.4 - Prob. 8ECh. 5.4 - Prob. 9ECh. 5.4 - Prob. 10ECh. 5.4 - Prob. 11ECh. 5.4 - Prob. 12ECh. 5.4 - Prob. 13ECh. 5.4 - In Exercises 126, sketch the graph of the given...Ch. 5.4 - Prob. 15ECh. 5.4 - Prob. 16ECh. 5.4 - Prob. 17ECh. 5.4 - Prob. 18ECh. 5.4 - In Exercises 126, sketch the graph of the given...Ch. 5.4 - Prob. 20ECh. 5.4 - Prob. 21ECh. 5.4 - Prob. 22ECh. 5.4 - Prob. 23ECh. 5.4 - Prob. 24ECh. 5.4 - Prob. 25ECh. 5.4 - Prob. 26ECh. 5.4 - Prob. 27ECh. 5.4 - Prob. 28ECh. 5.4 - Prob. 29ECh. 5.4 - Prob. 30ECh. 5.4 - Home Prices The following graph shows a rough...Ch. 5.4 - Prob. 32ECh. 5.4 - Prob. 33ECh. 5.4 - Prob. 34ECh. 5.4 - Consumer Price Index The following graph shows the...Ch. 5.4 - Consumer Price Index The following graph shows the...Ch. 5.4 - Prob. 37ECh. 5.4 - Prob. 38ECh. 5.4 - Prob. 39ECh. 5.4 - Average Cost: PlayStation 4s Assume that it costs...Ch. 5.4 - Prob. 41ECh. 5.4 - Subprime Mortgage Debt during the Housing Bubble...Ch. 5.4 - Prob. 43ECh. 5.4 - Prob. 44ECh. 5.4 - Prob. 45ECh. 5.4 - Prob. 46ECh. 5.4 - Prob. 47ECh. 5.4 - Prob. 48ECh. 5.4 - Prob. 49ECh. 5.4 - Prob. 50ECh. 5.5 - Rewrite the statements and questions in Exercises...Ch. 5.5 - Rewrite the statements and questions in Exercises...Ch. 5.5 - Prob. 3ECh. 5.5 - Rewrite the statements and questions in Exercises...Ch. 5.5 - Prob. 5ECh. 5.5 - Prob. 6ECh. 5.5 - Rewrite the statements and questions in Exercises...Ch. 5.5 - Rewrite the statements and questions in Exercises...Ch. 5.5 - Prob. 9ECh. 5.5 - Puddles The radius of a circular puddle is growing...Ch. 5.5 - Balloons A spherical party balloon is being...Ch. 5.5 - Prob. 12ECh. 5.5 - Prob. 13ECh. 5.5 - Prob. 14ECh. 5.5 - Sliding Ladders The base of a 50-foot ladder is...Ch. 5.5 - Prob. 16ECh. 5.5 - Prob. 17ECh. 5.5 - Descending Elevator You are situated 300 feet from...Ch. 5.5 - Average Cost The average cost function for the...Ch. 5.5 - Average Cost Repeat Exercise 19, using the revised...Ch. 5.5 - Demand Demand for your tie-dyed T-shirts is given...Ch. 5.5 - Supply The number of retro portable CD players you...Ch. 5.5 - Revenue You can now sell 50 cups of lemonade per...Ch. 5.5 - Revenue You can now sell 40 cars per month at...Ch. 5.5 - Oil Revenues Daily oil production by Pemex,...Ch. 5.5 - Oil Expenditures Daily oil imports to the United...Ch. 5.5 - Resource Allocation Your company manufactures...Ch. 5.5 - Resource Allocation Your company is the largest...Ch. 5.5 - Production The automobile assembly plant you...Ch. 5.5 - Prob. 30ECh. 5.5 - Prob. 31ECh. 5.5 - Demand The demand equation for rubies at Royal...Ch. 5.5 - Ships Sailing Apart The H.M.S. Dreadnaught is 40...Ch. 5.5 - Near Miss My aunt and I were approaching the same...Ch. 5.5 - Baseball A baseball diamond is a square with side...Ch. 5.5 - Baseball Refer to Exercise 35. Another player is...Ch. 5.5 - Prob. 37ECh. 5.5 - Prob. 38ECh. 5.5 - Prob. 39ECh. 5.5 - Prob. 40ECh. 5.5 - Prob. 41ECh. 5.5 - Grades A production formula for a students...Ch. 5.5 - Cones A right circular conical vessel is being...Ch. 5.5 - More Cones A circular conical vessel is being...Ch. 5.5 - Prob. 45ECh. 5.5 - Prob. 46ECh. 5.5 - Computers vs. Income In the 1990s the demand for...Ch. 5.5 - Computers vs. Income Refer back to the model in...Ch. 5.5 - Prob. 49ECh. 5.5 - Prob. 50ECh. 5.5 - Prob. 51ECh. 5.5 - Prob. 52ECh. 5.5 - Prob. 53ECh. 5.5 - Prob. 54ECh. 5.5 - Prob. 55ECh. 5.5 - Prob. 56ECh. 5.5 - Prob. 57ECh. 5.5 - Prob. 58ECh. 5.5 - Prob. 59ECh. 5.5 - Prob. 60ECh. 5.5 - Prob. 61ECh. 5.5 - Prob. 62ECh. 5.6 - Demand for Oranges The weekly sales of Honolulu...Ch. 5.6 - Prob. 2ECh. 5.6 - Prob. 3ECh. 5.6 - Prob. 4ECh. 5.6 - College Tuition An old study of about 1,800 U.S....Ch. 5.6 - Monorail Services The demand for monorail service...Ch. 5.6 - Tissues The consumer demand equation for tissues...Ch. 5.6 - Bodybuilding The consumer demand curve for...Ch. 5.6 - T-Shirts The Physics Club sells E=mc2 T-shirts at...Ch. 5.6 - Comics The demand curve for original Iguanawoman...Ch. 5.6 - E-Readers The demand for Amazons Kindle e-reader...Ch. 5.6 - Prob. 12ECh. 5.6 - Corn In the 1930s the economist Henry Schultz...Ch. 5.6 - Demand for Fried Chicken A fried chicken franchise...Ch. 5.6 - Paint-By-Number The estimated monthly sales of...Ch. 5.6 - Paint-By-Number Repeat Exercise 15 using the...Ch. 5.6 - Linear Demand Functions A general linear demand...Ch. 5.6 - Exponential Demand Functions A general exponential...Ch. 5.6 - Hyperbolic Demand Functions A general hyperbolic...Ch. 5.6 - Quadratic Demand Functions A general quadratic...Ch. 5.6 - Modeling Linear Demand You have been hired as a...Ch. 5.6 - Modeling Linear Demand You have been hired as a...Ch. 5.6 - Modeling Exponential Demand As the new owner of a...Ch. 5.6 - Prob. 24ECh. 5.6 - Prob. 25ECh. 5.6 - Prob. 26ECh. 5.6 - Income Elasticity of Demand: Broadband in 2010 The...Ch. 5.6 - Income Elasticity of Demand: Broadband in 2007 The...Ch. 5.6 - Prob. 29ECh. 5.6 - Prob. 30ECh. 5.6 - Prob. 31ECh. 5.6 - Price Elasticity of Supply Given a supply equation...Ch. 5.6 - Income Elasticity of Demand (based on a question...Ch. 5.6 - Prob. 34ECh. 5.6 - Complete the following: When demand is inelastic,...Ch. 5.6 - Prob. 36ECh. 5.6 - Prob. 37ECh. 5.6 - Given that the demand q is a differentiable...Ch. 5.6 - Prob. 39ECh. 5.6 - Prob. 40ECh. 5 - In Exercises 18, find all the relative and...Ch. 5 - Prob. 2RECh. 5 - Prob. 3RECh. 5 - Prob. 4RECh. 5 - Prob. 5RECh. 5 - Prob. 6RECh. 5 - Prob. 7RECh. 5 - Prob. 8RECh. 5 - Prob. 9RECh. 5 - Prob. 10RECh. 5 - Prob. 11RECh. 5 - Prob. 12RECh. 5 - Prob. 13RECh. 5 - Prob. 14RECh. 5 - Prob. 15RECh. 5 - Prob. 16RECh. 5 - Prob. 17RECh. 5 - Prob. 18RECh. 5 - Prob. 19RECh. 5 - Prob. 20RECh. 5 - Prob. 21RECh. 5 - Prob. 22RECh. 5 - Revenue Demand for the latest best-seller at...Ch. 5 - Revenue Demand for The Secret Loves of John O, a...Ch. 5 - Prob. 25RECh. 5 - Prob. 26RECh. 5 - Office Space Although still a sophomore at...Ch. 5 - Recreation Space As a result of complaints by the...Ch. 5 - Box Design The sales department at...Ch. 5 - Prob. 30RECh. 5 - Prob. 31RECh. 5 - Prob. 32RECh. 5 - Prob. 33RECh. 5 - Company Logos OHaganBooks.coms website has an ani-...Ch. 5 - Prob. 35RECh. 5 - Prob. 36RECh. 5 - Prob. 37RECh. 5 - Prob. 38RE
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- (10) (16 points) Let R>0. Consider the truncated sphere S given as x² + y² + (z = √15R)² = R², z ≥0. where F(x, y, z) = −yi + xj . (a) (8 points) Consider the vector field V (x, y, z) = (▼ × F)(x, y, z) Think of S as a hot-air balloon where the vector field V is the velocity vector field measuring the hot gasses escaping through the porous surface S. The flux of V across S gives the volume flow rate of the gasses through S. Calculate this flux. Hint: Parametrize the boundary OS. Then use Stokes' Theorem. (b) (8 points) Calculate the surface area of the balloon. To calculate the surface area, do the following: Translate the balloon surface S by the vector (-15)k. The translated surface, call it S+ is part of the sphere x² + y²+z² = R². Why do S and S+ have the same area? ⚫ Calculate the area of S+. What is the natural spherical parametrization of S+?arrow_forward(1) (8 points) Let c(t) = (et, et sint, et cost). Reparametrize c as a unit speed curve starting from the point (1,0,1).arrow_forward(9) (16 points) Let F(x, y, z) = (x² + y − 4)i + 3xyj + (2x2 +z²)k = - = (x²+y4,3xy, 2x2 + 2²). (a) (4 points) Calculate the divergence and curl of F. (b) (6 points) Find the flux of V x F across the surface S given by x² + y²+2² = 16, z ≥ 0. (c) (6 points) Find the flux of F across the boundary of the unit cube E = [0,1] × [0,1] x [0,1].arrow_forward
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