Implicit Differentiation Optimization Problems Homework

  • PROBLEM 1 : Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.

    Click HERE to see a detailed solution to problem 1.




  • PROBLEM 2 : Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen ?

    Click HERE to see a detailed solution to problem 2.




  • PROBLEM 3 : An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume ?

    Click HERE to see a detailed solution to problem 3.




  • PROBLEM 4 : A container in the shape of a right circular cylinder with no top has surface area 3 ft.2 What height h and base radius r will maximize the volume of the cylinder ?

    Click HERE to see a detailed solution to problem 4.




  • PROBLEM 5 : A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume ?

    Click HERE to see a detailed solution to problem 5.




  • PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse.

    Click HERE to see a detailed solution to problem 6.




  • PROBLEM 7 : Find the point (x, y) on the graph of nearest the point (4, 0).

    Click HERE to see a detailed solution to problem 7.




  • PROBLEM 8 : A cylindrical can is to hold 20 m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

    Click HERE to see a detailed solution to problem 8.




  • PROBLEM 9 : You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. What route will take the least amount of time ?

    Click HERE to see a detailed solution to problem 9.




  • PROBLEM 10 : Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 12 feet, what dimensions will result in the rectangle having largest possible area ?

    Click HERE to see a detailed solution to problem 10.




  • PROBLEM 11 : There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees ?

    Click HERE to see a detailed solution to problem 11.




  • PROBLEM 12 : Find the dimensions of the rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and graph of y=8-x3 . (See diagram.)

    Click HERE to see a detailed solution to problem 12.




  • PROBLEM 13 : Consider a rectangle of perimeter 12 inches. Form a cylinder by revolving this rectangle about one of its edges. What dimensions of the rectangle will result in a cylinder of maximum volume ?

    Click HERE to see a detailed solution to problem 13.




  • PROBLEM 14 : A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle of the movie screen is as large as possible ? (See diagram.)

    Click HERE to see a detailed solution to problem 14.




  • PROBLEM 15 : Find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in a sphere of radius 2.

    Click HERE to see a detailed solution to problem 15.




  • PROBLEM 16 : What angle between two edges of length 3 will result in an isosceles triangle with the largest area ? (See diagram.)

    Click HERE to see a detailed solution to problem 16.




  • PROBLEM 17 : Of all lines tangent to the graph of , find the tangent lines of mimimum slope and maximum slope.

    Click HERE to see a detailed solution to problem 17.




  • PROBLEM 18 : Find the length of the shortest ladder that will reach over an 8-ft. high fence to a large wall which is 3 ft. behind the fence. (See diagram.)

    Click HERE to see a detailed solution to problem 18.




  • PROBLEM 19 : Find the point P = (x, 0) on the x-axis which minimizes the sum of the squares of the distances from P to (0, 0) and from P to (3, 2).

    Click HERE to see a detailed solution to problem 19.




  • PROBLEM 20 : Car B is 30 miles directly east of Car A and begins moving west at 90 mph. At the same moment car A begins moving north at 60 mph. What will be the minimum distance between the cars and at what time t does the minimum distance occur ?

    Click HERE to see a detailed solution to problem 20.




  • PROBLEM 21 : A rectangular piece of paper is 12 inches high and six inches wide. The lower right-hand corner is folded over so as to reach the leftmost edge of the paper (See diagram.).

    Find the minimum length of the resulting crease.

    Click HERE to see a detailed solution to problem 21.

  • Handouts/Classwork:
    03/15: Exam 2
    ​03/13: Open Review for Exam 2
    03/12: Average Value of a Function
    03/09: FTC Part II
    03/08: FTC Part II 
    03/06: Linear Motion Revisited
    ​03/05: Total Distance 
    03/02: Integration Multiple Choice sheet
    03/01: Do Now , Relationship Between Area Under a curve and the Definite Integral
    02/28: Do Now , Area Approximations
    02/27: Intro to Area, Riemann Sums
    02/26: Additivity
    02/16: !997 Multiple Choice #s 1-15, 17 -19, 21, 22
    ​02/15: Do Now, Differential Equations with initial conditions, Additivity
    02/13, 02/14: Separable Differential Equations
    02/12: Evaluating Definite Integrals
    02/09: Exam 1
    02/08: Open Review for Exam 1
    02/06, 02/07: More Integrating with Trig Functions
    02/05: Antiderivative Practice
    02/01: Integrating Using U substitution, Integration Practice 2
    01/31: More Special Antiderivatives, Rewriting the Integrand
    01/30: Some Special Antiderivatives
    01/29: More Local Linear Approximations, Finding Antiderivatives
    01/26: Linear Approximations
    01/18: Midterm Review Key
    01/12: Exam 3
    ​01/11: Open Review for Exam 3
    01/10: Linear Motion Review 
    01/09: Motion packet 4
    01/08: Motion Packet 2, Motion Packet 3
    01/03: Motion Packet 1
    ​01/02: Mean value Theorem
    12/22: 2012 Multiple Choice #s 1, 2, 4, 5, 7, 9, 11, 14, 16, 18, 19, 20-22, 24, 27, 28
    12/21: Optimization sheet 4 (optional practice)
    12/20: Exam 2
    12/19: Open Review 
    12/18 and 12/20: More Optimization Practice
    12/15: Optimization sheet 3
    12/14: 1991 AB 5 , Optimization sheets 1 and 2 
    12/13: Practice Problems
    12/11, 12/12: 2003 Multiple Choice #s 1, 3, 4, 6, 9, 13-18, 20, 21, 24-28
    ​12/08: EVT graph, Extreme Value Theorem
    12/07: Analysis of Graphs graphic organizer, Additional Notes on Relative extrema, points of inflection ...
    12/06, 12/07: Second Derivative test
    12/05: Analysis of Functions, Testing for relative extrema and points of inflection
    12/04: Analyzing Graphs AP Questions
    12/01: Graphs of Derivatives Packet 2
    11/30: Exam 1
    11/29: Open Review
    11/28: Graphs of Derivatives Packet 1
    11/27: More Related Rates
    11/21: Related Rates Packet 3
    11/16, 11/17, 11/20: Related Rates Packet 2
    11/15: Related Rates Packet 1
    11/14: Related Rates Intro
    11/13: Derivatives of Inverse Functions, Inverse Function Theorem
    11/09: 11-09 Work
    11/08: Quarter Exam Part II 5 multiple choice calculator active questions, 1 free response question (25 minutes)
    11/06, 11/07:  Derivatives of Inverse Trig Functions
    11/03: Quarter Exam Part I (non-calculator) 15 multiple choice questions.  You must manage your time and keep it moving.  
    11/02: Open Review for Quarter Exam
    10/31, 11/01: Implicit Differentiation packet, Multiple Choice Review Questions
    10/30: Implicit Differentiation
    10/27: Derivatives of exponentials and logarithms
    10/26: Differentiability
    10/25: Exam 3
    ​10/24: Open Review for Exam 3
    10/20, 10/23: Calculator Active Questions
    10/19: Even More Derivative Multiple Choice Practice,  Derivative Practice Packet
    10/18: Motion Questions
    10/17: Motion Introduction 
    10/16: Chain Rule Charts
    10/13: More Multiple Choice Derivative Practice​
    10/12: Derivatives of e and ln
    10/11: More Chain Rule with Trig Derivatives Practice
    10/10: Exam 2
    ​10/06: Open Review
    10/05: Practice Multiple Choice 
    10/04: Scavenger Hunt
    10/03: Derivatives of Sine and Cosine , Trig Derivatives 
    10/02: Chain Rule , Chain Rule II
    09/29: Quotient Rule, When Not to Use the Quotient Rule
    09/28: Product Rule
    09/27: Applications of Derivatives
    ​09/26: Higher Order Derivatives
    09/25: Rules for Differentiation
    ​09/20: More Practice with Derivative Definition
    09/15: Limits of Difference Quotients
    09/13: Types of Discontinuity
    09/12: Even More Limits ,  Key to HW # 5
    09/11: More Limits Practice
    ​09/08: ​Trig Limits Practice , Key To Optional Limits Practice Packet
    09/07: Need To Know Trig Stuff 
    09/05: Library of Basic Graphs resources: library_of_basic_graphs_student.pdf , library_of_graphs_textbook.pdf
               Optional Limits Practice Packet  
               Limits Graphically Do Now


    ​​Homework Assignments:
                    Please note: You need to check the answers to the ODD numbered problems in the back of your textbook. The answers to the                                                                                    even numbered problems will be given in class.

    03/14: Study for Exam 2
    ​03/12: HW #84: DUE FRIDAY: Complete this sheet: Average Value HW and for tomorrow, please bring in any questions for Open Review
    03/09: HW #83: 1999 AB 5, 2002 AB 4, and 2002 AB4 Form B from the Using the FTC packet
    ​03/08: HW #82: Complete this sheet: FTC Part II HW . Here is the key to Tuesday's homework: HW #81 key.  Please check your answers and bring in any questions for tomorrow.  Exam 2 will be on Thursday. March 15th.
    ​03/06: HW #81: From the Linear Motion revisited packet: 1987 AB 1, 1991 BC 1, 1997 AB 1, 2006 AB 4 and 1989 AB 3
                                                                                 
                                                                                Topics for Exam 2, Thursday, March 15th:
                                             Evaluating Definite Integrals (including absolute value functions and piecewise functions)
                                                                 Separable Differential Equations (including ones with initial conditions)
                                                                                              Additivity
                                                                 Area Approximations (Riemann sums and Trapezoidal)
                                                                                     Exact Area Under a Curve
                                                                                Motion (total distance as an integral)
                                                                                               FTC Part II

    ​03/05: HW #80: Complete the following 3 sheets: 2003 AB 4 , HW #80 A,  HW  #80 B
    03/01: HW #79: DUE MONDAY: Complete this sheet: HW #79 AND finish the Area Approximations packet #s 4-6 and finish the 10 questions from the Integration Practice sheet.
    02/28: HW #78: From this sheet: Calc HW #78 #s 9-21 odd
    02/27: HW #77: From this sheet, Area Approximation HW, #s 1, 3 and 4  Exam 2: Friday, March 9th 
    02/26: HW #76: Finish the Additivity packet​
    ​02/16:  HW #75:  !997 Multiple Choice #s 1-15, 17 -19, 21, 22
    02/15:  HW #74:  Complete this sheet: Properties of Definite Integrals                                                                                                                                      
    02/13: HW #73: Complete this sheet for Thursday: Separable Diff Eqs HW  I will be adding on to this tomorrow.  You technically should be able to answer most of the problems on the sheet tonight.
    02/12: HW #72: Finish the Evaluating Definite Integrals sheet
    02/08: Study for tomorrow's exam.  Don't forget your calculator.
    02/07: HW #71: Complete this sheet: Even More Integration Practice
    02/06: HW #70: Complete the "More Practice" section of the "More Integration with Trig Functions" packet
                                                                                             Topics for Exam 1, Friday, February 9th
                                                                                                           Linear Approximation
                                                                                                       Integration (including u sub)

    02/05: HW #69: #s 16-30 from Thursday's homework sheet and #s 1, 2, 4-6, and 10 from the Practice with Antiderivatives sheet
    02/01: HW #68: #s 1-15 from the Integration Practice sheet AND p.342 Section 7.2 Exercises #s 1-6  Exam 1: Friday, February 9th
    01/31: HW #67: Complete this sheet: Linear Approx/Integration Practice
    01/30: HW #66: Complete the numbers specified on the top of the back of the Some Special Antiderivatives sheet
    ​01/29: HW #65: Complete #s 4-6, 8 from the More Local Linear Approximations sheet AND #s 7-17 odd from the "More Practice Exercises" section of the Finding Antiderivatives packet
    01/26: HW #64: Complete #s 2 and 3 from the Linear Approximation sheet
    01/19:Study for Midterm.  Midterm Exam:Tuesday, January 23rd 12;15-1:45 cafe.  Midterm Extra Help: Monday, January 22nd 10:30-12 Math Study center.
    01/18: Bring in any last minute questions for the Midterm Exam.  Here is the key to the packet:Midterm Review Key 
    01/17: HW #63: Complete the "Selected Questions from Anton packet" section of the Midterm packet.
    ​01/16: HW #62: Complete the "Other Practice Questions" section of the Midterm packet (#s 1-18)  Here is something that you might find helpful in your preparation, add to it as needed: Midterm Notes
    01/11: Study for Exam 3.  Here is the midterm review sheet if you want to start working on it. Calculus Midterm packet
    01/10: Bring in questions for tomorrow's open review
    01/09: HW #61:  Complete these questions Linear Motion AP problems
    01/08: HW #60: From this sheet #s 11-14  parts (d)
                                                                                           Topics for Exam 3, Friday, January 12th
                                                                                                            Optimization
                                                                                                       Mean Value Theorem
                                                                                                  Intermediate Value Theorem
                                                                                                                  Motion

    01/03: HW #59: From this sheet # 12 c and from this sheet #s 11-14 skipping parts (d)
    01/02: HW #58:  From this sheet #s 12-16 even, 19 and 20.  Key to Min/Max sheet 4.  Key to 2012 Multiple Choice
    12/22: HW #57:  Complete 2012 Multiple Choice #s 1, 2, 4, 5, 7, 9, 11, 14, 16, 18, 19, 20-22, 24, 27, 28.  Enjoy break!!!
    12/21: HW #56:  Complete #s 4 and 5 from the More Optimization sheet and ​from 2012 Multiple choice #s 76-78, 80, 82, 84, 85, 87, 88, 91
    12/19: Study for tomorrow's exam.
    ​12/18: HW #55: DUE THURSDAY Questions 2, 6 and 7 from the More Optimization sheet.  ​FOR TOMORROW, please bring in questions for our open review.
                                                                                Topics for Exam 2, Wednesday, December 20th:
                                                                                                  Graphs of Derivatives 
                                                                                                  Analysis of Functions
                                                  Finding relative and global extrema, points of inflection, changes in concavity, increasing, decreasing...
                                                                                              

    12/15: HW #54: Complete #s 1, 4 and 5 from Min/max sheet 3
    ​12/14: HW #53: Complete the next 3 problems in the packet (#s 1, 2 and 3 from Min/max sheet 2)
    12/13: HW #52: Finish the Practice Problems packet from class
    12/12: HW #51: From the 2003 Multiple Choice #s 87, 89 and these 2 questions HW
    12/11: HW #50: From the 2003 Multiple Choice #s 1, 3, 4, 6, 9, 13-18, 20, 21, 24-28
    12/08: HW #49: Extreme Value Theorem Homework
    12/07: HW #48: Second Derivative Test Homework
    12/06: HW #47: Complete this sheet: Practice Problems
    12/05: HW #46: First Derivative Test and Finding Points of Inflection HW
    12/04: HW #46: Finish the Analyzing Graphs packet skip 2000 AB 3 part d, 1979 and Review book question part e
    12/01: HW #45: Complete this sheet: HW #45 
    11/29: Study FYI: Exam 2 will be on Wednesday, December 20th
    ​11/28: HW #44 DUE FRIDAY Complete pages 5 and 8 from the Graphs of Derivatives Packet 1 AND for tomorrow bring in questions for open review.
    11/27: HW #43: pp. 256-257 #s 19-22
    11/22: HW #42: Finish 1995 AB 5 Enjoy your Thanksgiving!  Start preparing for the Exam on the 30th
    ​11/21: HW #41: From the Related Rates packet, 1985 AB 5, 1988 BC 3, and 1990 AB 4
                                                                              
    ​                                                                           Topics for Exam 1, Thursday, November 30th:
                                                                                         Derivatives of Inverse Trig Functions
                                                                                           Derivatives of Inverse Functions
                                                                                                      Related Rates
                                                                                              *** Implicit Differentiation*** (I know we hit this on the last exam but revisiting it here)
                                                                                               

    11/20: HW #40: Finish the Related Rates packet 2
    ​11/17: HW #39: From Related Rates Packet 2 #s 5, 6a, 9 and 11 Exam 1 Thursday, November 30th
    11/16: HW #38: From Related Rates Packet 2 #s 1, 2 and 3 AND p. 169 Quick Quiz #s 2, 4
    11/15: HW #37: Finish #s 1-6 from the Related Rates Packet 1
    11/14: HW #36: Complete the Homework section of the Related Rates Intro packet
    11/13: HW #35: #s 4-7, 9 and 10 from the Inverse Function Theorem sheet
    11/09: HW #34: Complete the "11-09 Work" sheet
    11/07: HW #33: Study for Quarter Exam Part II. Remember HW #32 is due tomorrow.
    ​11/06: HW #32: Complete this sheet Derivatives of Inverse Trig Functions Homework due Wednesday
    11/03: HW #31: Complete this sheet More Calculator Active Questions
    11/02: Study for Quarter Exam Part I.  Remember this part has 15 non calculator questions.
    11/01: Bring in questions for tomorrow's Open Review and DUE MONDAY: HW #31: From the Implicit Differentiation packet #s 3, 4, 5, 6, 12

                                                                  New Topics for Quarter Exam, Friday, November 3rd
                                                                                           Differentiability
                                                                                 Derivatives of logs and exponentials
                                                                                        Implicit Differentiation
                                                                                   Older Topics for Quarter Exam

                                                                                     Formal Definition of Derivative
                                                                             More Chain Rule (including chain rule charts)
                                                                      Derivatives of the natural exponent and the natural log
                                                                                                        Motion

                                             Derivative Rules (power rule, product rule, quotient rule, reciprocal rule, chain rule)
                                                                                                  Trig Derivatives
                                                                                          Applications of Derivatives
                                                                                           Higher Order Derivatives

                                                                                                        Limits
                                                                                                      Continuity


    10/30: HW #30: Complete the Homework section of the Implicit Differentiation packet
    10/27: HW #29: Complete this sheet: Derivatives of exp and logs hw
    10/26: HW #28: Complete this sheet 10/26 HW
    10/24: Study!!!!!!
                                                                          Topics for Exam 3, Wednesday, October 25th:
                                                                                    Formal Definition of Derivative

                                                                             More Chain Rule (including chain rule charts)
                                                                      Derivatives of the natural exponent and the natural log
                                                                                                        Motion
                                                                                 Remember Calculus is cumulative by nature.
    ​                                      
    10/23: For tomorrow bring in questions for Open Review and DUE THURSDAY HW #27, complete this sheet Calculator Active Questions HW
    10/19: HW #26: DUE MONDAY  Complete Set E from the Derivative Practice packet
    10/18: HW #25: Finish questions 71 and 71 from the sheet that you received today.  
    ​10/17: HW #24: Complete #1 on the back page of the Motion Introduction packet and p. 159 #s 56 and 58
    10/16: HW #23:  Complete the back side of the Chain Rule Charts sheet skipping #s 7,11 and 13.  Tentative remaining exam dates for Q1 are posted on Announcements page.  
    10/13:  HW #22: Complete the More Multiple Choice Derivative Practice packet
    ​10/12:  HW #21: Complete this sheet: Derivatives of e and ln Homework
    10/11:  HW #20: Complete this sheet: More Trig Derivatives Homework
    ​10/06:  Study!!!!!!!!!!!!!!!!!!!!
    ​10/05:  HW #19: Complete #s 2-22 even from this sheet: Derivative Practice
    10/04:  HW #18: Complete this sheet: Some Practice Derivative Multiple Choice Questions 
    ​                                                                          Topics for Exam 2: Tuesday, October 10th:
                                             Derivative Rules (power rule, product rule, quotient rule, reciprocal rule, chain rule)
                                             Trig Derivatives
                                             Applications of Derivative 
                                             Higher Order Derivatives
    ​​
    10/03: HW #17: Complete this sheet: Trig Derivatives Homework
    10/02: HW #16: Complete this sheet: Chain Rule Homework. Also please use the link below to sign up for the Remind notifications.  If you do not do so then that will count as an incomplete homework.
    09/29: HW #15: Complete this sheet: Quotient Rule Homework *there are textbook problems listed on the bottom of the sheet, complete those as well*  Also please sign up for the Remind app for this class by clicking here
    09/28: HW #14:Complete the Homework section of the Product Rule packet
                         **  Exam 2 Tuesday, October 10th** 
    09/27: HW #13:  Complete this sheet: Applications of Derivatives Homework
    09/26: HW #12:  Complete this sheet: Higher Order Derivative Homework
    09/25: HW #11: p. 105 Quick Review #s 1-4 and pp.124-125 #s 1-7, 11, 12, 53, 54
    09/20: HW #10: pp. 105-106 #s 1-4, 17-20
    09/18: Study for tomorrow's exam and remember HW #9 is due Wednesday
    ​09/15: HW #8:  The Do Now section of the Limits of Difference Quotient sheet (due Monday) and HW #9: The Classwork section of the Limits of Difference Quotients sheet (due Wednesday)
    09/13: HW #7: From the textbook  p.77 Quick Quiz #s 1-3 pp. 84-85 #s 3-16, 47-50, 59  p. 95 #2 (quick quiz section)
    09/12: HW #6: Complete the Even More Limits sheet
    09/11: HW #5: From the More Limits Practice sheet #s 1-4, 5-49 odd and 55
                                       **  Exam 1 Tuesday, September 19th** 
                                                                Topics:
                                                Limits (all types we discussed)
                                                Continuity
                               
    09/08: HW #4: Complete the Trig Limits Practice sheet
    ​09/07: HW #3: Complete page 5 of the Optional Limits Practice packet
    09/06: HW #2: Complete this sheet: Limits Review B
    09/05: HW #1: Please get course policies sheet signed for tomorrow.


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