Using Calculus to Solve Optimization Problems

Slides:



Advertisements
Similar presentations
Calculus Applications Math Studies 1. a)Find the local extrema and identify them as either a local maximum or a local minimum. b)Find the coordinates.
Advertisements

problems on optimization
Section 5.4 I can use calculus to solve optimization problems.
Lesson 2-4 Finding Maximums and Minimums of Polynomial Functions.
To optimize something means to maximize or minimize some aspect of it… Strategy for Solving Max-Min Problems 1. Understand the Problem. Read the problem.
Volume and surface area of solids
3.7 Optimization Problems
Pre – CalcLesson 2.4 Finding Maximums and Minimums of Polynomial Functions For quadratic functions: f(x) = ax 2 + bx + c To fin d the max. or min. 1 st.
Applications of Differentiation
A rectangular dog pen is constructed using a barn wall as one side and 60m of fencing for the other three sides. Find the dimensions of the pen that.
Sec 2.5 – Max/Min Problems – Business and Economics Applications
Optimization Practice Problems.
4.6 Optimization The goal is to maximize or minimize a given quantity subject to a constraint. Must identify the quantity to be optimized – along with.
Volume of Rectangular Prism and Cylinder Grade 6.
Limits “at Infinity”.  Deal with the end behavior of a function.
Section 3.7 – Optimization Problems. Optimization Procedure 1.Draw a figure (if appropriate) and label all quantities relevant to the problem. 2.Focus.
Optimization Problems
Perimeter and Area. Common Formulas for Perimeter and Area Square Rectangle s l s w A = lw P = 4sP = 2l + 2w Perimeter and Area of Rectangle.
Area of a Parallelogram Area of a Triangle Circumference & Area of a Circle.
4.4 Modeling and Optimization What you’ll learn about Examples from Mathematics Examples from Business and Industry Examples from Economics Modeling.
Lesson 4.4 Modeling and Optimization What you’ll learn about Using derivatives for practical applications in finding the maximum and minimum values in.
Applied Max and Min Problems Objective: To use the methods of this chapter to solve applied optimization problems.
4.7 Applied Optimization Wed Jan 14
CHAPTER 3 SECTION 3.7 OPTIMIZATION PROBLEMS. Applying Our Concepts We know about max and min … Now how can we use those principles?
Applied Max and Min Problems
1.Tim is painting his living room with a new coffee colored Paint. There are 3 walls in the living room that measure 15 ft by 8 ft each and a fourth wall.
{ ln x for 0 < x < 2 x2 ln 2 for 2 < x < 4 If f(x) =
Do Now: ….. greatest profit ….. least cost ….. largest ….. smallest
Linearization , Related Rates, and Optimization
Section 4.4 Optimization and Modeling
Volume word problems Part 2.
Calculus and Analytical Geometry
AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.4:
Agriculture Mechanics I.  Square measure is a system for measuring area. The area of an object is the amount of surface contained within defined limits.
Optimization Section 4.7 Optimization the process of finding an optimal value – either a maximum or a minimum under strict conditions.
VOLUME. So far, we have learned about length. This is a measure of 1 dimension.
Perimeter, Area, and Volume Geometry and andMeasurement.
Optimization Problems
Sullivan Algebra and Trigonometry: Section R.3 Geometry Review Objectives of this Section Use the Pythagorean Theorem and Its Converse Know Geometry Formulas.
College Algebra Section R.3 Geometry Review Objectives of this Section Use the Pythagorean Theorem and Its Converse Know Geometry Formulas.
Optimization Problems Section 4.5. Find the dimensions of the rectangle with maximum area that can be inscribed in a semicircle of radius 10.
Extra Optimization Problems “Enrichment Problems”.
Section 4.5 Optimization and Modeling. Steps in Solving Optimization Problems 1.Understand the problem: The first step is to read the problem carefully.
1. The sum of two nonnegative numbers is 20. Find the numbers
Optimization Problems
Finding Maximum and Minimum Values. Congruent squares are cut from the corners of a 1m square piece of tin, and the edges are then turned up to make an.
Optimization. First Derivative Test Method for finding maximum and minimum points on a function has many practical applications called Optimization -
Section 4.7. Optimization – the process of finding an optimal value- either a maximum or a minimum under strict conditions Problem Solving Strategy –
Optimization Problems Section 4-4. Example  What is the maximum area of a rectangle with a fixed perimeter of 880 cm? In this instance we want to optimize.
Back to menu Final jeopardy question Definitions The Round Let’s Cover Fill It The Whole Up It Up Thing
A25 & 26-Optimization (max & min problems). Guidelines for Solving Applied Minimum and Maximum Problems 1.Identify all given quantities and quantities.
4.4 Modeling and Optimization, p. 219 AP Calculus AB/BC.
Calculus 3-R-b Review Problems Sections 3-5 to 3-7, 3-9.
Ch. 5 – Applications of Derivatives 5.4 – Modeling and Optimization.
Optimization Problems
Sect. 3-7 Optimization.
Ch. 5 – Applications of Derivatives
GEOMETRY REVIEW.
5-4 Day 1 modeling & optimization
Honors Calculus 4.8. Optimization.
Applied Max and Min Problems
Optimization Chapter 4.4.
Chapter 5: Applications of the Derivative
4.6 Optimization The goal is to maximize or minimize a given quantity subject to a constraint. Must identify the quantity to be optimized – along with.
3.7 Optimization Problems
Using Calculus to Solve Optimization Problems
Optimization (Max/Min)
9.4 – Perimeter, Area, and Circumference
4.6 Optimization Problems
Presentation transcript:

Using Calculus to Solve Optimization Problems Section 5.4 Using Calculus to Solve Optimization Problems 5.3 Pick up packet out of folder

1. The sum of two nonnegative numbers is 20. Find the numbers (a) if the sum of their squares is to be as large as possible. Let the two numbers be represented by x and 20 – x. makes x = 10 a minimum. Maximum must occur at an endpoint. 0 and 20

1. The sum of two nonnegative numbers is 20. Find the numbers (b) If the product of the square of one number and the cube of the other is to be as large as possible Let the two numbers be represented by x and 20 – x. 12 20 + _ Max at 12, Min at 20 12 and 8

1. The sum of two nonnegative numbers is 20. Find the numbers (c) if one number plus the square root of the other is as large as possible. Let the two numbers be represented by x and 20 – x. therefore a max

A rectangular pen is to be fenced in using two types of fencing. Two opposite sides will use heavy duty fencing at $3/ft while the remaining two sides will use standard fencing at $1/ft. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a total cost of $3600? 3x 1y Therefore max The dimensions of a rectangular plot of greatest area are 300 x 900

3. A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. With 800 m of fence at your disposal, what is the largest area you can enclose? x y Therefore a max The largest area you can enclose is 80000

4. An open-top box with a square bottom and rectangular sides is to have a volume of 256 cubic inches. Find the dimensions that require the minimum amount of material. y x therefore a min 8 x 8 x 4

5. Find the largest possible value of 2x + y if x and y are the lengths of the sides of a right triangle whose hypotenuse is units long. x y 2 + _ Therefore x = 2 is a max

6. A right triangle of hypotenuse 5 is rotated about one of its legs to generate a right circular cone. Find the cone of greatest volume. x y 5 Therefore max

Determine the area of the largest rectangle that may be Inscribed under the curve 1 _ + Therefore max

Since f’ changes from neg to pos, we have a minimum 8. (calculator required) A poster is to contain 100 square inches of picture surrounded by a 4 inch margin at the top and bottom and a 2 inch margin on each side. Find the overall dimensions that will minimize the total area of the poster. 4 2 x y Since f’ changes from neg to pos, we have a minimum

9/2 + _ Therefore min

+ _ Therefore max

Since f’ changes from pos to neg, we have a maximum (calculator required) Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10. Since f’ changes from pos to neg, we have a maximum

Max at x = 7.46 since f ‘ changes from pos to neg At 7.46, r = 358.08, c = 194.34, or P = 163.14 Max profit is $163,000 which occurs when 7460 units are made

A tank with a rectangular sides is to be open at the top. It is to be constructed so that its width is 4 m and its volume is 36 cubic meters. If building the tank costs $10 per square meter for the base and $5 per square meter for the sides, what is the cost of the least expensive tank? y x 4

CALCULATOR REQUIRED Minimum since f ‘ (x) changes from neg to pos at –0.426

Since A ‘ changes from neg to pos, min area at t = 2

+ _ Therefore max

Since A ‘ changes from pos to neg at x = 0.860, max of A occurs at x = 0.860

Consider the set of all right circular cylinders for which the sum of the height and diameter is 18 inches. What is the radius of the cylinder with the maximum volume? X

_ +

Two possibilities where CALCULATOR REQUIRED Two possibilities where D ‘ changes from neg to pos, denoting a min

22. Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius 5. r 0.5h R h – height of cylinder r – radius of cylinder R – Given radius of sphere Therefore a max