aim: optimization problems course: calculus do now: aim: so, what is it this calculus thing can...

Aim: Optimization Problems Course: Calculus

Do Now:

Aim: So, what is it this calculus thing can really do to solve problems?

….. greatest profit

….. least cost

….. largest

….. smallest

Write a function based on two equations.


Finding Minimum

A manufacturing company has determined that the total cost of producing an item can be determined from the equation C = 8x2 – 176x + 1800, where x is the number of units that the company makes. How many units should the company manufacture in order to minimize the cost?

16 176 0dC

xdx

2

2 16d C

dx

11x

Find critical values of x

Looking for minimum: 2nd D. must be > 0

Must manufacture 11 units to min. costs

since C is a quadratic this 2nd

D. is always +


Finding Maximum

A rocket is fired into the air, and its height in meters at any given time t can be calculated using the formula h(t) = 1600 + 196t – 4.9t2. Find the maximum height of the rocket and at which it occurs.

196 9.8 0dh

tdx

Find critical values of x

Looking for maximum: 2nd D. must be < 02

2 9.8d h

dx

20t

h(20) = 1600 + 196(20) – 4.9(20)2 = 3560 m.

since h(t) is a quadratic this 2nd D. is always -


Finding Maximum Volume

A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

xx

h

V lwh2V x h

primary equation – contains the quantity to be optimized

Surf. Area = (area of base) + area of 4 sides2 4SA x xh = 180

maximize volume

secondary equation


Finding Maximum VolumeA manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

xxh

2V x h2 4SA x xh = 108

1. maximize Volume2. express V as a function of one variable

2 4 108x xh

solve for h in terms of x2108

4

xh

x

replace h in primary equation2

2 108

4

xV x

x

2 4 310827

4 4

x x xx

x



3

274

xV x

15

10

5

-5

-10

-15

-0.5 0.5

domain for function is all reals, but . . .

we must find the feasible domain

x must be > 0

Area of base = x2 can’t be > 108

feasible domain 0 108x



3

274

xV x

to maximize – find critical values23

274

dV x

dx = 0 x = ±6

evaluate V at endpoints of domain and 6

30

(0) 27 0 04

V 36

(6) 27 6 1084

V

3

108(6) 27 108 0

4V

xxh

663


Problem Solving Strategy

1. Assign symbols to all given quantities and quantities to be determined. Sketch

2. Write a primary equation for the quantity that is to be optimized.

3. Reduce primary equation to one having a single independent variable. This may involve the use of a secondary equation relating the independent variables of the primary equation.

4. Determine the feasible domain of the primary equation.

5. Use calculus to optimize


Finding Minimum Distance

Which points on the graph of y = 4 – x2 are the closest to the point (0, 2)?

4

2

-2

-4

-6

f x = 4-x2

(x, y)

2 2

2 1 2 1d x x y y

2 20 2d x y

(0, 2)primary equation

y = 4 – x2

secondary equation



Which points on the graph of y = 4 – x2 are closest to the point (0, 2)?

4

2

f x = 4-x2

(x, y) 2 2

0 2d x y

(0, 2)

primary equation

y = 4 – x2 secondary equation

22 2(4 ) 2d x x rewrite w/one independent

4 23 4d x x d is smallest when radicand is smallest

f(x) = x4 – 3x2 + 4 = 0



Which points on the graph of y = 4 – x2 are closest to the point (0, 2)?

f’(x) = 4x3 – 6x = 0

find critical numbers3

0,2

x

6

4

2

x = 0 is a relative maximum

(x, y)

(0, 2)

3are relative minima

2x

evaluate y = 4 – x2

for x =3

2

3 5,

2 2

min. distance from (0, 2)

f(x) = x4 – 3x2 + 4 = 0


Model Problem

Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area?

Maximize what? A of + A of 2 2A r x primary equation

2 4 4P r x secondary equation

rewrite w/one independent

2 1 xr

2

2 2 1 xA x

solve for r:


Model ProblemFour feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area?

2

2 2 1 xA x

22 4(1 )x

x

2 2 24(1 ) ( 4) 4 8x x x x

21( 4) 4 8x x

feasible domain? 0 < x < 1


find critical numbers

Model ProblemFour feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area?

2 4 80

xdA

dx

0 < x < 1 domain

4.56

4x

is only critical value in domain

A(0) 1.273 A(.56) .56 A(1)= 1

maximum area occurs at x = 02 2A r x

evaluate primary equation

max when all wire is used for circle!

+ 0


Model Problem

You are planning to close off a corner of the first quadrant with a line segment 15 units long running from (x, 0) to (0, y). Show that the area of the triangle enclosed by the segment is largest when x = y.


Model Problem

Find the points on the hyperbola x2 – y2 = 2 closest to the point (0, 1).


Do Now:

Aim: So, what is it this calculus thing can really do to solve problems?

A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

aim: optimization problems course: calculus do now: aim: so, what is it this calculus thing can...

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