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 of the absolute maximum and absolute minimum of the function in the interval

2060045)( 23 xxxxf

30,0

600903)(' 2 xxxf

6009030 2 xx

200300 2 xx

)20)(10(0 xx

First, differentiate.

Set f’(x) to 0

Solve for x

Factor

Hence, we have local extrema at x = 10 and x = 20

To identify them as either maxima or minima, we can use the derivative -

)9('f 33600)9(90)9(3 2

)11('f 27600)11(90)11(3 3

To identify them as either maxima or minima, we can use the derivative -

)19('f 27600)19(90)19(3 2

)21('f 33600)21(90)21(3 3

b) Check the endpoints, x = 0 and x = 30

)0(f 2020)0(600)0(45)0( 23

)30(f 452020)30(600)30(45)30( 23

2. A rectangular pen is to be fenced in using two types of fencing. Two opposite sides will use heavy duty fencing at $3/ftwhile the remaining two sides will use standard fencing at $1/ft.What are the dimensions of the rectangular plot of greatest areathat can be fenced in at a total cost of $3600?

3x

1y

A xy 2 3x 2 1y 3600 3x y 1800

A x 1800 3x 2A 1800x 3x

A ' 1800 6x 0 1800 6x

300 x

A " 6Therefore max

900 y

The dimensions of a rectangular plot of greatest area are 300 x 900

y = 1800-3x

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.

2

2

S x 4xy

V x y 256

2

2

256S x 4x

x

2

2

2

1024S x

x1024

S' 2xx

10240 2x

xx 8

y 4

3

2048S" 2 0

x

therefore a min

8 x 8 x 4

yx

x

y = 256/x²

Solve this problem.