Calculus Applications Math Studies 1. a)Find the local extrema and identify them as either a local...
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Transcript of Calculus Applications Math Studies 1. a)Find the local extrema and identify them as either a local...
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
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