1 applications of the calculus the calculus is a mathematical process with many applications. of...
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Applications of the Calculus
The calculus is a mathematical process with many applications.
Of interest are those aspects of calculus that enable us to calculate the maximum and minimum value of a function.
For example, the maximum value of this function is to be calculated: xxy 12003 2
200
321200
2
ab
xWe know this occurs at
2
x
y
200
tangents
The slope of tangents to any curve can be calculated using differentiation.
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At the functions maximum value, the tangent is horizontal and so its slope is 0.
This is true for all maximum and minimum values.
Using calculus then, involves searching for the places where the slope of the tangent is zero.
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Rules for Differentiation
Notation: The derivative of a function is denoted by
dxdy y xf
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nxy .1
3 . xyeg
cy .2
3 . yeg
1 nnxy
0 y
23 xdxdy
0 y
Rules for Differentiation
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xcfy .3
25 .1 xyeg
56
.2 x
yeg
xfcy
xxdxdy
1025 1
56
156
0 xdxdy
Rules for Differentiation
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xxfy g .4
xxyeg 2 .1
24 24 .2 xxyeg
33 416 xx
33 2244 xxy
xgxfy
12 xy
Rules for Differentiation
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Special Cases:
axey .5
xy ln .6
axaey
xy
1
Rules for Differentiation
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xxy 12003 2
120061120023 So 01 xxxy
represents the slope of the tangent to the curve y.y
1200 ,0 . yxeg
60012001006 ,100 yx
The introductory example has
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When the tangent is horizontal (slope = 0) then .0y
120060 x
2006
1200 x
The maximum occurs at x = 200.
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Using differentiation to find the maximum and minimum points is a more general method in that it “works” for all curves.
Using is a shortcut method that works
for quadratic equations (parabolas) only.ab
x2
The equation of the tangent is not difficult to determine. At a point on a curve, the slope is calculated using differentiation and the result is used in the equation 11 xxmyy
11, yx
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Example: Find the equation of the tangent to the curve when x = 2.
When x = 2,
and the point on the curve is (2,9)
523 xxy
9522 23 y
xxy 23 2
822232 2 y
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So using 11 xxmyy
289 xy
1689 xy
78 xy
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Example: Find all the points on the curve 123 xxxxf
where the slope is 6.
123 2 xxxf
6 xf5230 2 xx
5 ,2 ,3 cba
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53422 2 x 1or 35 x
1236 2 xx
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Example: Find the derivative of the following function
23 xxy
962 xxxy
xxxy 96 23
9123 2 xxy
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Example: Find the derivative of the following function
xx
xf2
xx
xxf
2
121 xxf
22 212 xxxf2
2or x
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Example: Find the derivative of the following function
xxx
y32
21
32 xxxy
212 62
xxxy
21
23
62 xxy
2
12
1
21
623
2 xxy
xxxxy
3333 2
12
1
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The Derivative as a Measure of Rate of Change
The derivative as a measure of the gradient of a tangent to a curve has been discussed.
A second interpretation of this process describes the rate of change of some variable.
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Consider the following physics example:
A car’s position (s) is described by the equation , where s is in metres and t is in seconds.
2ts
After 2 seconds, (t = 2), .422 s
The car is 4 metres from some starting point.
At t = 10, s = 102 = 100 and so on.
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Looking more closely at the case when t = 10, s = 100, the question is asked “How fast was the car travelling?”
One answer is 100m in 10 seconds equals 100/10 = 10 m/s. This describes the average velocity (velocity is a measure of the rate of change of position(s) ). But as with any car journey, the velocity is always changing.
Differentiation allows us to calculate the velocity at any time instant.
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dtds
sv or Velocity
tdtds
ts 2 then If 2
smdtds
t 422 ,2 Then when
smdtds
t 20102 ,10when
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t (secs) s (m)
1 1 2
3 9 6
5 25 10
7 49 14
10 100 20
v s m s
tdtds
svts 2 ,2
Average velocity = 100/10 = 10 m/s
The derivative describes the rate of change of position (instantaneous velocity) at any time.
In summary:
timetotaldistance total
velocityaverage
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Consider the following situation:
The total cost function is given by
20.5 10,000 1000C q q
1000 ,0 Cq
210, 0.5 10 10,000 10 1000
$100,950
q C
2
1000, 0.5 1000 10,000 1000 1000
$9,501,000
q C
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The average cost for producing q items is given by
( is the usual symbol for average)
That is,
Cc
q
100,95010, $10,095 per item
10q c
9,501,0001000, $9501 per item
1000q c
produced items no. totalcost total
cost average
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How can the derivative be used and interpreted?
describes how quickly cost is changing
A comparison shows that cost is changing at a faster rate when q = 10 than when q = 1000.
2If 0.5 10,000 1000C q q then 1 10,000C q
C
10 1 10 10,000 9990C
1000 1 1000 10,000 9000C
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These figures also give an approximate cost to produce the next item.
The 11th item will cost approximately $9990 to produce.
The 1001st item will cost approximately $9000 to produce.
When C describes the total cost, then represents the marginal cost.
.999010 C
.90001000 C
dqdC
C or
C is called the marginal cost function.
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Marginal cost function:
Marginal cost
77005.46.003.0 23 qqqC
5.42.109.0 2 qqdqdC
5.15.4102.11009.0 ,10 2 dqdC
q
5.165.4202.12009.0 ,20 2 dqdC
q
5.7845.41002.110009.0 ,100 2 dqdC
q
Example:
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Total cost function
Marginal cost function:
2 70000.001 0.3 40c q q
q
CC
c cqq
7000403.0001.0
7000403.0001.0
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2
qqq
qqC
406.0003.0 2 qqdqdC
Example:
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406.0003.0 2 qqdqdC
3.3440106.010003.0 ,10 2 dqdC
q
2.2940206.020003.0 ,20 2 dqdC
q
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There is an equivalent interpretation for the revenue function.
If R represents total revenue then is called the marginal revenue.
Marginal revenue describes two things:
(1) How quickly revenue is changing.
(2) Approximate revenue received by selling the next unit.
dqdR
R
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Example:
Marginal revenue:
22.0601.0302 qqqqR
qdqdR
4.060
56104.060 ,10 dqdR
q
52204.060 ,20 dqdR
q
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The following conclusions can be drawn from these results:
(1) Revenue is changing faster at q = 10 than at q = 20.
(2) The 11th item will generate approximately $56 and the 21st item will earn $52