Date post: | 02-Jan-2016 |
Category: |
Documents |
Upload: | victor-sanders |
View: | 221 times |
Download: | 0 times |
1
Chapter 2Differentiation: Basic Concepts
In this Chapter, we will encounter some important concepts.
The Derivative
Product and Quotient Rules, Higher-Order Derivatives
The Chain Rule
Marginal Analysis, Implicit Differentiation.
1.1. Constant Rule: Constant Rule: derivative of a constant is zeroderivative of a constant is zero
2.2. Power Rule: Power Rule: for any real n, for any real n,
3.3. Constant Multiple Rule: Constant Multiple Rule: for constant c and a for constant c and a differentiable function f(x), differentiable function f(x),
4.4. Sum Rule: Sum Rule: when f(x) and g(x) are both differentiable,when f(x) and g(x) are both differentiable,
0cdx
d
Review: Techniques of Differentiation
1][ nn nxxdx
d
)()( xfdx
dcxcf
dx
d
)]([)]([)]()([ xgdx
dxf
dx
dxgxf
dx
d
Exercise
3
2
23
5141
13
x
xy
x
xy
Solution
Exercise
Find the equation of the line that is tangent to the graph of the function at the point (-1,-8).
135 23 xxxy
Suggested Solution:
Suppose the equation of the tangent line is
m= at the point (-1,-8).
So at the point (-1,-8), Hence, , and therefore -8=(-10)(-1)+b, gives b=2.The equation of the tangent line to this function at point (-1,-8)
is
3103
135
2
23
xxdx
dym
xxxy
103)1(*10)1(*3 2 m
bmxy tt
dx
dy
bxy tt 10
210 tt xy
1.1. Relative rate of change:Relative rate of change:
2.2. Percentage rate of change:Percentage rate of change:
Review:
Q
dxdQ
xQ
xQ
Q(x)
/
)(
)(
of change
of rate Relative
%)(
)(100
)( of change of
rate Percentage
xQ
xQ
xQ
Exercise1.Find the relative rate of change of with respect to x for the value x=1.
452)( 23 xxxf
2.The gross annual earnings of a certain company were thousand dollars t years after its formation in 2004.
(1)At what rate were the gross annual earnings of the company growing with respect to time in 2008?(2)At what percentage rate were the gross annual earnings of the company growing with respect to time in 2008?
20101.0)( 2 tttA
1.Suggested Solution: The function is Its rate of change is
Its relative rate of change is
When x=1,
41
4
42*51*2
1*101*6
)1(
)1('
452
106
)(
)('
106)('
452)(
23
2
2
23
f
f
xx
xx
xf
xf
xxxf
xxxf
2.Suggested Solution:
(1) , which is the rate that company’s gross
annual earning changes (G.A.E) with respect to t years after 2004.
In 2008, t=4, the G.A.E will change at A’(4)=0.2*4+10=10.8 thousand dollars per year.
(2)The percentage rate of that of the company’s G.A.E changes is expressed as . So in year 2008, this value is
This means the company’s G.A.E increases 17.53% per year in the year 2008.
102.0)('
20101.0)( 2
ttA
tttA
%100*)(
)('
tA
tA
%53.17%100*204*104*1.0
8.10%100*
)4(
)4('2
A
A
Review:Product Rules: For function y=f(x) and z=g(x). If they are both
differentiable at x, then the derivative of their product is
Quotient Rules: For function y=f(x) and z=g(x). If they are both differentiable at x, then the derivative of their quotient is
iff z=g(x) ≠ 0
dx
dzyz
dx
dyzy
dx
d
xgxfxgxfxgxf
**)*(
)(')()()(')]'()([
2
2
**)(
)]([
)(')()()(']'
)(
)([
z
ydxdz
zdxdy
z
y
dx
d
xg
xgxfxgxf
xg
xf
Review:Second Order Derivative: is the derivative of the first derivative.
Notation as follow:
Example: Function S(t) PositionFirst Order Derivative S’(t)=V(t) VelocitySecond Order Derivative S”(t)=V’(t)=A(t) Acceleration
2
2
)('' )('dx
fdxf
dx
dfxf
Exercise:An efficiency study of the morning shift at a certain
factory indicates that an average worker arriving on the job at 8:00am will have produced units t hours later.
(a)Compute the worker’s rate of production.
(b)At what rate is the worker’s rate of production changing with respect to time at 9:00am?
ttttQ 158)( 23
Suggested Solution(a)Worker’s rate of production is
(b) The rate that worker’s rate of production changes is
So at 9:00am, t=1
hourper units 15163)(')( 2 tttQtP
hourper hour per units 166)(")(' ttQtP
hourper hour per units 10166)1(")1(' QP
Review: Chain Rule: For functions y=f(u), which is differentiable at
u, and z=g(x), which is differentiable at x.
The composition function y=f(z)=f(g(x)) is differentiable at x, which is
!!! Pay Attention to the phrase “WITH RESPECT TO”
)('*))(('))]'(([ xgxgfxgfdx
dz
dz
dy
dx
dy
Exercise Differentiate
Suggested Solution
12
13)(
x
xxf
2/32/1
22
)12()13(2
5)('
rule,chain by then
)12(
5
)12(
)13(2)12(3 and
1213
2
1
2
1then
12
13 ,
12
13)(
xxdx
du
du
dy
dx
dyxf
xx
xx
dx
dux-
xudu
dyx
xuwhereu
x
xyxf
Review:Marginal Cost: If C(x) is the total cost of producing x units of a commodity.
Then the marginal cost of producing units is the derivative ,
which approximates the additional cost incurred when the level of production is increased by one unit, from to
Marginal Revenue :
Marginal Profit :
Review:Marginal Cost: If C(x) is the total cost of producing x units of a commodity.
Then the marginal cost of producing units is the derivative ,
which approximates the additional cost incurred when the level of production is increased by one unit, from to
Marginal Revenue :
Marginal Profit :
10 x
)( 0xC
)()1( 00 xCxC
0x
0x
)(' 0xR
)(')(')(' 000 xCxRxP
Review:Approximation by Increment: If y=f(x) is differentiable
at , and △x is a small change in x, then
Approximation Percentage of change: if △x is a small change in x, then
xxfxfxxff
xxfxfxxf
)(')()(
)(')()(
000
000
oxx
)(
)('100
)(100in change Percentage
xf
xxf
xf
ff
Exercise
Use Incremental Approximation to approximate the value of 1.00380. Remember f(x+x) – f(x) = f f’(x)x
2
2
)0('')0(')0()( x
fxffxf
Suggested solution
• Let f(x) = x80, x=1, x = 0.003• f’(x) = 80x79.• When x = 1, f’(x) = 80• f 80x = 0.24• f(x+x) – f(x) = 0.24.• 1.00380
• =1+0.24=1.24
21
The ratio of Errors (output to input)
• Suppose x is the correct (or precise) input• And x+x is the incorrect (or approximate)
input.• Then x is called the error of the input.• and y = f(x +x) - f(x) is called the error of
the output• If x is small, then y’ y/ x.• Or y = y’ x.
Example 23
During a medical procedure, the size of a roughly spherical During a medical procedure, the size of a roughly spherical tumor is estimated by measuring its diameter and using the tumor is estimated by measuring its diameter and using the formula to compute its volume. If the diameter is formula to compute its volume. If the diameter is measured as measured as 2.52.5 cm with a maximum error of cm with a maximum error of 2%,2%, how how accurate is the volume measurement?accurate is the volume measurement?
3
3
4RV
Solution:Solution:
A sphere of radius A sphere of radius RR and diameter and diameter x=2Rx=2R has volume has volume
33333 cm 181.8)5.2(6
1
6
1)
2(
3
4
3
4 x
xRV
The error made in computing this volume using the diameter 2.5 , while the actual
diameter is 2.5+△x , is
(2.5 ) (2.5) (2.5)V V x V V x 22
Exercise
At a certain factory, the daily output is units, where K denotes the capital investment measured in units of $1,000. The current capital investment is $900,000. Estimate the effect that an additional capital investment of $800 will have on the daily output.
2/1600)( KKQ
Suggested SolutionThe current capital investment K₀=900 thousand dollars.The increase in capital investment K=0.8 thousand dollars.△To estimate the effect of this K on the daily output:△
This means an additional capital investment of 800 dollars would increase the daily output by 8 units.
units 88.0*)900(3008.0)900('
80900for
300K )('
2/1
-1/2
., Δ K
KKKQQ
Review:Differentials: • Differentials of x is dx= x△ (small increment)• If y=f(x) is a differentiable function of x, then the differential of y
is dxxfdy )('
Review:Implicit Differentiation: • Explicit Form: a function that can be written as y=f(x) “y is solved, and given by an equation of x”• Implicit Form: cannot express y as a equation of x.• How to differentiate an implicit equation? (1) Differentiate both side with respect to x you now have a function containing x and y and y’. (2) Express y’ in terms of x and y.
Exercise: Differentiate implicitly with respect to x.132 22 xxyy
Exercise: Differentiate implicitly with respect to x.
Suggested Solution:
132 22 xxyy
xyy
yy
yxyyy
xyyyyy
42
23'
23)42('
:equation above thegrearranginBy
3'42'2
:have weside,both on ateDifferenti
2
2
2
28
Example
Suppose the output at a certain factory is units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor. The current labor force consists of 30 hours of skill labor and 20 hours of unskilled labor.
Question: Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level.
3232 yyxxQ
Solution:
30
The manager of a company determines that when q hundred units of a particular commodity are produced, the cost of production is C thousand dollars, where . When 1500 units are being produced, the level of production is increasing at the rate of 20 units per week.
What is the total cost at this time and at what rate is it changing?
Example
42753 32 qC
31
312 22 ppxx
Exercise
When the price of a certain commodity is p dollars per unit, the manufacturer is willing to supply x thousand units, where
How fast is the supply changing when the price is $9 per unit and is increasing at the rate of 20 cents per week?
32
Exercise
A lake is polluted by waste from a plant located on its shore. Ecologists determine that when the level of pollutant is x parts per million (ppm), there will be F fish of a certain species in the lake, where
When there are 4000 fish left in the lake, the pollution is increasing at the rate of 1.4ppm/year. At what rate is the fish population changing at this time?
xF
3
32000