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MATH 27 LECTURE GUIDE
UNIT 1. DERIVATIVES OF AND INTEGRALS YIELDING TRANSCENDENTAL FUNCTIONS Objectives: By the end of the unit, a student should be able to
find derivatives of transcendental functions; find integrals of and integral forms yielding transcendental functions; find derivatives using logarithmic differentiation; and
evaluate limits of functions using L'Hopital's rule. __________________________ 1.1 Derivatives of and Integrals Yielding Trigonometric Functions
(TC7 163-166, 320-321 / TCWAG 173-176, 291-291)
The sine function defined by xsinxf and the cosine function defined by xcosxf are
continuous over the set of real numbers.
The tangent function ( xtanxf ), cotangent function ( xcotxf ), secant function
( xsecxf ) and the cosecant function ( xcscxf ) are continuous over their respective
domains.
Using the definition of a derivative,
h
xfhxflimx'fh
0, it can be derived that
xcosxsinDx and xsinxcosDx .
The sine and cosine functions are differentiable over the set of real numbers. The tangent, cotangent, secant and cosecant functions are differentiable over their respective domains.
TO DO!!! Deriving the derivatives of xtan and xcsc .
xtanDx
xcscDx
MUST REMEMBER!!! Derivatives of Trigonometric Functions
xcosxsinDx xsecxtanDx2 xtanxsecxsecDx
xsinxcosDx xcscxcotDx2 xcotxcscxcscDx
CHAIN RULE: Derivatives of trigonometric functions (in case of compositions) Let u be a differentiable function of x .
uDucosusinD xx uDusecutanD xx 2 uDutanusecusecD xx
uDusinucosD xx uDucscucotD xx 2 uDucotucscucscD xx
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REVIEW!!! From your MATH 26, if xfx'F , then CxFdxxf .
__________________________ 1.2 Derivatives of and Integrals Yielding Inverse Trigonometric Functions
(TC7 491-503 / TCWAG 503-513)
The inverse trigonometric functions are continuous given by xsinArcxf , xcosArcxf ,
xtanArcxf , xcotArcxf , xsecArcxf and xcscArcxf are continuous over their
respective domains except for some “boundary” points.
TO DO!!! Evaluate the following.
1. xcosxsinDx
2. xsecxtanDx 235
3. xcsccotDx
____________
Evaluate 22 xcosxsinDx ,
xcos
xtanxDx and xcotcscDx
2 .
MUST REMEMBER!!! Integrals Yielding Trigonometric Functions
Cxsinxdxcos Cxtanxdxsec 2 Cxsecxdxtanxsec
Cxcosxdxsin Cxcotxdxcsc 2 Cxcscxdxcotxcsc
TO DO!!! Evaluate the following.
1. xdxcosxsin
2. xdxsecxtan 2
3. xdxcot2
____________
Evaluate dxxsinx 2 , xdxtan2 and xdxcosxcot .
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KEEN MIND HERE!!!
Since 21
1
x
xsinArcDx
, then
CxsinArcdx
x21
1.
Also,
CxcosArcdx
x21
1. But, CxcosArcdx
x
dx
x
22 1
1
1
1.
HOW?
MUST REMEMBER!!! Derivatives of Inverse Trigonometric Functions
21
1
x
xsinArcDx
21
1
xxtanArcDx
1
1
2
xx
xsecArcDx
21
1
x
xcosArcDx
21
1
xxcotArcDx
1
1
2
xx
xcscArcDx
TO DO!!! Solve for dx
dy.
1. 21 xcosArcy
2. 3xsinsecArcy
_______________
Solve for dx
dy. xcoscotArcy 2 , 2xcscArcsecy , xtantanArcy
CHAIN RULE: Derivatives of trigonometric functions (in case of compositions) Let u be a differentiable function of x .
uD
u
usinArcD xx
21
1 uD
uutanArcD xx
21
1 uD
uu
usecArcD xx
1
1
2
uD
u
ucosArcD xx
21
1 uD
uucotArcD xx
21
1 uD
uu
ucscArcD xx
1
1
2
TO DO!!! Deriving the derivative of xsinArc .
Let xsinArcy . Hence, ysinx .
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MUST REMEMBER!!! Integrals Yielding Inverse Sine Function
CxsinArcdx
x21
1 If a is a constant,
Ca
xsinArcdx
xa 22
1
CxtanArcdxx21
1
C
a
xtanArc
adx
xa
11
22
CxsecArcdx
xx 1
1
2
Ca
xsecArc
adx
axx
11
22
TO DO!!! Evaluate
1.
dxx225
3 2.
62xx
dx
If u is a differentiable function of x and a is a constant,
Ca
usinArc
ua
du
22
C
a
utanArc
aua
du 1
22
Ca
usecArc
aauu
du 1
22
TO DO!!!
1.
dx
xsin
xcos
23
2.
dxxx 258
2
2 3.
42xe
dx
___________
Evaluate
dx
e
e
x
x
24
,
dxxtan
xsec
42
2
and
dx
xxx
x
321 242
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1.3 Derivatives of and Integrals Yielding Logarithmic Functions (TC7 451-456, 473 / TCWAG 449-454, 466)
The natural logarithmic function defined by xlnxf is continuous over ,0 .
Also,
xlnlimx 0
and
xlnlimx
.
Note that if 10 a,a , aln
xlnxloga .
MUST REMEMBER!!!
x
xlnDx1
and if u is a differentiable function of x , uDu
ulnD xx 1
TO DO!!! Solve for x'f .
1. xtanxseclnxf
2. xsinxlnxf
MUST REMEMBER!!!
xaln
xlogD ax11
and if u is a differentiable function of x , uDxaln
ulogD xax 11
TO DO!!! Evaluate .xloglogDx
2
110
HOW TO . . . derive the derivative of xln ! ! !
Alternative definition, x
dtt
xln1
1.
x
xx dtt
DxlnD1
1
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KEEN MIND HERE!!! Why Cxlndxx
1
, instead of xln ?
Problem: Domain of x
1: Domain of xln
Solution:
0
0
xifx
xifxx
xxlnDx
1 xlnDx
__________________________
1.4 Logarithmic Differentiation (TC7 447-448, 474-475/ TCWAG 449-450) Logartihmic differentiation is an alternative way of differentiating SUPER PRODUCTS, SUPER
QUOTIENTS and functions in the form of variable raised to variable like xxxf .
MUST REMEMBER!!!
Cxlndxx
1
and if u is a differentiable function of x , Culnu
du
TO DO!!! Evaluate the following
1. bax
dx where a and b are constants
2.
dx
x
x
16
4
2
3. xdxtan
_____________
Evaluate dxx
xln,
dxxcos
xsin
1 and dx
x
xtan.
MUST REMEMBER!!! Integrals of the “Other” Trigonometric Functions
Cxseclndxxtan Cxcsclndxxcot
Cxtanxseclndxxsec Cxcotxcsclndxxcsc
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HOW TO DO . . . logarithmic differentiation ! ! !
Given xfy .
1. Consider xfy . Get the natural logarithms of both sides of xfy , i.e.
xflnyln . Note that x
xDx1
.
2. Use properties of logarithms to express xfln as sums instead of products, as
difference instead of quotients and products instead of exponentiations.
3. Get the derivatives of both sides of xflnyln . Hence, xflnDdx
dy
yx
1
4. Solve for dx
dy by cross-multiplying y and expressing y in terms of x .
__________________________ 1.5 Derivatives of and Integrals of Exponential Functions
(TC7 462-463, 470-471/ TCWAG 458-460, 464)
The natural exponential function defined by xexf is continuous at every real number.
Also, 0
x
xelim and
x
xelim .
TO DO!!! Use logarithmic differentiation for the following.
1. If 12 xxsecxy , solve for dx
dy.
2. If xln
xsinxxf
, solve for x'f .
3. Evaluate xsinx xD .
___________
Try to evaluate xx xD .
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In general, if 1b , then
x
xblim and 0
x
xblim
if 10 b , then 0
x
xblim and
x
xblim
KEEN MIND HERE!!! What is xx eD ?
Let xey . ylnx ydy
dx 1
dx
dy
MUST REMEMBER!!! xxx eeD If u is a differentiable function of x , uDeeD x
uux
TO DO!!! Evaluate the following.
1.
xxxD
2210
2. xlnxsinx eD 3
TO DO!!! Evaluate the following.
1. dxx 124
MUST REMEMBER!!! Derivatives of Exponential Functions
alnaaD xxx If u is a differentiable function of x , uDalnaaD x
uux
MUST REMEMBER!!! Integrals of Exponential Functions
Cedxe xx Caln
adxa
xx
If u is a differentiable function of x , Cedue uu and Caln
adua
uu .
Refer to the graphs of exponential functions of
the form xbxf from
MATH 14 or MATH 17.
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_________________________
1.6 Some Application on Optimization, Related Rates and Laws of Natural Growth and Decay
(TC7 477-483, Examples of Chapter 5 / TCWAG 469-473, Examples of Chapter 7)
HOW TO SOLVE . . . maximization/minimization problems ! ! !
Given xfy . To solve for value/s of x that maximizes or minimizes y :
1. Determine the critical points of f (i.e. value/s of x where 0 x'fdx
dy.
2. If there are several critical points, compare function values at the critical points to determine the maximum or the minimum. If possible, use second-derivative test on the critical points.
If critical a is a critical point of f and 0x''f , then f has a maximum at a .
If critical a is a critical point of f and 0x''f , then f has a minimum at a .
Some related rates problem . . .
TO DO!!!
If R feet is the range of a projectile, then g
sinvR
22
, 2
0 , where v feet per second is
the initial velocity, g ft/sec2 is the acceleration due to gravity and is the radian measure of the
angle of projectile. Find the value of that makes the range a maximum.
________________________________
An individual’s blood pressure, P , at time t is given by tsinP 22590 . Find the values of the
maximum and minimum pressure. When do these values occur?
2. dxex x32
3.
dxx
x
12
2
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Some related rates problem . . .
Given a model xfy where x and y varies with respect to time. To solve for dt
dy, differentiate
both sides with respect to time. An exponential growth or decay is a phenomenon undergone by certain organisms and radioactive elements. It happens when a rate of growth (or decay) is proportional to the present population of an organism or the present quantity of a radioactive element.
KEEN MIND HERE!!! The Exponential Model of Growth and Decay Suppose an organism (or an element) grows (or decays) in such a way that rate of growth is proportional to the present quantity (or population). Let y be the quantity (or population) at time t .
Also, dt
dy is the rate of growth (or decay).
Hence, kydt
dy kdt
y
dy
kdty
dy
Cktyln
Solving for y , Cktey ktBey , where CeB is a constant.
Moreover, B is the quantity (or population) at 0t .
The examples will be on interpreting models. It will be assumed that this models were arrived at using the procedures above.
TO DO!!!
A woman standing on top of a vertical cliff is 200 feet above a sea. As she watches, the angle of depression of a motorboat (moving directly away from the foot of the cliff) is decreasing at a rate of 0.08 rad/sec. How fast is the motorboat departing from the cliff?
_________________________________
After blast-off, a space shuttle climbs vertically and a radar-tracking dish, located 800 m from the launch pad, follows the shuttle. How fast is the radar dish revolving 10 sec after blast-off if the velocity at that time is 100 m/sec and the shuttle is 500 m above the ground?
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__________________________ 1.7 Indeterminate Forms and the “L’Hopital’s” Rule
(TC7 634-649 / TCWAG 650-665)
This section is for limit problems involving the indeterminate forms 0
0 and
.
This is also applicable for one-sided limits and x .
MUST REMEMBER!!! L’Hopital’s Rule for 0
0
Suppose that 0
xflimax
and 0
xglimax
. Then,
x'g
x'flim
xg
xflim
axax .
TO DO!!!
1. A lake is stocked with 100 fish and the fish population P begins to increase according to the
model te
,P
191
00010, where t is measured in months.
Does the population have a limit as t increases without bound?
After how many months is the population increasing most rapidly?
2. The revenue R (in million dollars) for an international firm from 2000 to 2010 can be modeled
by te.t..P 0040521151296 , where 0t correponds to 2000. When did they reach the
maximum revenue within the period? Examine the validity of the model for the years beyond 2010.
_________________________________ On a college campus of 5000 students, the spread of flu virus through the student is modeled by
t.e
,P
8049991
0005
, where P is the number of students infected after t days. Will all students
on the campus be infected with the flu? After how many days is the virus spreading the fastest?
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This is also applicable for one-sided limits and x .
Other indeterminate forms: 0 , , 00 , 0 and 1 .
WHAT TO DO . . . in case of 0 ! ! !
Convert 0 to a form 0
0 or
by expressing as
0
1 or 0 as
1, respectively. Then,
use L’Hopital’s Rule on the converted form.
TO DO!!!
1. 2
1
21
xx
xlimx
2. 30 x
xxsinlimx
MUST REMEMBER!!! L’Hopital’s Rule for
Suppose that
xflimax
and
xglimax
. Then,
x'g
x'flim
xg
xflim
axax .
TO DO!!!
1. xx e
xlim
2
2. xln
xcotlim
x 0
MUST REMEMBER . . . NOT REALLY “L’Hopital’s” Rule was named after Guillaume Francois Antoine de L’Hopital but he is not who discovered it! The man behind this rule was Johann Bernoulli.
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WHAT TO DO . . . in case of ! ! !
Express the given as a single quotient. Then, use L’Hopital’s Rule if 0
0 or
is obtained.
WHAT TO DO . . . in case of 00 , 0 or 1 ! ! !
1. Consider xfy .
2. Get the natural logarithm of both sides of xfy so that xflnyln .
3. Use property of logarithms so that the form 00 can be converted to a form 0 .
4. By now, ylnlimax
is of the form 0 . Get ylnlimax
by resolving 0 .
5. Now, yln
axaxelimylim
.
The following are PSEUDO-indeterminate forms. These can be resolved using the techniques above without the use of L’Hopital’s Rule.
0
0 1
END OF UNIT 1 Lecture Guide
TO DO!!! Evaluate the following.
1.
x
xxlnlim
21
2. xsin
xxlim
0
______________
Evaluate: xsinx
xxtanlimx
0
xlnxlnxlimx
11
1 x
xxlim
0