math 27 lecture guide unit 1

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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


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


1.

xxxD

2210

2. xlnxsinx eD 3


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


1.

x

xxlnlim

21

2. xsin

xxlim

0

______________

Evaluate: xsinx

xxtanlimx

0

xlnxlnxlimx

11

1 x

xxlim

0

math 27 lecture guide unit 1

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