mat194_coursesummary(2013f)

8/11/2019 MAT194_CourseSummary(2013F)

1/28

2013: Fall Jeremy WangMAT194H1 F Stangeby, P. C. Engineering Science 1T7

Page 1 of 28

MAT194H1 F Course Summary

Instructor: P. C. Stangeby | LEC 0101 | TUT 0106 | Fall 2013 | 2013-2014

1.1 Course Overview

The difference between an intellectual and the average person is that truth for the lay personcomes from authority, but an intellectual will demand rigorous proof

Prof. P. C. Stangeby

Calculus I consists of a comprehensive study of the theory and application of differential

and integral calculus. The main topics covered in the differential calculus portion of the course

are: the axioms of mathematics; a review of the number systems and types of functions;

definition of the limit; Epsilon-Delta proofs; limit and derivative theorems; and curve sketching.

As for the integral calculus component, these are the main topics: Riemann Definition of the

Integral; the Fundamental Theorem of Calculus; definite and indefinite integrals; u-substitution;

inverse functions; natural log functions; LHopitals Rule, solving differential equations.

Specific applications of this knowledge, such as finding velocity, acceleration, areas,

volumes, work, and carrying capacities of ecological systems, are presented simultaneously.

Textbook: Calculus, 7th

Edition, James Stewart

Sections: 1; 2.1-2.8; 3; 4; 5; 6.1; 6.2*-6.4*; 6.5-6.8; 9.1; 9.3-9.5; 17.1-17.2

2.1-2.6 Evaluation Techniques, Procedural Guidelines, and Applications of Calculus

Each section below corresponds to a collection of analytic models that may assist in solving a

specific type of problem or otherwise executing a particular mathematical task encountered in

this course. Related theorems and other mathematical tools are included alongside each

systematic procedure and/or guideline. Also note that this text assumes the User understands the

theory behind the statements made herein.

The reader is also strongly encouraged to read through 3. Appendices at the end of this text.

Legend:

[START] Beginning of procedure[END] End of procedure

StepNof the present procedure

IFF If and only if

Logical implication


2/28


Page 2 of 28

2.1 Epsilon-Delta Proofs

Applications:

- Quashing frail intuitivmathematics with a rigorous thermonuclear warheadEpsilon-Delta Definition of a Limit:

lim ()= 0 0 Note: a limit exists IFFthe left- and right-handed limits exist and are equal

Systematic Procedurefor proving a Limit:

1. [START]Recognise that > 0is imposed on you2. Simplify (or re-express conveniently) the left-hand side of |() |< 3. Substitute cinto 0


3/28


Page 3 of 28

Q2.1B: Rigorously prove orrigorously disprove the following limit: lim 25 = 11


4/28


Page 4 of 28

2.2 Continuity & Evaluating Limit Expressions

Definition of a Continuity for a Function:

A function ()is cont. at = if lim ()exists.A function ()is cont. on a closed interval[,], if for every (,), lim () existsA function

()is a continuous function if it is continuous at every

in the domain.

Note: remember types of discontinuities (e.g. jump, removable, function DNE, limit DNE, etc.)

Basic Limit Theorems:

If lim ()= and lim ()= , then:Constant Limit Theorem: lim = Constant Multiple Theorem: lim ()= Polynomial Limit Thorem: lim ()= ()Additivity Limit Thorem: lim()+ () = L + MProduct Limit Theorem:

lim()() = LM

Rational Fcn. Limit Theorem: lim ()() = ; M 0Power Limit Theorem: lim[()] = ; Composite Fcn. Limit Theorem: lim[()] = (); () Basic Continuity Theorems:

If ()and ()are continuous functions on an interval [,], then:The Basic Limit Theorems can be applied to every [, ]to yield the associated continuitytheorems (e.g. Additivity Continuity Theorem states that (() + ())is also continuous)Property of Direct Substitution:

If ()is a polynomial or rational function and ais in the domain off, then:lim () = ()

Squeeze Theorem:

If () () ()when is near (except possibly at ), and:lim () = lim() () =

Then:

lim ()=

LHopitals Rule:

If ()and ()are two given, differentiable functions on some interval containing a, butnecessarily at a; () 0 this interval, but excluding = ; lim ()= lim ()=0 + ; lim ()() = exists and we know its value; then:


5/28


Page 5 of 28

lim()()= lim

()()=

Remember: if necessary, manipulate the limit expression so that LHopitals Rule can be used.

Note: an expression lim () may be manipulated into a form appropriate for LHopitals byusing

()= ()and applying the limit to

ln()to find

lim (); however, the

conditions for Composite Function Continuity Theorem must be met.

Intermediate Value Theorem:

If a function ()is continuous on an interval [a,b], and there is some number such that()< < () , then there is some number such that ()= .The Fundamental Trigonometric Limit:

limsin

= 1 Note: this can be used to derive lim = 0 Evaluation Techniques (rigid/explicit procedures not appropriate):

Remember: a limit exists IFFthe left- and right-handed limits exist and are equal

Note: at any time, if a real number is determined as the value of the limit expression, [END]!

1. Attempt direct substitution

2.

Simplify the limit expression as much as possible (factor and divide out terms, group

terms, etc.)

3. Multiply by 1 (e.g.

,

,.)

4.

Add zero (i.e. +[factor] [factor]) then factor

5. Try to apply one of the basic limit laws/theorems

6. Substitute in equivalent expressions (e.g. definitions, identities, etc.)

7. If an indeterminate form of , , is obtained and the conditions for LHopitals

Rule are present, use it (and you may repeatedly use it until a real number is produced so

long as the conditions for LHopitals Rule are present)

8. Attempt to use Squeeze Theorem

9. Split the limit into two a left-handed and right-handed limit and evaluate those

individually.

10.

Integrate (no pun intended) other definitions involving limits (e.g. if the limit expressionis actually a derivative)


6/28


Page 6 of 28

Q2.2A: Evaluate the following limit:

lim

4 sec

Q2.2B: Evaluate the following limit:

lim(sinh )


7/28


Page 7 of 28

Q2.2C: Evaluate the following limit:

lim(1 + )

Q2.2D: Evaluate the following limit:

lim ( )(())()

Q2.2E: Evaluate the following limit:

lim(tan)


8/28


Page 8 of 28

2.3 Finding the Derivative

Applications:

- Physics (force, velocity, accelleration, etc.)

- Implicit differentiation

- Optimisation and curve sketching

-

Approximating the value of non-integer numbers

- Related rates problems

- Any task requiring instantaneous rates of change

Definition of the Derivative of a Function:

The derivative of the function y = ()is given by:= ()= = lim

( + ) ()

Note: the derivative is the instantaneous rate of change of = ()at a given Definition of Differentiability:

A function ()is differentiable at a point = if ()exists.A function ()is differentiable on an interval [a,b] if for every ( , ), ()exists.A function ()is a differentiable function if it is differentiable for every in the domain.Note: a function is diff. at a given point IFFboth the left- and right-handed derivatives exist.

Basic Derivative Theorems:

Power Derivative Theorem: () = ; Constant Derivative Theorem:

[()]

=

()

Polynomial Derivative Theorem: ()= + ( 1) + + 2 + Additivity Derivative Theorem: ( + )()= ()+ ()Product Derivative Theorem: [()()] = ()() + ()()Quotient Derivative Theorem: ()()

= ()()()()[()] ; () 0 Reciprocal Derivative Theorem: ()

= ()[()] ; () 0Composite Function Derivative Theorem (Chain Rule):

()

=

Note: Chain Rule is used during implicit differentiation, and always applies when differentiating.

Continuity Implication from Differentiability:

If ()is differentiable over an interval (a,b), then it is also continuous over that interval.Note: the converse is not always true.


9/28


Page 9 of 28

Extreme Value Theorem:

If ()is continuous on an interval [a,b], then ()has some absolute maximum, (), andsome absolute minimum (), within the interval, where , [,].Fermats Theorem:

If ()has a local max./min. at x = c, then:()= 0 () Note: the converse is not always true.

Mean Value Theorem:

If ()is continuous on [,]and ()is differentiable on (,), then there is at least one (,)such that:

()= () () Derivatives of Trigonometric Functions:

(sin)= (csc)= csc cot (cos)= (sec)= sec tan (tan)= sec (cot)= csc Derivatives of Inverse Trigonometric Functions:

sin

= 1

cos = 1 tan

=

+

Note: these can be obtained by forming a right angle triangle such that = (), wherefis thefunction for which the inverse is being sought andyis an interior angle of the triangle, then

taking the derivative of that expression.

Derivative of an Inverse Function:

()= 1()

Derivative of an Exponential Function: = ln


10/28


Page 10 of 28

Derivative of a Logarithmic Function: log =

1 l n

Differentiation Techniques (rigid/explicit procedures not appropriate):

Note: [END]when a the desired form of the derivative is found!

1. Simplify the limit expression as much as possible (factor and divide out terms, group

terms, etc.)

2. Apply a derivative theorem/known trigonometric derivative

3. Multiply by 1 (e.g. ,

,.)

4. Add zero (i.e. +[factor] [factor]) then factor

5. Substitute in equivalent expressions (e.g. definitions, identities, etc.)

Systematic Procedure to Calculate Differential Approximations for Numbers:

1.

[START]Define a function (or use a given fcn.) = (), such that = ( + ) ()is equal to the number for which the approximation is being made2. Choose =[your chosen increment]3. Find

= (), which can be rearranged into = ()

4. Substitute values into = ()|and solve for 5. Use = [ ] to approximate the number of interest. [END]

Q2.3A Evaluate the following:

[10

]

Q2.3B Find ()where ()= 4


11/28


Page 11 of 28

Q2.3C Find ()where:()= lim ()

Q2.3D Evaluate the following: |, = [ (3 + 4)], (7)=

14

Q2.3E Approximate the value of 51using differentials.


12/28


Page 12 of 28

2.4 Optimisation and Curve Sketching

Applications:

- Graphical visualisations of data and models

- Maximisation/Minimisation optimisation problems

Definition of Absolute/Local Maximum/Minimum:

For a function = ()that is defined for an interval [,], and for a [,]:There is an absolute maximum at ()if () () for all in the domain.There is an absolute minimum at ()if() () for all in the domain.There is a local maximum at ()if () () for all , [, ].There is a local minimum at ()if () () for all , [,].Derivative Analysis and Shapes of Graphs:

For a function = ()that is differentiable on an interval (,), and for a (,):Quick Test 1: is increasing if > 0 is decreasing if < 0 Quick Test 2: If < 0 to the left of and > 0 to the right of = , ()is a local min.

If > 0 to the left of, and > 0 to the right of = ()is a local max.If is the same on both sides of = , then ()is not a local min/max.

Quick Test 3: If > 0 on the interval, then is concave up.If < 0 on the interval, then is concave down.

Quick Test 4: If ()< 0 , then ()is a local max.If ()> 0 , then ()is a local min.If

()= 0, this does not suggest anything conclusive about

().

Systematic Procedure for finding Local Maxima/Minima:

Note: [END]whenever all the local maxima/minima have been found!

1. [START]Find all ccritby setting = 0and solving for or find where .2. If Quick Test 4 conditions are met, use it.

3. If conditions for are not met, but those for Quick Test 2, use it. But, it is more work

(we must consider sign of f above/below ccrit, as well as continuity).

4. If conditions are not met for Quick Test 2 either, use the basic definition of a local

maximum/minimum (usually, this is the most work of all).

Systematic Procedure for finding the Optimal Value of a Function:

1. [START]Find the local maxima and minima.

2. Evaluate the endpoints of the function on the interval of interest.

3. Select the critical point that optimises the function in the desired manner. [END]


13/28


Page 13 of 28

Systematic Procedure for Curve Sketching:

1. [START]Try to alter the form of the function to make it more manageable for curve

sketching, if possible (i.e. factoring)

2. Find the domain and range, endpoints, horizontal/vertical/slant/other asymptotes of the

function.

3.

Find the x-intercept(s) and y-intercept(s).

4. Identify any special properties (i.e. even/odd function, periodicity).

5. Use QT1 and QT2 to identify intervals where the function is increasing/decreasing (and

from there, identify any local max./min. as well as abs. max./min.).

6. Use QT3 and QT4 to identify intervals where the function is concave up/down (and from

there, identify any points of inflection as well as confirm max./min. found in ).

[END]

Q2.4A Sketch the curve = sin , .


14/28


15/28


Page 15 of 28

2.5 Integral Operations

Applications:

- Physics (e.g. finding change in position from velocity, work done by a force, etc.)

- Areas between curves

- Volumes of revolution

-

Differential equations (see 2.6 Solving Differential Equations)

- Any tasks involving calculation of: areas; change in a function using its derivative

Riemann Definition of the Definite Integral:

For a function ()that is continuous, or is continuous except for a finite number of jumpdiscontinuities, on an interval [a,b], the net area under the curve (the definite integral) is:

()

= lim = ()

where

is called the Riemann Sum,

is some point in the each subinterval of width

,

and = .Note: many integral properties (such as the sign change due to interchanging the limits of

integration, or the properties of integrals of odd/even functions from 0to ) can be relativelyeasily deduced via conceptual manipulation of the Riemann definition.

See 3.2 Additive Series of Powers to find equivalent expressions for some sums of powers.

The Fundamental Theorem of Calculus, Part I:

For a function ()that is continuous, then the function (), defined as:

()= ()

is continuous anddifferentiable on (a,b) and ()= () () = () The Fundamental Theorem of Calculus, Part II:

For a function ()that is continuous on an interval [a,b], then: ()

= () ()

where F(x) is any anti-derivative of f(x).

Note: this demonstrates that the integral of a rate of change of a function is the net change of the

value of that function.

Definition of the Indefinite Integral:

For a function ()that is continuous on an interval [a,b], then the indefinite integral is:()= () =[ ]+

where is a constant produced during integration.


16/28


Page 16 of 28

Note: the indefinite integral represents a family of functions that differ by constant values.

Note: the indefinite integral is also dubbed the general anti-derivative.

Note: the definite integral is a number while the indefinite integral is a function.

Substitution Rule:

For a function = ()that is differentiable on an interval , such that a function ()iscontinuous on the interval , then:

()() = ()Note: Substitution Rule tells us that it is acceptable to operate with and found in theintegrand as thoughthey were differentials.

Mean Value Theorem for Integrals:

For a function ()that is continuous at [,], there exists at least one [, ]such that:( )()= ()

Systematic Procedure for finding the Area between Curves:

Note: [START]by considering or simultaneously.

1. a)Divide the interval of interest into subintervals, if necessary.

Note: each subinterval must have a function that is always greater in value (say,

()) relative to a function that is always lesser in value (say, ()).b) If you do not have functions, but relations, then rearrange the equations into in termsof

, if it eases the task at hand.

2. Find () () for each subinterval [,].3. Add the definite integrals together to find the total area between the curves. [END]

Systematic Procedure for finding a Volume of Revolution using Disc Method:

1. [START]Formulate an expression for in terms of a variable whose axis is orthogonalto the plane of the flat bottom of the discs; should be in the form of:

= 2. Sum all the terms that comprise the volume by calculating a definite integral of

over the interval of interest. [END]

Systematic Procedure for finding a Volume of Revolution using Shell Method:

1. [START]Formulate an expression for in terms of a variable whose axis is parallel tothe plane of the flat bottom of the shells (hollow cylinders); should be in the form of:

= 2. Sum all the dV terms that comprise the volume by calculating a definite integral of

over the interval of interest. [END]


17/28


Page 17 of 28

Q2.5A For > 0, let be the volume created by revolving the region bounded by = and = around the axis = . The units of , and are [m]. The units of are [m1/2]. The valueof remains fixed. Find ().

Q2.5B How much work is done when:

a) a weightless cable is used to life 200kg of water in weightless bucket up from well that is

60m deep?

b) a bucket that weighs 50kg and a weightless cable are used to lift water up from a well

60m deep. The bucket is initially filled with 200kg of water and is pulled up at a rate of

2m/s but the water leaks out of the bucket from a hole at a rate of 2kg/s.


18/28


Page 18 of 28

Q2.5C Suppose is a continuous function on (,). Calculate the following in terms of :) lim

1 ()

)lim ( ()

()

Q2.5D Calculate the following integral:

sin2cos + sin


19/28


Page 19 of 28

Q2.5D (Continued)


20/28


21/28


Page 21 of 28

Compound Interest Formula:

The amount of money, (), in a bank account undergoing compound interest is:()= 1 +

where

is the amount of time elapsed in units of

(

is a unit of time, e.g. month, year, etc.),

is the number of compounds per , and is the interest rate per .1OLDE: First-Order Linear Differential Equation:

+ () = ()where (),()are given, continuous functions. If we define an integration factor, , to be:

()= ()Then the solution to the differential equation is:

()= 1 Note: alternatively, multiply both sides of the 1OLDE by I and then integrate.

2OLDE: Second-Order Linear Differential Equation:

() + () + () = ()where (), (), (), ()are given, continuous functions.H2OLDE:Homogenous Second-Order Linear Differential Equation:

() + () + () = 0where (), (), () are given, continuous functions.H2OLDE Theorem for Specific Solutions:

If (), ()are two possible solutions to the H2OLDE, then:()= () + ()is also a solution, for any constants , .H2OLDE Theorem for General Solutions:

If (), ()are two linearly independent solutions of the H2OLDE, then:()= () + ()is the general, complete solution to the H2OLDE, for constants , determined by theboundary conditions.

Eulers Relation:

cos + sin Note: this is true by definition.


22/28


Page 22 of 28

H2OLDE w/ CCs: Homogeneous Second-Order Linear Differential Equation with Constant

Coefficients:

+ + = 0 where , ,are constants. Then the associated characteristic or auxiliary equation is:

+ + = 0

: = 42

Now, we assume asolution is ()= because it satisfies the differential equation:Case 1: 4 > 0 , are real and distinct (so, ()and ()are linearlyindependent). By the H2OLDE Theorem for General Solutions:

()= + Case 2: 4 = 0 = and we need an alternative method to find two linearlyindependent solutions. ()= is an appropriate candidate, hence:

()= +

Case 3: 4 < 0 = + and = , where = , =

. Then, weuse ()= + along with Eulers Relation to arrive at:

()= ( cos + sin)Non-H2OLDE w/ CCs:Non-Homogeneous Second-Order Linear Differential Equation with

Constant Coefficients:

+ + = () where the associated complementary equation (an H2OLDE w/ CCs) is:

+ + = 0 The general, complete solution to the Non-H2OLDE w/ CCs is given by:

()= () + ()where ()is a particular solution to the Non-H2OLDE w/ CCS, and ()is thegeneral, complete solution to the complementary equation.

Systematic Procedure for solving Separable Differential Equations:

1. [START]Collect the terms with only on one side, and collect the terms with only y onthe other side.

2. Integrate both sides (dont forget about constants!)

3. Simplify and isolate for in terms of (or vice-versa, depending on the task at hand).[END]


23/28


Page 23 of 28

Method of Undetermined Coefficients:

1. [START]Find the general, complete solution to the complementary equation (these can

always be found).

2. Attempt to solve the coefficients of a ()in the form of ()by substituting yourchosen

()into the differential equation.

Note: In order for the left-hand side to be equal to the right-hand side, the

coefficients for each power of must be the same (otherwise, the polynomials arelinearly independent and no solution can be found).

[END]

Method of Variation of Parameters:

1. [START]Find two solutions, and , to the complementary equation (these canalways be found).

2. Find ()by setting ()= + , where , are unknown functionsto be found in .

3. Solve the following system of equations:

+ = 0 + = ()

Note: if we cannot integrate , , we cannot solve the differential equation.[END]

Systematic Procedure for solving Linear Differential Equations:

1. [START]Identify the type of linear differential equation (1OLDE, H2OLDE, H2OLDE

w/ CCs, non-H2OLDE w/ CCs).

2. Use one of the known theorems and/or methods associated with the particular situation

and solve for the differential equation. [END]


24/28


Page 24 of 28

Q2.6A Solve the differential equation 2 3 = + 2 , with the boundary conditions(0)= (0)= 0 using Method of Undetermined Coefficients.


25/28


Page 25 of 28

Q2.6B Find the solution of the differential equation 3 = 6.


26/28


Page 26 of 28

3. Appendices

3.1 Trigonometric Identities

sin + cos = 1

sin( )= sin cos cos sin (also yields double-angle, product identities)

cos( )= cos cos sin sin (also yields double-angle, half-angle, product identities)tan( ) = tan tan 1 tan tan cosh sinh = 1sinh( + ) = sinh cosh + cosh sinh3.2 Numerical Methods to find the Roots of an Equation

3.2.1 Method of Successive Bisections

1. [START]Choose two values,

, in the domain, such that

() > 0and

() < 0

2. Find midpoint between and , call it 3. a) If () > 0, then for next step, =

b) If () < 0, then for next step, = 4. Repeat Steps and as many times as desired [END]

3.2.2 Newtons Method

1. [START]Guess as the x co-ordinate of a root.2. Find (), ();find the eqn of the tangent line

a.

() = () + ()( )3. Find such that () = 04. Set 0 = ()+ () = ()()5. Repeat = ()()as many times as desired [END]

Note: with Newtons Method it is possible to produce a series of approximations for the of theroot that do not converge. In this case, choose a new that is offset from the present set, orproceed to use Method of Successive Bisections.


27/28


Page 27 of 28

3.3 Additive Series of Powers

= ( + 1)2

= ( + 1)(2 + 6)6

= ( + 1)2

3.4 Inverse Function Relationships

Definition of the Inverse Function

For a function (), the inverse of (), denoted as ()is defined as the function such that:() = () =

where () = ()and () = () Note: an inverse function may be geometrically interpreted as the reflection of the originalfunction across the line = .Definition of the One-to-One (1-1) Function

A function ()is said to be one-to-one IFF:()= ()

Note: a function is 1-1 on an interval IFFits inverse exists

Note: if function is strictly increasing or strictly decreasing on interval, then it is 1-1 on that

interval and the inverse function exists.

Systematic Procedure to find ():1. [START]Establish that ()is one-to-one on the interval of interest2. a) If you can guess via trial and error the value of (), then test it:

does () = ?[END]b) Start with = ()and try algebraic manipulation to yield = ()andset ()= () . [END]

3.5 ln and RelationshipsDefinition of ln : ln = 1

, > 0 = exp(ln) =

Note: ln is the inverse function of Note: ln = log , however this course focuses on calculus and so only the former is used.Note: by FTC Part I, ln is continuous and differentiable for > 0


28/28


Properties of ln :Operational Properties:

ln 0 = 1 = 1 ln = ln + ln

ln 1 = ln ln = ln ln ln = ln ; basis of logarithmic differentiationLimit Properties:

lim ln = lim ln =

()() = ln |()| + ln|()|= ()()

()() = () +

3.6 Hyperbolic Trigonometric Functions

sinh = 2

cos =

+

2 tanh =sinh cosh= +

mat194_coursesummary(2013f)

Documents