MATH MAJOR 14 – DIFFERENTIAL CALCULUSTopics.

1. Before calculus (precalculus)2. Limits3. Differential calculus4. Special functions and numbers5. Numerical integration6. Lists and tables7. Multivariable8. Series9. History10. Nonstandard calculus

• Calculus – is that branch of mathematics thatdeals with growth (development), motion(process or power of changing place or position),maxima (greatest quantity) and minima (leastquantity).

• Calculus - the branch of mathematics that dealswith the finding and properties of derivatives andintegrals of functions, by methods originallybased on the summation of infinitesimaldifferences.

• Calculus - a particular method or system ofcalculation or reasoning.

The two main types of Calculus:

1. differential calculus - a branch of mathematics concerned chiefly with thestudy of the rate of change of functions with respect to their variablesespecially through the use of derivatives and differentials

• Differential Calculus - First Branch of mathematical ANALYSIS, devisedby ISAAC NEWTON and G.W. LEIBNIZ, and concerned with the problem offinding the rate of change of a FUNCTION with respect to the variable onwhich it depends. Thus it involves calculating DERIVATIVEs and using them tosolve problems involving nonconstant rates of change. Typical applicationsinclude finding MAXIMUM andMINIMUM values of functions in order to solvepractical problems in OPTIMIZATION. Known Use of DIFFERENTIAL CALCULUS1702

2. Integral calculus - a branch of mathematics concerned with the theory andapplications (as in the determination of lengths, areas, and volumes and inthe solution of differential equations) of integrals and integration FirstKnown Use of INTEGRAL CALCULUS circa 1741

• Invention and development of the calculus is credited to SirIsaac Newton (1642-1727) an English man who wrote hisfirst scientific treaties called fluxions in 1670.

• Integral calculus Branch of CALCULUS concerned with thetheory and applications of INTEGRALs. While DIFFERENTIAL

CALCULUS focuses on rates of change, such as slopes of tangentlines and velocities, integral calculus deals with total size orvalue, such as lengths, areas, and volumes.

• The two branches are connected by the FUNDAMENTAL THEOREM

OF CALCULUS, which shows how a definite integral is calculatedby using its antiderivative (a FUNCTION whose rate of change,or derivative, equals the function being integrated).

1 - Before calculus (precalculus) • In American mathematics education, precalculus, is an

advanced form of secondary school algebra, and afoundational mathematical discipline. It is also calledIntroduction to Analysis. In many schools, precalculus isactually two separate courses: Algebra and Trigonometry.

• Algebra - the part of mathematics in which letters and othergeneral symbols are used to represent numbers andquantities in formulate and equations.

• Trigonometry - the branch of mathematics dealing withthe relations of the sides and angles of triangles and withthe relevant functions of any angles.

Basic Algebra Examples:

1. Arithmetic, Geometric & Exponential Patterns Examples:

The first four triangle numbers are 1, 3, 6, and 10. They are calledtriangular because they can be arranged in dots as triangle, like so:

What will the 10th triangular number be?

There are many ways to work out a pattern. We like to start by arranging the numbers in a row to see how they relate. Underneath each number write what needs to be added, multiplied, or divided to get the next number.

1 3 6 10

+2 +3 +4Notice a pattern? The number 5 would be added next, then 6,

then 7 and so on.

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

1 3 6 10 15 21 28 36 45 55

+2 +3 +4 +5 +6 +7 +8 +9 +10

So, the 10th triangular number is 55

II - Evaluating Algebraic Expressions

Evaluate the expression when x = 2, y = -3, and z = -1;

in 2x + 3y = z

III - Combining Like Terms

Simplify: 3xy + 2x2y - 6xy + 7xy2

IV - Multiplying Monomials: (x)( 2x)(3x)(4x)

First multiply all the coefficient, then all variables and combine together

V - Multiplying Binomials: (5y + 3x)(8y – 1)

VI- Dividing Polynomials: 48x2 – 10xy – 2y2 ÷ 8x + y

VII - Solving One-Step Equations: x – 8 = 20

+ 8 + 8

x = 28

4x – 12 = 16

+ 12 +12

4x = 28

4 4

x = 7

• IX - Solving More Complex Equations

• X - Solving Equations with Variables on Both Sides

• XI - Graphing Inequalities

• XII - Graphing Lines

Intercepts

Graphing Lines By Plotting Points

Slope-Intercept Form

• XIII - Solving Multiple Equations by Graphing

Calculus – is that branch of mathematics that deals with growth, motion, maxima and minima.

>Design of airplanes growth- development

>Rocketships motion – process or power of changing place or position

>Ocean liners

>Automobiles maxima – greatest quantity//minima least quantity

Variable - is a quantity whose value may change during the given discussion or problem denoted by x, y, z or any letter

Constant – is a quantity whose value is fixed during the given discussion. Constant is just a value, a fixed value that doesn't change.

Numerical or absolute constant – always has a fixed value. Ex: 3, 5, √ 2 , √ 5

Arbitrary constant – are letters which represents fixed numerical values.

• arbitrary constant is a value that is fixed throughout multiple functions you pick for ease of calculations. A symbol to which various values may be assigned but which remains unaffected by the changes in the values of the variables of the equation

A parameter is a limit of sorts on a function or the solutions to a problem.

Constant:g = gravitational acceleration constant on earth = 9.8 m/s^2, this is a constant because it never changes.

C = constant of the anti-derivative of a function, the C does not have a fixed value, instead you can pick any value for it but once you do, you must stick with that value throughout a problem.

• Mathematical Analysis - is, simply put, the study of limits and how they can be manipulated. Starting with an exhaustive study of sets, mathematical analysis then continues on to the rigorous development of calculus, differential equations, model theory, and topology. Topics including real and complex analysis, differential equations and vector calculus can be discussed in this category.

Example: 1.) In the equation of a line Ax + By + C = 0A – coefficient of x; B – coefficient of y and C the constant

2. ) in the slope intercept form a straight line3: y = mx + bx and y = are the variable coordinates of a point moving along their linem and b = arbitrary constant

Function – when two variables are so related that the value of the first variable depends on the value of the second, then the said variable is said to be a function of the second.

Example: Area of a circle : A = πr2

A = 1st variable – dependent variabler = 2nd variable – independent variable

Function is always a relation but not all relations are functions.Relations are function if no vertical line intersect the graph in more than one point. It

is an association between elements of two sets where a change of one variable may result in a corresponding variable change in the value of another variable. A relation in which no two pairs in a set of ordered pairs have the same first number is a function.

Example:1.) The interest on deposit is a function of the amount of deposit. It means that interest depends in the amount deposited.

2.) The circumference of a circle is a function on its radius.

Functional Notation – the symbol f(x) is used to express a function of x and is read as f of x.

Letters u, v, w are used to denote function of x.U = f(x)V = g(x) 3 different function of xW = ϕ(x)

Examples: 1.) f(x) = x2 – 4x + 5f(sin ϴ ) = sin2ϴ - 4 sinϴ + 5

2.)g(x) = x3 – 2x + 3g(s – 1) = (s -1)3 – 2(s – 1) + 3= s3 – 3s2 – 3s + 1

3.) ϕ (x) = x3 + 5x – 3ϕ(cosϴ) = cos3ϴ + 5 cosϴ - 3

Graph of Function: Continuity

The graph of function is continuous if for all the independent variable, there is a corresponding value for dependent variable. Example: y3

if x = -2, -1, 0, 1, 2x y y

-2 -8-1 -10 0 x1 12 8

Function – is a rule which establishes a correspondence between a set X ofreal number x to a set Y of real number y, where the number y is uniquefor a specific value of x. We call y a function of x and write y = f(x), g(x) …the letter f, g, symbolizing the function while f(a), g(a) …denote the valueof the function at x = a. A function is also an ordered pairs (x, y).

Function is a relation between two sets such that to each element of thedomain (input) there corresponds exactly one element of the range(output).

Another way of showing a function is through an arrow diagram:

x y

f

x f(x)

a f(a)

f: x y

The set of values which x can assume is called the domain of the function f; yis the range of f

Example:

1.) X2 + Y = 1 2.) Y2 – X = 1

Solving for y

X2 + Y = 1 y2 – x = 1

y = 1 – x2 y2 = 1 + x

y = ± √ 1 + x

Relation – id defined by a set of ordered pairs or by a rule that determineshow the ordered pairs are found

Domain Range

1 3

2 6

3 9

Limits - is the value that a function or sequence "approaches" as the input or index approaches some value.

Suppose f is a real-valued function and c is a real number. The expression

Lim f(x) = L read as the limit of f(x) as f(x)

x a approaches “a” is L. Zero maybe used as a limit, approach both direction.

Theorems on Limits:

1 – the limit of the algebraic sum of several function is equal to their sum of their limit

II-The limit of the product of several functions is equal to the product of their limits

III-The limit of the quotient of two functions is equal to the quotient of their limits, provided the denominator is not zero.

Supposed u, v and w are functions of variable x

Lim u = L1 lim v = L2 limit w = L3

x a x a x a

Theorem 1= lim (u + v + w) = L1 + L2 + L3

x a

Theorem II = lim (u. v. w) = L1 . L2 . L3

x a

Theorem III = Lim u/v = L1/L2 , provided L2 is not zerox a not zero.

If K is a constant and L2 is not zero

Lim (u + K) = L1 + K, lim KU = KL , lim K/v = K/L2

x a x a x a

Proof of Theorem

Let lim f1(x) = L1

x a

and lim f2(x) = L2

x a

Then lim f3(x) = L3

x a

Examples: Evaluate lim (x2 + 3x)

x a

Answer: lim x = 1, lim x2 = 1 , lim 3x = 3

x 1 x 1 x 1

Therefore: lim (x2 + 3x) = 1 + 3 = 4

x a

2.Evaluate: lim x2

x 3

Therefore: lim x2 = limx . Limx = 3 . 3 = 9

x 3 X 3 X 3

3. Evaluate: lim y2 – 4

y 4 y – 2

= lim (42 – 4) = 12

y 4

= lim (4 – 2) = 2

y 4

Therefore: lim y2 – 4 = 12 = 6

y 4 y – 2 2

Exercises: Evaluate the given operations

1.) f(x) = x3 – 3x, x = 4

2.) f(x) = x2 + 8, x = 9

3.) f(x) = x4, x = 5

4.) f(x) = y3 – 8, x = 3

y – 2

5.) f(x) = (x3 - 3x2 + 4), x = - 1

Differentiation and ApplicationIncrement of a variable is the very small change in the value of

the variable

An increment of x is denoted by the symbol ∆x and is called “delta x”

∆y denotes and increment of y

∆ϕ denotes an increment of ϕ

∆t denotes an increment of time

Ratio of increment of ∆y and ∆x

Consider the function y = x2, let x take on an increment of ∆x an y = x2 1

Y + ∆y = (x + ∆x)2

expanding Y + ∆y = ( x2 + 2x∆x + ∆x2) 2

Subtract y = x2 1

∆y = 2x∆x + ∆x2

We get ∆y in terms of x and ∆x

To find the ratio of the increments, divide both numbers of the equation by ∆x; ∆y = 2x + ∆x

∆x

If ∆x is very small approaching zero, the ratio ∆y will approach 2x

∆x

In symbols: lim ∆y = 2x

∆x 0 ∆x

If the fixed value of x is 2; lim ∆y = 4

∆x 2 ∆x