1 / 52

PREAMBLE OF MATHEMATICS II EAS-203

PREAMBLE OF MATHEMATICS II EAS-203. PREAMBLE STRUCTURE HOLLISTIC FIX KEY CONCEPT KEY RESEARCH AREA KEY APPLICATION INDUSTRIAL APPLICATION RESEARCH HOW WE STUDY KEY JOBS PROJECTS ONE CAN DO TRENDS. INDEX. Introduction To Mathematics II Why? What? Where? How?.

nardo
Download Presentation

PREAMBLE OF MATHEMATICS II EAS-203

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PREAMBLE OFMATHEMATICS IIEAS-203

  2. PREAMBLE STRUCTURE • HOLLISTIC FIX • KEY CONCEPT • KEY RESEARCH AREA • KEY APPLICATION • INDUSTRIAL APPLICATION • RESEARCH • HOW WE STUDY • KEY JOBS • PROJECTS ONE CAN DO • TRENDS INDEX

  3. Introduction To Mathematics II Why? What? Where? How?

  4. 1. NAME OF THE INSTRUCTOR 2. CABIN LOCATION 3. TELEPHONENO. 4. EMAIL-ID 5. MEETING HOURS TEACHERS INTRODUCTION

  5. PREREQUISITES • Basic Knowledge • Trigonometry, Series & Progression, Logarithms, Permutations & Combinations (10th & 12th standard) • Differentiation ( 12th standard) • Integration ( 12th standard) • Partial Differentiation (B. Tech I Sem) HOLLISTIC FIX OF MATHEMATICS II

  6. Basics in TrigonometryAngles, Arc length, Conversions Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative. Coterminal Angles: Angles that have the same terminal side. 60°, 420°, and –300° are all co terminal. Degrees to radians: Multiply angle by radians Radians to degrees: Multiply angle by Note: 1 revolution = 360° = 2π radians. Arc length = central angle x radius, or Note: The central angle must be in radian measure.

  7. Right Triangle Trig Definitions sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a B c a A C b

  8. Special Right Triangles 30° 45° 2 1 60° 45° 1 1

  9. Basic Trigonometric Identities Quotient identities: Even/Odd identities: Even functions Odd functions Odd functions Reciprocal Identities: Pythagorean Identities:

  10. All Students Take Calculus. Quad II Quad I cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0 Quad IV Quad III

  11. Reference Angles Quad I Quad II θ’ = 180° – θ θ’ = θ θ’ = π – θ θ’ = θ– 180° θ’ = 360° – θ θ’ = θ– π θ’ = 2π – θ Quad III Quad IV

  12. Unit circle Radius of the circle is 1. x = cos(θ) y = sin(θ) Pythagorean Theorem: This gives the identity: Zeros of sin(θ) are where n is an integer. Zeros of cos(θ) are where n is an integer.

  13. Graphs of sine & cosine Fundamental period of sine and cosine is 2π. Domain of sine and cosine is Range of sine and cosine is [–|A|+D, |A|+D]. The amplitude of a sine and cosine graph is |A|. The vertical shift or average value of sine and cosine graph is D. The period of sine and cosine graph is The phase shift or horizontal shift is

  14. Sine graphs y = sin(x) y = sin(x) + 3 y = 3sin(x) y = sin(3x) y = sin(x – 3) y = 3sin(3x-9)+3 y = sin(x) y = sin(x/3)

  15. Graphs of cosine y = cos(x) y = cos(x) + 3 y = 3cos(x) y = cos(3x) y = cos(x – 3) y = 3cos(3x – 9) + 3 y = cos(x) y = cos(x/3)

  16. The Binomial Theorem The Binomial Theorem:

  17. Differentiation Rules

  18. Differentiation Rules

  19. TABLE OF INTEGRATION FORMULAS

  20. TABLE OF INTEGRATION FORMULAS

  21. TABLE OF INTEGRATION FORMULAS

  22. TABLE OF INTEGRATION FORMULAS

  23. Evaluating Definite Integrals By the fundamental theorem we can evaluate easily and exactly. We simply calculate

  24. Definition of Even and Odd Functions Analytically, f is an even function if its domain contains the point –x whenever it contains x, and if f (-x) = f (x) for each x in the domain of f. See figure (a) below. The function f is an odd function if its domain contains the point –x whenever it contains x, and if f (-x) = - f (x) for each x in the domain of f. See figure (b) below. Note that f (0) = 0 for an odd function. Examples of even functions are 1, x2, cos x, |x|. Examples of odd functions are x, x3, sin x.

  25. Arithmetic Properties The following arithmetic properties hold: The sum (difference) of two even functions is even. The product (quotient) of two even functions is even. The sum (difference) of two odd functions is odd. The product (quotient) of two odd functions is even. The product (quotient) of an odd and an even function is odd.

  26. Integral Properties If f is an even function, then If f is an odd function, then

  27. Definite Integral Properties

  28. ordinary differential equations Definition: A differential equation is an equation containing an unknown function and its derivatives. Examples:. 1. 2. 3. y is dependent variable and x is independent variable, and these are ordinary differential equations

  29. Partial Differential Equation Examples: 1. u is dependent variable and x and yare independent variables, and is partial differential equation. 2. 3. u is dependent variable and x and tare independent variables

  30. Order of Differential Equation The order of the differential equation is order of the highest derivative in the differential equation. Differential Equation ORDER 1 2 3

  31. Degree of Differential Equation The degree of a differential equation is power of the highest order derivative term in the differential equation. Differential Equation Degree 1 1 3

  32. Linear Differential Equation • A differential equation is linear, if • 1. dependent variable and its derivatives are of degree one, • 2. coefficients of a term does not depend upon dependent variable. Example: 1. is linear. Example: 2. is non - linear because in 2nd term is not of degree one.

  33. First Order Ordinary Differential equation

  34. Origin of Differential Equations Solution • Geometric Origin 1. For the family of straight lines the differential equation is . 2. For the family of curves A. the differential equation is B. the differential equation is

  35. Physical Origin 1. Free falling stone where s is distance or height and g is acceleration due to gravity. 2. Spring vertical displacement where y is displacement, m is mass and k is spring constant 3. RLC – circuit, Kirchoff ’s Second Law q is charge on capacitor, L is inductance, c is capacitance. R is resistance and E is voltage

  36. Physical Origin • Newton’s Law of Cooling where is rate of cooling of the liquid, is temperature difference between the liquid ‘T’ and its surrounding Ts 2. Growth and Decay y is the quantity present at any time

  37. Solving First Order Differential Equations Special Case 1:y = f(x) (a function of x only) Method: Just integrate both sides indefinitely (or definitely using the Fundamental Theorem of Calculus)

  38. Separable First Order DEs Special Case 2:y = dy/dx = g(x)H(y) Method: Separate x and dx on one side from y and dy on the other, then perform an indefinite integration.

  39. Linear First Order DEs Special Case 3:y + P(x)y = Q(x) Method: I.F.=eP(x)dx. Solution y I.F. = Q(x)I.F.dx + constant

  40. Complex Number : Definition A complex numberz is a number of the form where x is the real part and y the imaginary part, written as x = Re z, y = Im z. j is called the imaginary unit If x = 0, then z = jy is a pure imaginary number. The complex conjugate of a complex number, z = x + jy, denoted by z* , is given by z* = x – jy. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

  41. Complex Plane A complex number can be plotted on a plane with two perpendicular coordinate axes The horizontal x-axis, called the real axis The vertical y-axis, called the imaginary axis Represent z = x + jy geometrically as the point P(x,y) in the x-yplane, or as the vector from the origin to P(x,y). The complex plane x-y plane is also known as the complex plane.

  42. Polar Coordinates With Note that : z takes the polar form: r is called the absolute value or modulus or magnitude of z and is denoted by |z|.

  43. Complex plane, polar form of a complex number Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x-axis to OP in the above figure. From the figure,

  44. θ is called the argument of z and is denoted by arg z. Thus, For z = 0, θ is undefined. A complex number z≠ 0 has infinitely many possible arguments, each one differing from the rest by some multiple of 2π. In fact, arg z is actually The value of θthat lies in the interval (-π, π] is called the principle argument of z (≠ 0) and is denoted by Arg z.

  45. Euler Formula – an alternate polar form The polar form of a complex number can be rewritten as : This leads to the complex exponential function : Further leads to :

  46. Hyperbolic Functions Definition tanh cosh and sinh The graph of the cosh function is a catenary curve. It represents a freely hanging cable.

  47. Hyperbolic Identities Formulae

  48. Differentiation of Hyperbolic Functions Formulae

  49. Inverse Hyperbolic Functions Formulae

More Related