MA1506 LECTURE NOTES CHAPTER 1 DIFFERENTIAL EQUATIONS 1.1 Introduction A differential equation is an equation that contains one or more derivatives of a differentiable function. [In this chapter we deal only with ordinary DEs‚ NOT partial DEs.] The order of a d.e. is the order of the equation’s highest order derivative; and a d.e. is linear if it can be put in the form any (n)(x)+an−1y (n−1)(x)+· · ·+a1y (1)(x)+a0y(x) = F‚ 1 where ai‚ 0 ≤ i ≤ n‚ and F are all functions of x. For example
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Since the third letter of my last name is the letter L‚ I have to solve problem 46 on page 636 and problem 48 on page 646. This first problem will be solved by factoring the polynomial formula of ax^2+bx+c to get a*c(-30) and to add up to (-1) we have a=2‚b=-1‚and c=-15=0. The new equation is (2m^2-m-15) Original equation: 2m^2-m-15=0 Step 1. (2m^2-6m+5=15=0) To start we separate the fraction. Then solve both sides of the original
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I. Greek Mathematicians Thales of Miletus Birthdate: 624 B.C. Died: 547-546 B. C. Nationality: Greek Title: Regarded as “Father of Science” Contributions: * He is credited with the first use of deductive reasoning applied to geometry. * Discovery that a circle is bisected by its diameter‚ that the base angles of an isosceles triangle are equal and that vertical angles are equal. * Accredited with foundation of the Ionian school of Mathematics that was a centre of learning and research
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CCE for SSLC 2014 - 15 Subject : Mathematics Dimension 1 sl.no. Unit no.of periods marks 1 Real numbers 2 2 Sets 2 3 Progressions 4 Permutations and combinations 5 5 Probability 3 6 Statistics 4 7 Surds 3 8 Polynomials 4 9 Quadratic equations 10 10 10 Similar triangles 6 11 Pythagoras theorem 4 12 Trigonometry 6 13 Co-ordinate geometry 4 14 Circle - chord properties 1 15 Circles - tangent properties 9 Dimension – 2 Weightage to objectives 1 2
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slightly less ignorant 8th grader who barely made USAJMO. In 7th grade‚ I did not even know what the nine-point circle was‚ and now in 8th grade‚ I am able to prove its existence. In 7th grade‚ I could not work with polynomials and roots very well‚ and now in 8th grade‚ I can use polynomials quite efficiently. In 7th grade‚ I failed the AMC 10 with a pathetic score of 123‚ and now in 8th grade‚ I was able to score well enough on the AIME to qualify me for USAJMO. In 7th grade‚ I was complete garbage
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Finite Element Analysis Introduction Finite element analysis (FEA) has become commonplace in recent years‚ and is now the basis of a multibillion dollar per year industry. Numerical solutions to even very complicated stress problems can now be obtained routinely using FEA‚ and the method is so important that even introductory treatments of Mechanics of Materials – such as these modules – should outline its principal features. In spite of the great power of FEA‚ the disadvantages of computer
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Quadratic equation is a squared plus b squared = c squared. It’s used to find the length of three sides of a triangle. The theory is the same as any other polynomial‚ and the Greeks found out that‚ this formula holds true‚ regardless of the different types or lengths of the sides. So‚ by using the same method you use to solve a typical polynomial‚ you can solve this equation as well. For the formula to actually work you must have your equation in this form‚ quadratic=0. The 2a in the bottom of the
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the value for x into the equation y = (1/2)(4) + b Now put value of y in the equation 2 = (1/2)*(4) + b Now switch b so it can be on the opposite side of the equation (1/2) *(4) + b = 2 Multiply ½ by 4 to come up with 2 2 + b = 2 Reorder polynomials b + 2 = 2 Find be value: b = 2-2 b = 0 Finally‚ to get the equation of the line use the values of the slope (m) and y-intercept (b) in the formula y = mx + b to find the equation of the line. So the end result is y = x/2 Here is a graph
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Outline Rene Descartes I. All about Rene Descartes A. Rene Descartes is born B. Family C. Education D. Interests & Hobbies E. Passing F. Why he was important G. Philosophical Work‚ Scientific Discoveries‚ & Mathematical Findings II. Rene Descartes is born 1. When & where he was born A. March 31‚ 1596 B. La Haye en Touraine‚ Touraine (present-day Descartes‚ Indre-et-Loire)‚ France C. Parents: Jeanne Brochard (mother)‚ Joachim (father) D. Father
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outside c) on d) as vertex of viii) Sutherland – Hodgman algorithm is used for _______________. a) line clipping b) point clipping c) polygon clipping d) hybrid clipping d) Newton polynomials d) animation ix) The blending functions of Bezier curves are_______________. a) Splines b) Bernstein polynomials c) Lagrangian polynomials x) Z-buffer algorithm is used for _______________. a) frame buffer removal b) visible surface detection xi) Refresh rate is
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