Mathematics-1 Lecture Notes PDF

MATHEMATICS-1 LECTURE 2 2019-1ST LEVEL BIO/SWE PROGRAMS Dr. Ahmed Kafafy Operation Research & Decision Support Dept. Faculty of computers & information, Menoufia University Course outlines – General Pre-calculus review – Limits and continuity – Differentiation – The Mean value theorem – Integrat...

MATHEMATICS-1 LECTURE 2 2019-1ST LEVEL BIO/SWE PROGRAMS Dr. Ahmed Kafafy Operation Research & Decision Support Dept. Faculty of computers & information, Menoufia University Course outlines – General Pre-calculus review – Limits and continuity – Differentiation – The Mean value theorem – Integration and its application – Techniques of integration General Pre-calculus review – Review of elementary math. – Inequalities – Coordinate plane – analytical geometry – Functions and their combinations – Function composition Functions ‰ The fundamental process of calculus (differentiation & integration) are applied to functions. ‰ we will concern with real valued functions of real variables, ‰ A function f is a process of mapping each element x in its domain to a unique image f(x) in its range Domain and Range Example: Squaring function ݂ ‫ ݔ‬ൌ ‫ ݔ‬ଶǡ ‫ܴא ݔ׊‬ Dom( f ) Î ܴ ൌ ሺെλ,, λሻ Î ܴା ൌ Ͳǡ λ Range(f)Î Example: The function g(x) maps [0,6] onto [2,4] Domain and Range Some functions are definedd piecewise x) mapss ሺെλ,, λሻ ֜ ሺെλ,, λሻ h(x) A more familiar example is the absolute value function x) mapss ሺെλ,, λሻ ֜ ሾͲ,, λሻ f(x) Remark, this form is often used: ‫ݕ‬ൌ݂ ‫ݔ‬ ‫ ݕ‬is the dependent variable ‫ ݔ‬is the independent variable Graph of functions The graph of a function f with domain D is the graph of the equation ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ with ‫ ݔ‬restricted to ‫ܦ‬ šƒ’Ž‡ǣ ࢌ ࢞ ൌ ࢞૛ ǡ ࢞ ‫ א‬ሺെλǡ λሻ Graph of functions The graph of a functions Is a function or not ? Is this graphs for a function or not? Hint: Use vertical line Test is a function is not a function is not a function f(x) is a function iff Each ‫ ܦ א ݔ‬has only one image f(x) Even & Odd functions , Symmetry For even integer ݊ǡ ሺെ‫ݔ‬ሻ௡ = ‫ ݔ‬௡ , But for odd integer ݊ǡ ሺെ‫ݔ‬ሻ௡ = െ‫ ݔ‬௡ Even :Symmetric about y-axis y- Odd: Symmetric about origin Convention on domain If the domain of a function f is not given, then we take as domain the maximal set of real numbers x which f(x) is a real number. Example: give the domain of each function Solution: (a) f(x) is real number oŽ› ‹ˆ ‫ ݔ‬ଶ ൅ ‫ ݔ‬െ ͸ ് ͲǤ Since ‫ ݔ‬ଶ ൅ ‫ ݔ‬െ ͸ ൌ ‫ ݔ‬൅ ͵ ‫ ݔ‬െ ʹ ֜ Thus ‫ ܴ א ݔ‬െ െ͵ǡʹ (b) For g(x) to be a real number, we need: ૝ െ ࢞૛ ൒ ૙ and ࢞ ് ૚Ǥ Since ૝ െ ࢞૛ ൒ ૙ iff ࢞૛ ൑ ૝ iff െ૛ ൑ ࢞ ൑ ૛ Thus: ࢞ ‫ א‬െ૛ǡ ૛ െ ૚ ֜ ሾെ૛ǡ ૚ሻ ‫ ׫‬ሺ૚ǡ ૛ሿ The Elementary functions The functions that figure most prominently in single-variable calculus are the polynomials, the rational functions, the trigonometric functions, the exponential functions, and the logarithm functions. Polynomials : A function of the form: where the coefficients ܽ௡ ǡ ܽ௡ିଵ ǡ ǥ ǡ ܽ଴ ‫ܴ א‬ǡ and ܽ௡ ് Ͳ is called a (real polynomial of degree ݊) 0 degree Î‫ ݕ‬ൌ ܿ line 1st degree Î‫ ݕ‬ൌ ܽ‫ ݔ‬൅ ܿ line 2nd degree Î ‫ ݕ‬ൌ ܽ‫ ݔ‬ଶ ൅ ܾ‫ ݔ‬൅ ܿ quadratic function 3rd degree Î ‫ ݕ‬ൌ ܽ‫ ݔ‬ଷ ൅ ܾ‫ ݔ‬ଶ ൅ ܿ‫ ݔ‬൅ ݀ cubic function The Elementary functions-Rational Rational function: A function of the form: ௉ሺ௫ሻ ܴ ‫ ݔ‬ൌ ொሺ௫ሻ where ܲ & Q are polynomials. dom(ܴሻ ൌ ሼ‫ݔ‬ǣ ܳ ‫Ͳ ് ݔ‬ሽ Note: every polynomial is rational function ܲ ‫ ݔ‬ൌ ܲሺ‫ݔ‬ሻȀͳ The Elementary functions Trigonometric functions: ‰ Degree measure, traditionally used to measure angles. ‰ Another way of measuring angles is: measuring angles in radians. ‰ In degree measure a full turn is effected over the course of 360°. ‰ In radian measure, a full turn is effected during the course of ʹߨ radians. (The circumference of a circle of radius 1 is ʹߨ ) Thus The Elementary functions Cosine & Sine: ࢟ ൌ ‫ܖܑܛ‬ሺࣂሻ Note the unit circle (radius =1, ‫ݕ‬ centered (0,0)) Let ߠ be any real number. The rotation ‫ݔ‬ ߠ takes ‫ ܣ‬ሺͳǡ Ͳሻ to some point ܲሺ‫ݔ‬ǡ ‫ݕ‬ሻǤ ࢞ ൌ ‫ܛܗ܋‬ሺࣂ) ‘• ߠ ൌ ‫ ݔ‬ǡ •‹ ߠ ൌ ‫ݕ‬ 9ࡼ has the coordinates ሺ‫ࣂ ܛܗ܋‬ǡ ‫ࣂ ܖܑܛ‬ሻ Ǥ 9 •‹ ߠ ൅ ʹߨ ൌ •‹ሺߠሻ ‘•ሺߠ ൅ ʹߨሻ ൌ ‘•ሺߠሻ 9 •‹ െߠ ൌ െ•‹ሺߠሻ ‘•ሺെߠሻ ൌ ‘• ߠ (even & odd) The Elementary functions Cosine & Sine: c) d) Sine is even & cosine is odd Tangent, cotangent, secant cosecant: ௬ ୱ୧୬ሺఏሻ Tangent –ƒ ߠ ൌ ֜ ௫ ୡ୭ୱሺఏሻ ଵ ଵ Cosecant • ߠ ൌ ֜ ௬ ୱ୧୬ሺఏሻ ଵ ଵ Secant •‡ ߠ ൌ ֜ ௫ ୡ୭ୱሺఏሻ ௫ ୡ୭ୱሺఏሻ Cotangent ‘– ߠ ൌ ֜ ௬ ୱ୧୬ሺఏሻ The Elementary functions Trigs in terms of right triangle The Elementary functions Trigs Identities i) Unit circle ii) Periodicity iii) Odd & even vi) Sine & cosine v) double- angle Graph of trigonometric y sin( x) !  1 d sin( x) d 1 y cos( x) !  1 d cos( x) d 1 y tan(x) y cot(x) Graph of trigonometric b) g ( x) cos( x) y csc( x) 1 / sin( x) y sec( x) 1 / cos( x) Combination of functions Algebraic Combinations On the intersections of their domain, Functions can be: => added => subtracted => Multiplied => form the quotient | g(x) ് 0 => form the reciprocal| g(x) ് Ͳ (αf )( x) α f ( x) => Multiplied by a scalar ߙ ‫א‬ ܴ (Df  Eg)( x) D f ( x)  Eg( x) => Linear combination by ߙ&ߚ ‫ܴ א‬ Combination of functions Example: given: & a) we can form ‫ ݔ‬൅ ͵ iff ‫ ݔ‬൅ ͵ ൒ Ͳ ֜ ‫ ݔ‬൒ െ͵ ֜ ‫ א ݔ‬ሾെ͵ǡ λሻ we can form ͷ െ ‫ ݔ‬െ ʹ iff ͷ െ ‫ ݔ‬൒ Ͳ ֜ ‫ ݔ‬൑ ͷ ֜ ‫ א ݔ‬ሺെλǡ ͷሿ b) ݀‫ ݂ ݉݋‬൅ ‰ ൌ ݀‫ ֜ ‰ ݉݋݀ ת ݂ ݉݋‬െ͵ǡ λ ‫ ת‬ሺെλǡ ͷሿ ൌ ሾെ͵ǡͷሿ ݂ ൅ ‰ ‫ ݔ ݂ ֜ ݔ‬൅ ‰ š ֜ ‫ ݔ‬൅ ͵+ ͷ െ ‫ ݔ‬െ ʹ ௙ c) For ݀‫݉݋‬ , we must exclude from െ͵ǡͷ , ‫ ݔ‬at which ‰ሺ‫ݔ‬ሻ=0 => ‫ ݔ‬ൌ ͳǤ ୥ ௙ ݀‫݉݋‬ ‫ א ݔ ֜ ݔ‬െ͵ǡͷ െ ͳ ֜ ሾെ͵ǡͳሻ ‫ ׫‬ሺͳǡͷሿ ୥ ݂ ݂ሺ‫ݔ‬ሻ ‫ݔ‬൅͵ ‫֜ ݔ‬ ֜ ‰ ‰ሺ‫ݔ‬ሻ ͷെ‫ݔ‬െʹ Vertical/horizontal shifts ‰ Adding/subtracting a positive constant c to a function raises/lowers the graph by c units. ‰ Adding/subtracting a Positive constant c to the argument shifts the graph left/right c units. ‰ Stretching the graph Composition of functions Composition of functions Example 1: Let ‰ ‫ ݔ‬ൌ ‫ ݔ‬ଶ & ݂ ‫ ݔ‬ൌ ‫ ݔ‬൅ ͵ : Find ݂‫ݔ ‰ל‬ & ‰‫ݔ ݂ל‬ ‰ ݂‫ݔ ‰ ݂֜ ݔ ‰ל‬ ֜ ݂ ‫ݔ‬ଶ ֜ ‫ ݔ‬ଶ ൅ ͵ Î first squares then adds 3 ‰ ‰‫ݔ ݂ ‰֜ ݔ ݂ל‬ ֜ ‰ ‫ ݔ‬൅ ͵ ֜ ሺ‫ ݔ‬൅ ͵ሻଶ Î first adds 3 then squares Example 2: Let: ݂ ‫ ݔ‬ൌ ‫ ݔ‬ଶ െ ͳ & ‰ ‫ ݔ‬ൌ ͵ െ ‫ ݔ‬, Find ݂ ‫݂ ל ‰ & ‰ ל‬ ‰ dom(‰ሻ ൌ (െλǡ ͵ሿǡ ݂is every where defined, Î thus ݀‫݉݋‬ሺ݂ ‫( =)‰ ל‬െλǡ ͵ሿ ݂‫ݔ ‰ ݂֜ ݔ ‰ל‬ ֜ ሺ ͵ െ ‫ݔ‬ሻଶ െ ͳ ֜ ͵ െ ‫ ݔ‬െ ͳ ൌ ʹ െ ‫ݔ‬ ‰ Since , ‰ ݂ሺ‫ݔ‬ሻ ൌ ͵ െ ݂ሺ‫ݔ‬ሻ, It must have ݂ሺ‫ݔ‬ሻ ൑ ͵ , thus x ‫ א‬െʹǡʹ Ǥ ‰‫ݔ ݂ ‰֜ ݔ ݂ל‬ ֜ ‰ ‫ ݔ‬ଶ െ ͳ ֜ൌ ͵ െ ሺ‫ ݔ‬ଶ െͳሻ ൌ Ͷ െ ‫ݔ‬ଶ Composition of functions Composition can be performed on more than two function, for example, the triple ݂ ‫ ݄ ל ‰ ל‬consists of first ݄ then ‰ and then ݂ : ݂‫ݔ ݄ ‰ ݂֜ ݔ ݄ל‰ל‬ ଵ Example 3: Let ݂ ‫ ݔ‬ൌ , ‰ ‫ ݔ‬ൌ ‫ ݔ‬ଶ ൅ ͳ ǡ ݄ ‫ ݔ‬ൌ ‘•ሺ‫ݔ‬ሻ thus: ௫ ૚ ݂ ‫݄ ‰ ݂ ֜ ݔ ݄ ל ‰ ל‬ሺ‫ݔ‬ሻ ֜ ݂ ‰ ‘•ሺ‫ݔ‬ሻ ֜ ݂ሺ ‘• ଶ ሺ‫ݔ‬ሻ ൅ ͳሻ ֜ ࢉ࢕࢙૛ ሺ࢞ሻା૚ ଵ ଵ ଵ ݂ ‰ ݄ሺ‫ݔ‬ሻ ֜ ֜ ֜ ୥ ௛ሺ௫ሻ ௛ሺ௫ሻమ ାଵ ௖௢௦ మ ሺ௫ሻାଵ ଵ Example 4: Find the function ݂ ǡ ‰ Ƭ ݄ •— Š–Šƒ–݂ ‫ ݄ ל ‰ ל‬ൌ ሺ‫ݔ‬ሻ ൌ ௫ ାଷ ሺ‫ݔ‬ሻ first takes the absolute value, adds 3 and then inverts ଵ Let ݄ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬, ‰ ‫ ݔ‬ൌ ‫ ݔ‬൅ ͵ and ݂ ‫ ݔ‬ൌ ௫ ଵ ଵ ଵ Thus ݂ ‫ ݄ ל ‰ ל‬ൌ ݂ ‰ ݄ ‫ݔ‬ ൌ ൌ = ୥ ௛ ௫ ୥ ௫ ௫ ାଷ