====== Definitions: FSc Part 2 (Mathematics): PTB ====== On this page, all the definitions of "Calculus and Analytic Geometry, MATHEMATICS 12" (Mathematics FSc Part 2 or HSSC-II), Punjab Textbook Board (PTB) Lahore, Pakistan are given. We are very thankful to **Muhammad Waqas Sulaiman** for his valuable contribution. Definitions by Mr. [[people:Muzzammil-Subhan]] are given at [[:fsc-part2-ptb:definitions-Muzzammil-Subhan]] NEW ===== Unit 01 (Functions and Limits) ===== * **Function:** A function is a rule or correspondence, relating to two sets in such a way that each element in the first set corresponds to one and only one element in the second set. Or \\ A function from //X// to //Y// is a rule that assigns to each element //x// in //X// a unique element //y// in //Y//. \\ e.g. $A=x^2$, that is, //A// is a function of //x//. * **Domain:** In a function $f:X\to Y$ the set $X$ is called the domain of function. * **Range:** In a function $f:X\to Y$ the set of corresponding elements $y$ in $Y$ is called range of function. * **Independent and Dependent Variables:** If a variable //y// depends on a variable //x// in such a way that each value //x// determines exactly one value of //y//, then we say that //y// is function of //x//. The variable //x// is called independent variable and //y// is called dependent variable.\\e.g. in $y=ax+b$, the $x$ variable is called independent and $y$ is called dependent variable of function. * **Algebraic Function:** The function which are defined by variable expressions. e.g. $f(x)=2x-6$. * **Polynomial Function:** A function of the form $p(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} + ... + {a_1}x + {a_0}$, where ${a_0},\,{a_1},\,{a_2},...,{a_n}$ are real number and exponent are non-negative integers is called polynomial function. * **Linear Function:** If degree of polynomial function is one it is called linear function. e.g $f(x)=ax+b$. * **Identity Function:** For any set $X$, a function $I:X\to X$ of f. * **Constant Function:** Let $X$ and $Y$ be the set of real numbers. A function $C:X \rightarrow Y$ defined by $C(x)=a$ for all $x \in X$, $a \in Y$ and fixed, is called a constant function. * **Rational Function:** A function $R(x)$ of the form $\frac{P(x)}{Q(x)}$, where both $P(x)$ and $Q(x)$ are polynomial functions and $Q(x)\neq 0$, is called rational function. * **Exponential Function:** A function, in which variable appears as exponent (power) is called an exponential function. The functions, $y=e^{ax}$, $y=e^x$, $y=2^x$, etc are called exponential functions. * **Logarithmic Function:** If $x=a^y$, then $y=log_ax$, where $a<0$, $a\neq 1$. * **Explicit Function:** If $y$ is easily expressed in terms of the independent variable $x$, then $y$ is called an explicit function of $x$. ===== Unit 02: Differentiation ===== * ** Differentiation:** Instantaneous rate of change of one variable with respect to other variable is called derivative or differentiation. Its denoted by $\frac{dy}{dx}$. * ** Increasing function:** A function $f$ is defined on an interval $[a,b]$ is said to be increasing function on $[a,b]$ if $f(x_1)>f(x_2)$ whenever $x_10$. * ** Turning point:** A stationary point is called turning point if it is either a maximum point or a minimum point. * ** Point of inflection:** A stationary point is called point of inflection if a function have neither local maxima nor local minima at that point. ===== Unit 03: Integration ===== * ** Integration or Anti-derivative:** Inverse process of differentiation is called integration. * ** Differential equation:** An equation containing at least one derivation of a dependent variable w.r.t an independent variable. e.g. $y\frac{dy}{dx}+x=0$. * ** order of differential equation:** The order of a differential is the order of the highest derivative in the equation. ===== Unit 04: Analytical geometry ===== * ** Coordinate Axes:** In a mutually perpendicular number lines $X'OX$ and $Y'OY$, one horizontal and the other vertical, are called the coordinate axes. * ** Coordinate of point:** Coordinate of a point represents an ordered set of numbers that defines the position of a point in a plane or space. * ** Analytic geometry:** Analytic geometry (coordinate geometry) is the branch of mathematics. Analytical geometry is a fusion of algebraic equation and geometry curves. * ** Centroid of a triangle:** The centroid of a triangle is the point of intersection of its medians. * ** Ortho-center:** The ortho-center of a triangle is the point of intersection of the lines through the vertices and perpendicular to the opposite side (altitudes) of the triangle. * ** Circum-center:** The point of intersection of the perpendicular bisectors of the sides of a triangle is called cirum-center. * ** In-center:** The point of intersection of the internal bisectors of the angles of a triangle is called its in-center. It is the center of the circle, which touches the sides of the triangle. * ** Inclination of a line:** The angle $\alpha(0<\alpha<180)$ measured counter clockwise from +ve axis to a non-horizontal straight line is called the inclination of line. * ** Slope of line:** The slop $(m)$ of the non-vertical straight line with $\alpha$ as inclination is define by $m=\tan \alpha$. * ** Homogeneous equation:** An equation in two variable $x$ and $y$ is said to be homogeneous equation if sum of exponents of $x$ and $y$ in each term remains same. The equation in this form $ax^2+2hxy+y^2=0$ (where $a$, $b$ and $h$ simultaneously not zero) is called general form of Homogeneous equation. ===== Unit 05: Linear Inequalities and Linear Programming ===== * ** Inequality:** An inequality involving any one of the symbols $<$(less then), $>$(greater then), $\leq$(less then equal to), $\geq$(greater then equal to) is called inequality. If $ax+b

Download PDF (160KB)

{{tag>FSc FSc_Part2}}