Prerequisites for Calculus

Coordinate Geometry Increments Increments Slope Slope Parallel and Perpendicular Lines Parallel and Perpendicular Lines Standard equations of Lines Standard equations of Lines Regression analysis Regression analysis

Contd. Increments in x = Δx= x 2 – x 1 Increments in x = Δx= x 2 – x 1 Slope = m = rise/run = (y 2 – y 1 )/(x 2 – x 1 ) Slope = m = rise/run = (y 2 – y 1 )/(x 2 – x 1 ) Two lines are parallel if m 1 = m 2 Two lines are parallel if m 1 = m 2 Two lines are perpendicular if m 1 m 2 = -1 Two lines are perpendicular if m 1 m 2 = -1 Equation of a vertical line through (a,b) is x = a Equation of a vertical line through (a,b) is x = a Equation of a horizontal line through (a,b) is y = b Equation of a horizontal line through (a,b) is y = b

Contd. Point-slope equation y = m(x – x 1 ) + y 1 Point-slope equation y = m(x – x 1 ) + y 1 Slope-intercept equation y = mx + b Slope-intercept equation y = mx + b General equation of line Ax + By = C General equation of line Ax + By = C If A(2, 3) and B(5, -7), find Equation of line vertical through A Equation of line vertical through A Equation of line through A and B Equation of line through A and B Equation of line perpendicular to AB passing trough B Equation of line perpendicular to AB passing trough B

Functions and Graphs Function – Vertical line test Function – Vertical line test Domain and Range Domain and Range Recognizing graphs: Linear, Quadratic, Cubic, Rational, Exponential, Logarithmic, and Trigonometric Functions. Recognizing graphs: Linear, Quadratic, Cubic, Rational, Exponential, Logarithmic, and Trigonometric Functions. Even and Odd Functions Even and Odd Functions Piecewise functions Piecewise functions Absolute value functions Absolute value functions Composite functions Composite functions

Contd. A vertical line intersects the graph of a function in no more than one point. A vertical line intersects the graph of a function in no more than one point. All the x-values constitute the Domain All the x-values constitute the Domain All the y-values constitute the Range. All the y-values constitute the Range. An even function is symmetric about the Y-axis. An even function is symmetric about the Y-axis. An odd function is symmetric about the origin. An odd function is symmetric about the origin.

Contd. In a piece-wise function, different formulas are used to define the function in different parts of the domain. In a piece-wise function, different formulas are used to define the function in different parts of the domain. Absolute value function y = abs(x) is defined as a piece-wise function y = -x, x<0 and y = x, x ≥0 Absolute value function y = abs(x) is defined as a piece-wise function y = -x, x<0 and y = x, x ≥0 The composite function of g and f is defined as f(g(x)) and notation for this is f o g. The composite function of g and f is defined as f(g(x)) and notation for this is f o g.

Logarithmic and Exponential Properties log m + log n = logmn log m + log n = logmn log m – log n = log(m/n) log m – log n = log(m/n) mlogx = logx^m mlogx = logx^m a^x ·a^y = a^(x+y) a^x ·a^y = a^(x+y) a^x /a^y = a^(x– y) a^x /a^y = a^(x– y) (a^x)^y = a^xy (a^x)^y = a^xy a^-1 = 1/a a^-1 = 1/a

Functions and graphs Linear : y = ax + b : Straight line Linear : y = ax + b : Straight line Quadratic: y = ax^2 + bx + c Parabola Quadratic: y = ax^2 + bx + c Parabola Absolute value: y = | x | Absolute value: y = | x | : v – shaped Trigonometric Functions Trigonometric Functions Exponential and logarithmic function Exponential and logarithmic function

Answer the following Factorize: Factorize: Simplify: Simplify: Graph the given functions: Graph the given functions: Solve: Solve:

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