Introduction to Calculus

Raymund T. de la CruzMAEd - Mathematics

Pure vs Applied Mathematics

Pure Mathematics - the branches of mathematics that study and develop the principles of mathematics for their own sake rather than for their immediate usefulness.

Applied Mathematics - the branches of mathematics that are involved in the study of the physical or biological or sociological world

Pure vs Applied MathematicsSo what’s the difference?...The difference between Pure Mathematics and Applied Mathematics lies in application and use. Pure Mathematics are studied with no consideration of necessity or application; they develop the principles of Mathematics for the sake of the principles of Mathematics. Research in the Fibonacci Sequence is an example of this: the Fibonacci Sequence has almost no useful application to mankind. Applied Mathematics, on the other hand, are studied purely for the sake of application. Examples of this lie in Economics, Computer Science, and Engineering.

Review of basic mathematics concept and definition used in numerical analysis…

EXPONENTSSimplify the following.(a) (b) (c) (d)

ALGEBRAIC IDENTITIES & ALGEBRAIC EXPRESSIONS(a) Expand (b) Simplify

SOLVING LINEAR EQUATIONS(a) Solve (b) Solve

Review of basic mathematics concept and definition used in numerical analysis…

SOLVING QUADRATIC EQUATIONSSolve the following:(a) (b)

SOLVING LINEAR INEQUALITIESSolve the following:(a) (b)

SOLVING QUADRATIC INEQUALITIESSolve the following:(a) (b)

Review of basic mathematics concept and definition used in numerical analysis…

SETSA. Let A = {2, 3, 5}, B = {2, 5, 6, 8}, and C = {1, 2, 3}. Find the following sets.(1) A B (2) A B (3) (A B) CB. Let and let A = { is a prime number}B = {is an even number}C = { is divisible by 3}.(1) (A C) (B C) (2) (AB)’ (3) A’ B’

Graphical representation of different types of the rectangular coordinate and polar coordinate system…A RECTANGULAR COORDINATE SYSTEM consists of two perpendicular real number lines called coordinate axes, that intersect at their origins. Generally one line is horizontal and called the x-axis, and the other is vertical called the y-axis. The axes divide the coordinate plane into four parts, called quadrants. Points on the axes are not in any quadrant.A ONE-TO-ONE CORRESPONDENCE exists between ordered pairs of numbers ( and points in the coordinate plane. Thus,1. To each point there corresponds an ordered pair of numbers called coordinates of is called the abscissa; is called the ordinate.2. To each ordered pair of numbers there corresponds a point, called the graph of the ordered pair.

𝑃 (𝑎 ,𝑏)

THE DISTANCE BETWEEN TWO POINTS and in the plane coordinate system is given by the distance formula:Example: Find the distance between (3, -2) and (5, 3).

V(+,-)I(-,-)

(-,+) (+,+)𝑥

𝑦

THE GRAPH OF AN EQUATION in two variables is the graph of its solution set, that is, of all ordered pairs that satisfy the equation. Since there are ordinarily an infinite number of solutions, a sketch of the graph is generally sufficient.INTERCEPTS: The coordinate of the points where the graph of an equation crosses the x-axis or y-axis. To find the x-intercept, set y = 0 and solve for x and to find the y-intercept, set x = 0 and solve for y.Example: Find the intercepts of the graph of the equation .Set x = 0; then Hence the y-intercept is 4.Set y = 0. If then thus Hence 2 & -2 are the x-intercepts.THE MIDPOINT OF A LINE SEGMENT with endpoints and is given by the midpoint formula. Midpoint of Example: Find the midpoint between (6, -3) and (4, -5).Midpoint =

A point P in the Cartesian plane can be described, apart from using the known coordinates (x, y), by polar coordinates (r, θ ), which are defined as follows. Denote by r the distance of P from the origin O. If r > 0 we let θ be the angle, measured in radians up to multiples of 2π, between the positive x-axis and the half-line emanating from O and passing through P.

𝑥

𝑦

𝑂

𝑃=(𝑥 , 𝑦)

𝑟

𝜃

It is common to choose θ in (−π, π], or in [0, 2π). When r = 0, P coincides with the origin, and θ may be any number.The passage from polar coordinates (r, θ) to Cartesian coordinates (x, y) is given by x = r cos θ , y = r sin θ .

Example:1. Let have Cartesian coordinates = As .The polar coordinates of are then

2. Take of polar coordinates this time; in the Cartesian systemy

EXERCISES:A. Find the distance between the following pair of points in the plane coordinate system.1. (-3, 4) and (-5, -1) 2. (6, 2) and (-2, 3) 3. (1, -5) and (-2, 2)B. Determine the intercepts of the given equations.1. 2. 3. C. Determine the midpoint from the given endpoints of a line.1. (-4, 7) and (6, -3) 2. (1, 11) and (-3, -3) 3. and D. Find the polar coordinates of the following points in the plane:1. 2. E. Find the rectangular coordinates from the given polar coordinates:1. 2.

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