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Need an account? Click here to sign up. Download Free PDF. MTH Handouts full pdf. Umair Ali. A short summary of this paper. Well, it is the study of the continuous rates of the change of quantities. It is the study of how various quantities change with respect to other quantities.
For example, one would like to know how distance changes with respect to from now onwards we will use the abbreviation w. You want to know how this happens continuously. We will see what continuously means as well. We will go through the history of REAL numbers and how they popped into the realm of human intellect. The simplest numbers are the natural numbers Natural Numbers 1, 2, 3, 4, 5,… They are called the natural numbers because they are the first to have crossed paths with human intellect.
Think about it: these are the numbers we count things with. So our ancestors used these numbers first to count, and they came to us naturally! The natural numbers form a subset of a larger class of numbers called the integers. This could be a collection of oranges, apples, cars, or politicians. In mathematical notation we say A is subset of B if. Set The collection of well defined objects is called a set. We will get into the basic notations and ideas of sets later.
Going back to the Integers. Well, they are an artificial construction. They also have a history of their own. But that's not all. There is more. The integers in turn are a subset of a still larger class of numbers called the rational numbers. With the exception that division by zero is ruled out, the rational numbers are formed by taking ratios of integers. So every integer is also a rational. Why not divide by 0? Well here is why: If x is different from zero, this equation is contradictory; and if x is equal to zero, this equation is satisfied by any number y, so the ratio does not have a unique value a situation that is mathematically unsatisfactory.
So we have some logical inconsistencies that we would like to avoid. I hope you see that!! Hence, no division by 0 allowed! Now we come to a very interesting story in the history of the development of Real numbers. Pythagoras was an ancient Greek philosopher and mathematician. He studied the properties of numbers for its own sake, not necessarily for any applied problems. Now Pythagoras got carried away a little, and developed an almost religious thought based on math. Now rational numbers have a unique property that if you convert them to decimal notation, the numbers following the decimal either end quickly, or repeat in a pattern forever.
All is well. But this idea was shattered in the fifth century B. The hypotenuse of this right triangle can be expressed as the ratio of integers. It gave a way of describing algebraic formulas by geometric curves and, conversely, geometric curves by algebraic formulas. The developer of this idea was the French mathematician, Descartes. Well, he is said to have seated himself in a 17th century furnace it was not burning at the time!
In this world of cold and darkness, he felt all his senses useless. But he could still think!!!! In analytic geometry , the key step is to establish a correspondence between real numbers and points on a line. We could have done it the other way around too.
Moreover, this has now become a standard in doing math, so anything else will be awkward to deal with. The positive direction is usually marked with an arrowhead so we do that too. Then we choose an arbitrary point and take that as our point of reference. So we have made our first correspondence between a real number and a point on the Line. Now we choose a unit of measurement, say 1 cm. It can be anything really.
We use this unit of measurement to mark of the rest of the numbers on the line. Now this line, the origin, the positive direction, and the unit of measurement define what is called a coordinate line or sometimes a real line. The real number corresponding to a point on the line is called the coordinate of the point. Example 1: In Figure we have marked the locations of the points with coordinates -4, -3, To describe this fact we say that the real numbers and the points on a coordinate line are in one-to-one correspondence.
The SIZE of a real number a makes sense only when it is compared with another real b. A little more about inequalities. So there are two conditions here. As one moves along the coordinate line in the positive direction, the real numbers increase in size. The following properties of inequalities are frequently used in calculus. We omit the proofs, but will look at some examples that will make the point. Now we shall assume in this text that you are familiar with the concept of a set and fully understand the meaning of the following symbols.
However, we will give a short explanation of each. Recall the example we did of the Set of all politicians! So here the pattern is that the set consists of the even numbers, and the next element must be 8, then 10, and then so on. When it is inconvenient or impossible to list the members of a set, as would be if the set is infinite, then one can use the set- builder notation.
When it is clear that the members of a set are real numbers, we will omit the reference to this fact. So we will write the above set as Intervals. We have had a short introduction of Sets. Now we look particular kind of sets that play a crucial role in Calculus and higher math. These sets are sets of real numbers called intervals.
What is an interval? But if things were only this simple! Intervals are of various types. For example, the question might be raised whether a and b are part of the interval? Or if a is, but b is not?? Or maybe both are? The square brackets indicate that the end points are included in the interval and the parentheses indicate that they are not.
Here are various sorts of intervals that one finds in mathematics. In this picture, the geometric pictures use solid dots to denote endpoints that are included in the interval and open dots to denote endpoints that are not. An interval that goes on forever in either the positive or the negative directions, or both, on the coordinate line or in the set of real numbers is called an INFINITE interval.
Such intervals have the symbol for infinity at either end points or both, as is shown in the table An interval that has finite real numbers as end points are called finite intervals. A finite interval that includes one endpoint but not the other is called half-open or sometimes half-closed. Let's remember this fact for good!
PAUSE 10 seconds. Let's talk some more. First Let's look at an inequality involving and unknown quantity, namely x. So the set of all solutions of an inequality is called its solution set. The process of finding the solution set of an inequality is called solving the Inequality. Let's do some fun stuff, like some concrete example to make things a bit more focused Example 4. We shall use the operations of Theorem 1.
Thus, the solution set is the interval shown in Figure 1. This concept plays an important role in algebraic computations involving radicals and in determining the distance between points on a coordinate line. It is called a non-negative number. However this need not be so, since a itself can represent either a positive or negative number. In algebra it is learned that every positive real number a has two real square roots, one positive and one negative.
The positive square root is denoted by a For example, the number 9 has two square roots, -3 and 3. Although this equality is correct when a is nonnegative, it is false for negative a. A result that is correct for all a is given in the following theorem. More precisely, for any n real numbers, a1,a2,a3,……an, it follows that a 1 a 2 ….. This is the content of the following very important theorem, known as the triangle inequality.
Well, there are two types of real numbers. What are they?? We begin with the Coordinate plane. Just as points on a line can be placed in one-to-one correspondence with the real numbers, so points in the PLANE can be placed in one-to-one correspondence with pairs of real numbers. What is a plane? Each line is a line with numbers on it, so to define a point in the PLANE, we just read of the corresponding points on each line.
For example I pick a point in the plane By an ordered pair of real numbers we mean two real numbers in an assigned order. Every point P in a coordinate plane can be associated with a unique ordered pair of real numbers by drawing two lines through P, one perpendicular to the x-axis and the other to the y-axis. Now this idea will enable us to visualise algebraic equations as geometric curves and, conversely, to represent geometric curves by algebraic equations. Labelling the axes with letters x and y is a common convention, but any letters may be used.
If the letters x and y are used to label the coordinate axes, then the resulting plane is also called an xy-plane. In applications it is common to use letters other than x and y to label coordinate axes. Figure below shows a uv-plane and a ts- plane. The first letter in the name of the plane refers to the horizontal axis and the second to the vertical axis. In a rectangular coordinate system the coordinate axes divide the plane into four regions called quadrants.
These are numbered counter clockwise with Roman numerals as shown in the Figure below. The GRAPH of an equation in two variables x and y is the set of all points in the xy-plane whose coordinates are members of the solution set of the equation. When a graph is obtained by plotting points, whether by hand, calculator, or computer, there is no guarantee that the resulting curve has the correct shape. For example, the curve in the Figure here pass through the points tabulated in above table.
As illustrated before, intersections of a graph with the x-axis have the form a, 0 and intersections with the y-axis have the form 0, b. The number a is called an x-intercept of the graph and the number b a y-intercept. Example: Find all intercepts of Solution is the required x-intercept is the required y-intercept Similarly you can solve part b , the part c is solved here In the following figure, the points x,y , -x,y , x,-y and -x,-y form the corners of a rectangle.
Here is what it is. So put only positive x-values in given equation and evaluate corresponding y-values. Since graph is symmetric about y-axis, we will just put negative signs with the x-values taken before and take the same y-values as evaluated before for positive x-values. The ideas we develop here will be important when we discuss equations and graphs of straight lines.
We will assume that you have sufficient understanding of trigonometry. Slope In surveying, slope of a hill is defined to be the ratio of its rise to its run. Consider a particle moving left to right along a non vertical line segment from a point P1 x1,y1 to a point P2 x2,y2. As shown in the figure below, the particle moves y2-y1 units in the y-direction as it travels x2-x1 units in the positive x-direction.
The vertical change y2-y1 is called the rise, and the horizontal change x2-x1 the run. By analogy with the surveyor's notion of slope we make the following definition. Definition 1. We make several observations about Definition 1. Speaking informally, some people say that a vertical line has infinite slope. When using formula in the definition to calculate the slope of a line through two points, it does not matter which point is called P1 and which one is called P2, since reversing the points reverses the sign of both the numerator and denominator of 1 , and hence has no effect on the ratio.
But the rise is the change in y value of the point and the run is the change in the x value, so that the slope m is sometimes called the rate of change of y with respect to x along the line. Angle of Inclination If equal scales are used on the coordinate axis, then the slope of a line is related to the angle the line makes with the positive x-axis. The following theorem, suggested by the figure at right, relates the Slope of a line to its angle of Inclination.
Theorem 1. This agrees with the fact that the slope m is undefined for vertical lines. Equations of Lines Lines Parallel to the Coordinate axes We now turn to the problem of finding equations of lines that satisfy specified conditions.
This line consists precisely of those points whose x-coordinate is equal to a. Similarly, a line parallel to the x- axis intersects the y-axis at some point 0, b. This line consists precisely of those points whose y-coordinate is equal to b. However, if we specify the slope of the line in addition to a point on it, then the point and the slope together determine a unique line.
Let us see how we can find the equation of a non-vertical line L that passes through a point P1 x1,y1 and has slope m. We often use the equations of lines called Linear equations to study such motions Many important quantities are related by linear equations. One we know that a relationship between two variables is linear, we can find it from the any two pairs of corresponding values just as we find the equation of a line from the coordinates of two points.
Slope is important because it gives us way to say how steep something is roadbeds, roofs, stairs. The notion of slope also enables us to describe how rapidly things are changing. For this reason it will play an important role in calculus. Distance between two points in the plane As we know that if A and B are points on a coordinate line with coordinates a and b, respectively, then the distance between A and B is b-a.
We shall use this result to find the distance d between two arbitrary points P1 x 1 ,y 1 and P2 x 2 ,y 2 in the plane. Example Show that the points A 4,6 , B 1,-3 ,C 7,5 are vertices of a right triangle.
To derive the midpoint formula, we shall start with two points on a coordinate line. Therefore, the midpoint of two points on a coordinate line is the arithmetic average of their coordinates, regardless of their relative positions. Example Find an equation for the circle of radius 4 centered at -5,3. Another version of the equation of circle can be obtained by multiplying both sides of above equation by a nonzero constant A.
Thus, the solution set of the equation is empty, and the equation has no graph. Thus, the graph of the equation is the single point 1, In spite of the fact these degenerate cases can occur, above equation is often called the general equation of circle. Depending on whether a is positive or negative, the graph, which is called a parabola, has one of the two forms shown below In both cases the parabola is symmetric about a vertical line parallel to the y-axis.
This line of symmetry cuts the parabola at a point called the vertex. Here is an important fact. With the aid of this formula, a reasonably accurate graph of a quadratic equation in x can be obtained by plotting the vertex and two points on each side of it.
Instead we shall give a graphical solution. From the s-coordinate of the vertex we deduce the ball rises Here is a table Above table shows that each value assigned to x determines a unique value of y. So this equation does not describe a function. This tells right away which variable is independent and dependent. It is just for expressing functional relationship Functions are used to describe physical phenomenon and theoretical ideas concretely.
We can also replace x with another variable representing number. Formula structure matters, not the variables used. EXAMPLE Suppose that a square with a side of length x cm is cut from four corners of a piece of cardboard that is 10 cm square, and let y be the area of the cardboard that remain. Therefore, even though it is not stated explicitly, the Here x can not be negative, because it denote length and its value can not exceed 5.
Many functions have no physical or geometric restrictions on the independent variable. However, restrictions may arise from formulas used to define such functions. This is called the natural domain of the function.
It is common procedure in algebra to simplify functions by canceling common factors in the numerator and denominator. However, the following example shows that this operation can alter the domain of a function. This is the range of g. The set of all possible y values is not at all evident from this equation. Example The cost of a taxicab ride in a certain metropolitan area is 1. After one mile the rider pays an additional amount at the rate of 50 paisa per mile.
Since it is complicated to solve it for y in terms of x, it may be desirable to leave it in this form, treating y as the independent variable and x as the dependent variable. Sometimes an equation can be solved for y as a function of x or for x as a function of y with equal simplicity. This is just a line with y-intercept 2 and slope 1. So there is a HOLE in the graph. This is a straight line with slope 0.
Suppose the graph of f x is known. Must it be the graph of a function?? Figure 2. This gives you two points on the graph namely a,b and a,c But this cannot be a function by the definition of a function.
Various vertical lines cross the graph in more than 2 places. So the graph is not that of a function which means that equations of Circles are not functions x as a function of y.
A given graph can be a function with y independent and x dependent. We will look at it intuitively, without any mathematical proofs. These will come later. We would like something like this to be our definition of a tangent line.
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