Understanding Calculus

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  Table of Contents

  1. Why Study
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  8. Understanding
  9. Derivative
  10. Integration
  11. Understanding
  12. Differentials

  Inverse Functions
  Applications of
  Sine and Cosine
  Sine Function
  Sine Function -
  Differentiation and
  Oscillatory Motion
  Mean Value
  Taylor Series
  More Taylor Series


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Chapter 3 - The Mathematical function and its Graph

Section 3.6 - The Graph of a Function

When the independent dimension is free to take on any value, the dependent dimension will depend on what that value is. As the independent dimension changes, the dependent dimension will change accordingly. For example the function for the total kinetic energy (energy of motion) of a moving object is expressed in terms of its mass and velocity

Using a constant mass of 4 kg, the function reduces to the following two-dimensional function.

The total energy of the body is now entirely dependent only on the square of its velocity, whatever it may be. If the velocity were 1 m/s, the energy would be 2 Joules. If it were 2 m/s, the energy would be 8 Joules. As the velocity of the body increases, its kinetic energy also increases. This makes sense, since the faster a body is moving, the more energy it has.

The way to visualize the changing relationship between the independent and dependent dimension is by graphing the function on a set of perpendicular axes. The horizontal axis is labeled after the independent dimension (condition), while the vertical axis is labeled after the dependent dimension (situation). The graph of a function is a continuous line that reflect a horizontal change in the independent dimension with a vertical change in the dependent dimension. Each point on the graph represent a unique situation . The graph of the function would be:

It is, however, not obvious as to how the graph of a function is a well-defined continuous line through all the points at which the function exists. To better understand the important relationship between a function and its graph, consider a hypothetical function . In this function, f represents an imaginary situation dependent on some condition x. Therefore x, is our independent dimension, while f is our dependent dimension.

For the next few chapters we will only be studying abstract fs and xs. This will allow our study of the mathematical function to be more specialized by avoiding the confusion that arises from dealing with so many, Fs, ms, as, ls, Es etc. It is difficult for the mind to keep track of so many abbreviations, much less understand where they were derived from.

With intervals of 1 the difference between each evaluated point of our independent dimension x is 1. This gives a fairly generalized idea of how the dependent dimension, f, changes. From x=0 to x=1, observe that f is zero at both these points. The conclusion is that the functions value is approximately zero from x=0 to x=1. After x=1 notice that f gets substantially larger, particularly between x=2 and x=3. This leads to the observation that as x increases f gets larger and larger.

Also notice that over each sub-interval the net change in y is different.

From x=0 to x=1, f goes from 0 to 0 or the net change in f is 0

From x=1 to x=2, f goes from 0 to 2 or the net change in f is 2

From x=2 to x=3, f goes from 2 to 6 or the net change in f is 6 - 2 = 4

In each case the change in x is 1 but the corresponding change in f increases as x increases.

Until now our discussion of the function has been limited to just four points, x= 0,1,2, and 3. The concept to understand is that a function is defined at every point and the continuous line connecting the points corresponds to the graph of the function. In order to gain a better understanding of the function, it needs to be analyzed at more points.

Though before continuing, an important symbol will be introduced, the Greek letter Delta, . is a symbol for a quantifiable change, not metaphorically speaking! It represents a definite difference or change between two values. For example if x changed from 8 to 21, instead of saying the change in x is 21-8 or 13, it is more accurate to say x is 13 or x = 13 . This symbol will be used throughout the book to reflect the amount a dimension changes by.

Returning to our example, , we observed how as x increased, f increased at a greater rate. However, between x=0 and x=1 there appeared to be some inconsistency in the function since its value was 0 at both endpoints. Let us now analyze the function at intervals of x differing by or x = .333. By reducing x from 1 to .33, the function can be analyzed at ten points instead of only four.

The change in our independent dimension x will be .333 and we will be evaluating points at this intervals. Plotting these points gives us:

The pattern from x=1 to x=3 remains the same with f increasing as x increases. However, from, x=0 to x=1 something new appears. Notice how the function seems to decrease until x = 0.5, and then increases afterward. In analyzing these different graphs (the first graph consisting of only four points) of the same function we can state an important observation.

The behavior of a function as reflected by its graphs is only as accurate as the number of points plotted over the interval of interest.

That should mean if x were infinitely small, then the graph of the function would be the line connecting the infinite amount of points. However, even this does not tell much. Through an infinite number of points we get a more and more accurate look at the functions graph or behavior, but between any two points x and (x + x) where x is the infinitely small distance separating the two points, there exists a further infinite amount of points such that we cannot be sure that our graph takes them into consideration? In other words how do we know if the graph is continuous over the small interval separating the points? Our graph could suddenly jump up or down as it did between x=0 and x=1.

To prove that the graph of a function is continuous over an infinitely small interval consider a simpler function, An infinitely small interval implies a change in x, x, equal to nearly zero. Therefore if is defined at any value for x, gives the f value of a point located a distance x from x.

If the distance, x between the two points x and (x + x) is allowed to go to zero, what we are doing is called taking the limit as x goes to a number that gets closer and closer to zero. Mathematically this is expressed as:

Taking this limit:

On the interval from x to x+ x the function or y-value changes negligibly such that

This proves that points on the graph that are in between two infinitely close points are all close to the values of the endpoints and do not jump up and down in a random fashion. Each successive point in this interval differs from the previous point by an amount that is related to the change in the independent dimension. The graph is uniquely defined as a function of the independent dimension such that for a small x the f is proportionally small.

Intuitively it is not difficult to understand why the graph of a function is continuous over an infinitely small interval. By definition the mathematical function expresses a relationship between dimensions, such that one dimension is said to be dependent on the value of the other dimension. The graph of a function reflects this relationship by relating a horizontal change of the independent dimension with a vertical change in the dependent dimension. Now if an imaginary function were to change by an infinitely small amount then the f value would also change a similar minimal amount, because the value of f is dependent on the value of x.

The graph of a function is then the continuous line drawn through an infinite set of points. For the graph to be continuous it only needs to be defined at all points where it is being studied.

The graph of can now be drawn as a continuous line that corresponds exactly to the behavior of the function, f, as the independent dimension, x, changes:


Whenever an object is thrown in the air towards some direction, the path it follows is called a trajectory and the study of its motion is the focus of projectile mechanics. The motion of a projectile is dependent on several conditions such as its initial velocity when set in motion, the angle of inclination, the force of gravity, air-resistance etc. What makes projectile motion so interesting is that gravity only restricts the vertical rise of the body, but has no effect on its horizontal motion parallel to the earths surface. This situation gives rise to the curved trajectory or path of the projectile.

The goal of the following problem is to find the equation for the trajectory (graph) of a ball struck in motion by a bat at an angle of 45 degrees with an initial velocity of 50 meters per second.

The height of the ball in motion is dependent on the horizontal distance covered and is given by the equation:

Substitute the known values for the given conditions (use g=9.8) to reduce the function to one of two dimensions, involving vertical height as a function of horizontal distance covered.

Get a rough idea of the graph of this projectile motion by plotting points at intervals of d = 1 meter. As soon as the height is negative stop plotting.

Use a d=.5 m, and determine the maximum height reached and the distance from the firing point when it hits the ground.

Use an angle of 60 degrees and see how the graph changes.

Double the initial velocity while using the same 45 degree initial angle of inclination.

Next section -> Section 4.1 - Rate of Change


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