Understanding Calculus

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  Table of Contents

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

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

  Links
  Contact

 
 
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CHAPTER 4

Chapter 4 - The Derivative

Section 4.4 - The Derivative

Calculus involves analyzing instantaneous changes with reference to the entire system. We saw how using a small interval of = 2 caused great differences in the rate of change of the function over the same interval. The function did not change nearly as much from x = 0 to x = 2 as it did from x = 4 to x = 6. The rate at which the function is changing must be dependent on the value of the function. In other words the rate of change must be defined at each interval and this value must be unique to that interval.

We can approximate the graph from x = 6 to x = 8 more closely by using a smaller interval of = 0.5. This will give an even more accurate description of the graph of the function over that interval. Keep in mind that the rate of change of the function varies with x, and the only way to accurately analyze this change is by breaking the graph up into smaller and smaller sub-intervals and study what is happening over these infinitely small intervals:

First let us define a method for calculating the rate of change between any two points on the graph of f(x): If the change in f of a function from a point x to another point (x+ ) is given by:

The rate of change of f(x) over this interval is found by dividing by

Remember the on the left side is the same as the in the f(x + ) expression.

From x = 6 to x = 6.5 the rate of change is:

From x = 6.5 to x = 7 the rate of change is:

Similarly the rate of change from x = 7 to x = 7.5 and x = 7.5 to x = 8 is 14.5 and 15.5 respectively. Notice how the rate of change of the function slightly increases as x increases.

This implies that there must be a way to instantaneously define the rate of change of function through “one point” on the graph. Before continuing, take a look at how the average rate of changes or slopes just calculated, approximate the graph from x = 6 to x = 8.

As you can see from the above graph, the smaller interval of = 0.5 gives a more accurate approximation of the graph of f(x). Our results are only averages because the rate of change of f(x) is increasing with x and is not constant over an interval To define the instantaneous rate of change, we need to define as an infinitely small distance separating two point x and (x+ ) on the graph of f(x). The rate of change calculated through these two points will then give us the exact rate of change of f(x) at that point, x.

Since through any one point there exists an infinite many lines, we are in actuality giving the slope of the line through the two points  x and (x+) where goes to zero. The two points are assumed to be infinitely close to each other such that the rate of change f(x) over that interval is constant.

As we let these two points (x) and (x+ ) come closer and closer together, the slope of the line through these two points corresponds to the actual graph itself. For example with = 2, the line through x = 6 and x = 8 only roughly represented the actual graph. However, by breaking up the interval from x = 6 to x = 8 into four sub-intervals of = 1/2, we got a better approximation to the actual graph itself.

The instantaneous rate of change can now be defined as the rate of change of the line through a point x and (x+ ) where is near to zero. This is called taking the limit as goes to zero. In the graph this looks like.

By letting the difference or between the two points go to zero we are able to accurately describe the behavior of the function at any large we can only approximate the functions behavior over the interval. This infinitely small allows us to define the rate of change of f(x) through any one point on the graph, where point is by definition an infinitely small interval between x and x + over which the rate of change is assumed to be constant or does not change.

Thus, the instantaneous rate of change can also be thought of as the slope of the tangent to the graph at any point. The point is actually two simultaneous points on the graph, separated by an infinitely small distance.

For example, the slope of the tangent to the graph at x = 6 is given by:

We can now go on to define the rate of change at any point x of the function as:

This function is called the derivative and gives us the rate of change or tangent to graph of

at any point x. It is extremely important to realize that the function’s derivative is also a function of x. In other words its value changes with x to reflect a change in the rate of change of the original function. This may sound a bit confusing but I encourage you to spend a few minutes and think how it is so.

You are probably asking yourself how can we assume that the derivative is constant over an infinitely small interval ? Since the function’s rate of change is changing then it can only be constant over an infinitely small interval or point. It is through this point that we define the derivative by taking the limit as goes to zero. In effect we define the graph of the function through a series of connected lines with changing slopes. In the following paragraph we will mathematically prove that the derivatives value converges to the exact value as all errors go to zero as goes to zero. Since the graph of the function is continuous over an infinitely small interval then so will its derivative be defined for that interval or point.

Note that and no longer refer to discrete values. When we let go to zero, its value becomes infinitely small such that:

Here df and dx represent an infinitely small difference in x, where:

We’ll now see how to find the derivative of any simple function, . Since the derivative is also a function , we shall call it f’(x), “f prime of x” where,

Our results can now be generalized for any function where n is a positive integer.

From the Binomial expansion theorem we get: ( The theorem is just algebra and you can find a proof of it by clicking here ).

This simplifies to

We can replace all the , , with , which are constants:

Take note of how x goes from x to n’th power to the 0 power while ∆x goes from the 0 power to the n’th power in the last term. If we divide through by ∆x we get:

As we take the limit as ∆x goes to zero, every term after the first one goes to zero and hence cancels out; This leaves us with:

The rate of change of a function at any point x is given by the function’s derivative evaluated at that point x.

Next section -> Section 5.1 - Differentiating Functions

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