Understanding Calculus

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  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

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Chapter 8 - Understanding the Derivative

Chapter 8 - Understanding the Derivative

Section 8.1 - Using the First Derivative

In this chapter we will take a close look at the definition of the derivative and its relation to its anti-derivative. We will see how the graph of the anti-derivative can be accurately described by just looking at the derivative function only.

From the definition of the derivative, is the derivative of a function . The derivative of a function tells us how fast f is changing relative to the independent variable, x. Thus the derivative refers specifically the rate of change of the anti-derivative function with respect to x. Rate of change is also synonymous with the slope of the tangent line to the graph at a particular point.

Therefore, a function and its derivative are closely related and knowing just the derivative can tell us a great deal about the behavior of its anti-derivative. Since the derivative of a function is derived from the definition of a derivative as: , we can work with the definition to find the anti-derivative when only the derivative is known. Also recall how differentiation is based on a limiting or subtracting process and dividing; therefore working backwards would tell us that we should be adding and multiplying.

We will develop on this later, however, let us first look at how we can use f’(x) to obtain equilibrium points on the graph of the anti-derivative or f(x).

Equilibrium points are by definition,  points  on the graph refer to static situations where the rate of change is zero. Thus a change in the independent variable results in no change in the dependent variable. Equilibrium sate generally occurs when a situation has reached a critical maximum value and then decreases or where a situation has reached a critical minimum value then increases.

Equilibrium and critical values of a function can refer to different things depending on the phenomena being studied. Therefore we will restrict ourselves to the geometric interpretation of an equilibrium as a point on the graph where the rate of change is zero. Since the rate of change is zero, the tangent to the graph at this point will be a horizontal line. 

A horizontal tangent tells us that the derivative’s value at the equilibrium point is zero. Such situations occur when the graph has reached a maximum or minimum value.

Horizontal tangent may also exist, but not necessarily, when the concavity of a graph changes.

A change in concavity occurs at points on the graph called inflection points. As we shall soon study, the derivatives value at an inflection does not have to be zero. Therefore we will restrict our definition of equilibrium points to reflect either maximum or minimum values on the graph.

For example to find equilibrium points for the function, . we first need to differentiate it to get, . The derivative, , tells us the instantaneous rate of change of function, , at any point x. Since we want to find points where the rate of change of f(x) is zero, we need to set equal to zero  to find those values of  x which the satisfy the equation, . Doing this for results in:

This tells us that at both x=0 and x=2/3 there exists an equilibrium point, which is confirmed by the graph of the function,

Having found the critical points, how do we classify them as either maximums or minimums? Obviously this can be done by just looking at the graph of , but the purpose of this chapter is to understand how to only use the derivative, to approximate the behavior of the function . To determine whether our equilibrium points are either maximum or minimums we need to evaluate points left and right of the equilibrium points to determine where the function, , is increasing or decreasing.

A minimum is defined as the bottom of a U-shaped or concave up graph. If the graph is concave up then the slope or rate of change is positive to the right of an equilibrium point and the function is increasing to the right of that point. To the left of the equilibrium point,  the slope is negative which means the function is decreasing till it reaches the equilibrium point. To better understand this, look at the following graph of a concave up portion of a graph. Notice how the function decreases till it reaches the equilibrium point then rises after passing it.

Similarly if the rate of change of were negative on  the right side and positive on the left side of the equilibrium point, then we get an inverted U shaped or concave down graph. A concave down graph thus reflects a maximum value at the equilibrium point.

If the slope is both positive or negative on either side of the equilibrium point then we get an inflection point that represents where the concavity changes from a U shape to an upside down U shape or vice-versa. The following graph summarizes the above conclusions.

Next section -> Section 8.2 - Using the Second Derivative

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