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

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Chapter 8 - Understanding the Derivative

Chapter 8 - Understanding the Derivative

Section 8.3 - Systematic use of the Derivatives

We shall now look at a systematic and orderly way of interpreting our knowledge of a functions first and second derivative. Before continuing let us return to the concept of the inflection point. An inflection point is a point on the graph where the concavity shifts from being concave up to concave down or vice-versa. Since a negative second derivative reflects a concave down graph while a positive second derivative represents a concave up graph then inflection point occur where, is equal to zero. Note that inflection points do not have to exist at equilibrium points. For example in the graph of , the derivatives value at the inflection point (x=0.45) is not equal to zero.

In some rare cases, the value of the derivatives can be of the same sign before and after an equilibrium point. In such cases the graph of the function, has to be determined by carefully looking at both the functions first and second derivative. To make this analysis simpler, let us go through a step by step process for predicting the behavior of , using only and

To begin drawing f(x), first find maximum and minimum values by setting f(x) equal to zero, and solving for x. Next find inflection points by setting f(x) = 0 and solving for x. Then plot two number lines for both f(x) and f(x) with plus and minus signs to indicate where they are positive and negative. First look at f(x) to understand where the graph is increasing or decreasing. Next look at f(x) to find out how the graph is increasing or decreasing, concave up or down. The number line for the example is:

Having drawn this important number line, how do we make interpret it. Remember positive rate of change implies that is increasing while the reverse is true for negative rate of change. Similarly a positive second derivative implies that the graph is concave up while negative values represent concave down. Then use equilibrium points to draw the graph through them.

Next section -> The Derivative and its Approximations

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