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 8.2 - Using the Second Derivative

The first derivative allows us to define equilibrium points on the graph of a function, . By evaluating points to the left and right of the equilibrium point we can classify these points as either maximums or minimums and thus determine the concavity of the graph. Without having found the equilibrium points it is extremely difficult to determine the behavior of a function over an interval. The sign of the first derivative only tells us if a function is increasing or decreasing; however, a function can increase or decrease in two way. For example consider the graphs of the following two different functions.

In both cases the function is increasing and the first derivative is always positive; however each function increases in a different way i.e. one increases concave up and the other increases concave down. Using the first derivative only, we would have to know not only where its positive or negative but also how the first derivative is changing i.e. positive and increasing, negative and increasing etc.

For the first graph is positive and increasing thus the graph of f(x) is increasing and concave up. For the second graph, is also positive but is decreasing. Thus the graph of

is concave down.The process of looking only at the graph of the first derivative to understand how behaves is an extremely abstract and difficult one. To quicken and simplify our work we can use the functionís second derivative to conclude where the graph is concave up or down. This information along with the fact that the derivative is either positive or negative over an interval will be enough to accurately determine a functionís behavior.

Recall that a property of a concave up part of a graph is that its slope or rate of change is always increasing.

On the left side the slope is negative; however, as x increases the slope gets less and less, -5, -3, -2, till it reaches 0, from where on it increases to 1, 5, 6, etc. We can then conclude that the rate at which the slope is changing must be positive or the graph of the derivative is increasing. Since the derivativeís value is constantly increasing, then the rate of change of the derivative, given by will be positive. Remember that positive rate of change implies that the function is increasing over that interval, while a negative rate of change implies the function decreases as x increases.

In a concave up graph the derivative is increasing, such that the second derivative over this interval will be positive. Working in reverse we arrive at an important conclusion. If the second derivative is positive over an interval, then the first derivative is increasing, implying that the graph of the original function, is concave up. This is true because the rate of change of a concave up graph is always increasing.

The reverse is true for concave down graphs. If the second derivative is negative then the first derivative is decreasing, implying that the original functions graph is concave down over the interval. The following graph summarizes the conclusions:

Though all the information concerning the behavior of f(x) can be obtained from studying its derivative, we can quicken and confirm our sketches by looking at the functions second derivative. Without having found any equilibrium points we can accurately determine the behavior of over an interval by using the signs of both the functionís first and second derivative simultaneously. Four possibilities may exist for the signs of the derivatives.

Both and are positive over an interval. Therefore is increasing and concave up.

2. is positive but is negative. Thus is increasing and concave down

3. and are negative, in which case is decreasing and concave down.

4. is negative but is positive, thus is decreasing and concave up.

Next section -> Section 8.3 - The Systematic Use of the Derivatives

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