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 6 - Applications of the Derivative

Chapter 6 - Applications of the Derivative

Section 6.1 - Motion, Velocity and Acceleration

We have developed the essential theory for defining and analyzing the mathematical function. The derivative of a function was defined as the instantaneous rate of change of a function at any given point or moment.Geometrically the derivative was simply the slope of the tangent to the graph at a particular point.

As a student you are probably still a little confused by the concept of instantaneous rate of change. To strengthen your understandingof the derivative, let us scientifically analyze a physical phenomena and see how exactly the derivative is defined for the situation.The situation we will study is that of the dynamic motion of a body.

Motion is characterized by a changing distance between a reference point and the object itself. Distance is measured in the dimension of length with units of meters, miles, or feet. We know how to measure distance but how do we measure a changing distance? Whenever a dimension changes with respect to itself it constitutes an action that defines time. Thus a changing distance has to be measured relative to a change in time.

We can define velocity to be a measure of how fast an object in motion moves. Thus velocity is the rate at which the distance is changing relative to time. A simpler definition is that velocity is the distance covered per unit time. If an object moves 100 m in one second, then its velocity for that interval is . The units of velocity are meters per second or change in distance per change in unit time. Consider afor a sports car moving at a constant velocity of 100 . If we plot graph distance covered from a reference point, we get.

The graph is a linearly increasing straight line, where the distance covered increases directly with time. The steepness or rate of change of this graph is by definition the change in distance over the change in time:

Since motion is characterized by a change in position relative to time, then velocity is only defined over that definite time interval. If an object moves from a to b, the velocity of the object over this interval is this intervaldistance covered, divided by the time taken to cover this distance, . Thus, velocity is the rate of change of distance covered with respect to time:

For example, the car above takes one hour to go from a station400 kilometers down the highway to another station 500 kilometers down the same highway. The velocity of the car is then

Since the graph of the carís distance covered with respect to time is a straight line, it has a constant rate of change, such that = v, where v is a constant velocity. This tells us that the velocity over the entire journey is a constant or . The graph of this velocity function is just a straight line, or is the same at any time t.

This confirms logic since throughout the journey the speedometer reads a constant 100 km./hr. Therefore, distance covered is directly related to the elapsed time, in each hour it will cover a 100 more kilometers.

Now let us define acceleration. Acceleration is the rate of change of velocity with respect to time. Acceleration is to velocity as velocity is to distance. The concept of acceleration refers to a changing velocity per unit time. The units of acceleration are

For example if the acceleration of a Ferrari is 4 km/hr per second, all this means is that each second the velocity increases or changes by 4 km/hr. If the velocity at t=3 seconds is 48 km/hr, then the velocity at t=6 seconds will be:

Lets see how this applies to a car moving with constant acceleration. Since the acceleration is constant, the graph is a horizontal line, where the acceleration of the car is at any time t is the same.

The graph of velocity as a function of time, will increase directly with time for a car moving with constant acceleration.

Acceleration is therefore the rate of change of velocity with respect to time or . An acceleration of 4 m/s/s means that every second the velocity increases by 4 m/s.

The concept of acceleration and velocity are fairly obvious to understand when dealing with constant accelerations and velocities. We now need to define a more precise way of explaining velocities as the derivative of the position or distance function with respect to time and acceleration as the derivative of the velocity function with respect to time.

Next section -> Section 6.2 - Instantaneous Velocity and Acceleration

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