Understanding Calculus

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  Testimonials
  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 3

Chapter 3 - The Mathematical function and its Graph

CHAPTER 3

Section 3.4 The Mathematical Function

The mathematical function expresses the relationship between a situation and the conditions that define it. The form of the mathematical function is thus:

This is read as, a situation is a function of certain conditions. The conditions are called dimensions. For example, consider the mathematical function for calculating the force acting on a body of mass, m.

Force = Mass * Acceleration

The force on a body is a situation defined by two conditions, mass and acceleration. In a closed system , the force acting on an accelerating mass, is the product of the mass and its acceleration. Thus, force is a function of two dimensions, mass and acceleration while force is the third dimension dependent on m and a of the system. Therefore force is called the dependent dimension while mass and acceleration are the independent dimensions that together define the dependent dimension. The dependent dimension, F, is placed outside the brackets, while the independent dimension are inside the brackets. Remember . the language of mathematics is as concise as possible. The function is more commonly written as:

For example, the force required to accelerate a 5 kg object at 3 meters per second per second ( acceleration is by definition the change in speed per unit time ) is 15 N, where N represents Newtons, the unit of measurement representing force:

One is free to use any value of m and a to calculate the corresponding force. For this reason, m and a are called independent dimensions, while F is the dependent dimension defined by them. Note that m and a must come from the same system, and the calculated force refers to that situation only.

What if I were given the acceleration of a body and the force acting on it and wanted to calculate its mass? In other words, given F and a, what is m? This is done by solving the force function for m or:

Therefore, mass as a function of its acceleration and the force acting on it, is the force divided by the acceleration. Here mass is the dependent dimension dependent on force and acceleration. Once again, the three dimensions, mass, force, and acceleration must all refer to the same situation.

It is not uncommon in engineering to encounter twelve to fifteen dimensional functions. For example, consider the five dimensional function for calculating the elongation of a cable being stretched with some tensional force.

The total elongation of the cable is dependent on four conditions. The tension acting on it, its length, the cross-sectional area, and a material property called elasticity. The point to understand is that the n'th condition is entirely dependent on the other ( n - 1 ) independent conditions. The ( n - 1 ) independent dimensions represent a set of fixed or constant conditions that together define the n'th dimension .

Questions

  • The distance covered by a falling object near the surface of the earth is given by:

    Simplify this function using d for distance, t for time etc.

    The acceleration can assumed to be a constant 9.8 m/s/s. Substitute this constant into the function and write the expression for distance as a function of time.

    To find approximately how tall you are, drop an object from the top of your head and using a stopwatch, find the time taken for it to hit the ground. Substitute this time into the distance function just derived. The answer should be close to your height. Try this at least ten times to get a close approximation.

    Calculate your height in feet using the following function:

You might think your answer will be different if you had dropped a metal object , paper or a balloon. You are correct, however, it is due to air resistance which is a function of amount of surface area exposed to the direction of travel. So you should get the same answer no matter how heavy your object is as long as it is relatively immune to air resistance.

Next section -> Section 3.5 - Two Dimensional Functions

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