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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Numbers and their uses

Chapter 2 - Understanding Numbers

Section 2.2 - Interpreting Numbers

Two times zero is still zero and half of infinity is still infinity. If you want to develop good engineering judgment you need to understand the profound meaning of this perspective on zero and infinity. When dealing with very small numbers, it makes no difference in the greater scheme of things whether the number is doubled or not. The number will still be a small number and needs to be interpreted that way.

Similarly when dealing with large numbers, even if you halve the number you still are left with a large number. For all numbers in between you can use relative comparisons , however, for small and large numbers, one needs to understand that relative comparisons may not hold true.

You are probably confused as to what defines a really large and small number. Relative does not have any absolute number interpretation. Relative comparison only has meaning in relation to what you are comparing to. You have to use engineering judgment to see what range most numbers fall within.

For example, a transportation engineer measuring average traffic flow on a highway will see most speeds fall within 45-65 mph. The engineer only needs to focus on speeds in that range and ignore the really small speeds close to zero. The slowest car in one highway may be 30 mph and in another highway could be 10 mph. This does not mean the first highway has a 300% higher average speed of slow cars. It means both those speeds are well below the average speed and cannot be compared to each other. The example next will explain this in more detail.

Consider an engineer who is given a half-sphere of material and is asked to build a stool from it that will be supported at three points. The engineer needs to find the optimal load path to support a person sitting on the top of the sphere and supported at three points equidistant apart at the base. This type of analysis is called 'Topology Optimization' and requires building a virtual model on the computer and uses equations of calculus and material science to solve a computationally intensive simulation on a supercomputer.

The picture above on the right is called the 'Optimal load path' . The way to interpret it is areas in red indicate areas of highest stress, so need material there. Areas in green are also under stress, while areas in blue have almost no stress. Actually most of the sphere is blue, however, all the blue parts of the model were taken out of the sphere so you could see the internal green and red parts of the model.

Before continuing, it will be useful to understand what stress means. Stress is an engineering concept and is simply the load or force divided by the area over which it is applied. Materials fail when the internal stress from external loads exceeds their allowable yield stress. Materials like steel have a high yield stress while those like rubber have a low yield stress. Think of this as a pin pricking you. A pin with a sharp point will be a lot more painful than a pin with a blunt point. The reason is the same load is going through a small area, thereby increasing the stress you feel.

Now suppose the same problem of designing the stool were given to another engineer. This engineer will use a different software and methodology to build his virtual model. In areas where the sphere is taking no load, the results from both engineer's model should be blue or the stress should theoretically be close to zero. However one engineer says the stress in the blue area is 1 MPa, while the other says it is 2 MPa ( MPa is a measure of force per unit area ). Does this mean the second engineer is producing results with 50% error ? No , it means both engineer's analysis are outputting very small numbers which are close to zero and therefore the SAME number.

Now consider you are looking at load near one of the three support points at the base. Theoretical results say the stress from load should be infinity as you have a high load going through an infinitely small point. Now one engineer says it is 6000 MPa, while the other engineer's analysis shows 3000 MPa. Again does this mean the results have 50% error? No, it means both engineer's analysis are outputting very large numbers which correlate to the trend of infinite stress and are therefore the SAME number.

Too many engineers only focus on the absolute numbers when comparing analysis results. Good engineering judgment involves looking at relative differences and also seeing how large or small the numbers are when making comparisons. Another way of thinking of this is in terms of order of magnitude. Small or very large numbers within the same order of magnitude are essentially the same number for all practical purposes. This is not only true in science but all life in general.

It is important to understand the concept of interpreting numbers as even many engineers with PhD's do not get it. Often those in the scientific community are too focused on the absolute details and fail to see the bigger picture around them.

Another problem scientists have is reporting results with too many significant digits. For example an atmospheric scientist who says the temperature is 97.6756 Farenheit or 22.22323 Celsius is only confusing you by adding so many significant digits. They may be more 'accurate', however by increasing confusion they defeat the purpose behind reporting the number! Only report just enough digits that are truly significant. So one should report temperature to within 1 whole degree as few people can perceive differences of a tenth of a degree.

Of course there are many situations where modeling the absolute response is of critical importance. It takes good engineering judgment to know when one needs to look at absolute numbers and when one needs to only look at relative differences in numbers. You are now in a much better position to interpret numbers now that you understand how relative comparisons work for large and small numbers.

Next section -> Section 2.3 - Risk Assessment

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