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 2.3 - Risk Assessment

" If it were up to me to add some classes to the grade school curriculum, I think I'd put more emphasis on these skills: public speaking, risk assessment, bullshit detecting, social skills, decision-making, managing your own body, and influencing people. " The Dilbert Blog - Smarter Than a 5th Grader

The ability to calculate risk is what distinguishes the smart from the not so smart. As Dilbert humorously put it, risk assessment is a critical skill that is overlooked by traditional school education. In the previous section we discussed how really small numbers can often be treated the same. However, in this section we are going to learn that differences in really small numbers can mean the difference between life or death.

First, what do we mean by risk assessment? Risk assessment is the ability to internally calculate the likelihood of one occurrence based on the cumulative probabilities of many interaction conditions occurring simultaneously over a period of time. Read the following article that appeared in The New Yorker by eminent surgeon, Dr. Atul Gawande. In the article, Dr. Warwick is explaining to his Cystic Fibrosis patient, Janelle, why experimenting with skipping medication is a bad idea even though it appears to not make any difference.

"Let's look at the numbers, he said to me, ignoring Janelle. He went to a little blackboard he had on the wall. It appeared to be well used. A person's daily risk of getting a bad lung illness with Cystic Fibrosis ( CF ) is 0.5 per cent. He wrote the number down. Janelle rolled her eyes. She began tapping her foot. The daily risk of getting a bad lung illness with CF plus treatment is 0.05 per cent, he went on, and he wrote that number down. So when you experiment you're looking at the difference between a 99.95 % chance of staying well and a 99.5 % chance of staying well. Seems hardly any difference, right? On any given day, you have basically a 100 % chance of being well. But, he paused and took a step toward me, it is a big difference. He chalked out the calculations. Sum it up over a year, and it is the difference between an 85 % chance of making it through 2004 without getting sick and only a 15 % chance.

He turned to Janelle. How do you stay well all your life? How do you become a geriatric patient?, he asked her. Her foot finally stopped tapping. I can't promise you anything. I can only tell you the odds.

In this short speech was the core of Warwick's world view. He believed that excellence came from seeing, on a daily basis, the difference between being 99.5 % successful and being 99.95 % successful. Many activities are like that, of course: catching fly balls, manufacturing microchips, delivering overnight packages. Medicine's only distinction is that lives are lost in those slim margins.

Let us look at another example of how to apply risk assessment to determine a safe speed to drive. Look around the next time you are on the road and you will notice at least 85% of your fellow motorists ignoring the "3 second rule," which requires a minimum of 3 seconds to account for reaction time, steering time, brake application and braking distance in case of the unexpected. It gets ignored one way or another on every street at every speed, every day.

Motorists who ignore the three second rule are confusing dumb luck with skill during every commute. Most of these motorists are choosing a 1 or 2 second rule at best; a recipe for an unavoidable crash, especially in fog. To heed the 3 second rule at 70 Mph ( 110 Kmh ), one must back off over 100 yards. Most people follow at only a few car lengths while ignoring the laws of physics. Instead they rely on their perception of supernatural braking power or angels overhead. Neither are respected by Murphy and his law.

Those who apply rational and predictive driving skills behave the opposite of the speed-at-any-cost mentality. Once a person truly comprehends risk, a calmness takes over, speeds drop and following distances increase. It is a rare thing to see: someone valuing staying alive over the perception of "lost time." Once you die, there is no time to contemplate how much time 90 MPH saved you on Monday!

The truth is that most motorists are essentially ignorant people in a big mob, hurrying all the time. They do not understand basic risk assessment. There is no rhyme or reason to it, just impatience and self-indulgence. But the excuses flow like water!

Successfully applying risk assessment involves making minor changes to your behavior with the understanding that your overall risk of a tragedy will be greatly reduced. Most people are unable to perform risk assessment because they are unable to look at long-term benefits. Instead they focus on short-term gains, i.e driving fast gets me there x minutes faster so it must be better.

Unfortunately, there are no hard and fast rules in life. As this article and the prior article attempt to prove completely opposite points using the same facts. That is why it is so important to understand what you learn and not memorize blanket statements and generalizations. Everything has meaning only in the context to which it applies. If you continuously seek to educate yourself on everything around you then you will be in a much stronger position to make independent decisions based on reason and experience.

Next section -> Section 3.1 - The Scientific Method

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