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-------------------- 2514 - CALCULUS - Everyday Calculus
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- If you make it simple it is simple to understand. “You can understand if you care”. The two things Calculus does is Differentiation and Integration. These are big words, but very simple ideas. Differentiation simply means division , or ratio, or the ”rate of change” of one variable versus another. Integration simply means multiplication or a summation of a bunch of multiplications.
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- Let’s start with the speedometer in your car. Miles per hour is simply the derivative of distance over time. velocity = ds / dt. Where “ds” is the change in distance, length, or space. Where “dt” refers to the change in time. Velocity is the rate of change of distance over time, v = ds / dt
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- We are using velocity as an example but derivatives apply to any time you have the rate of change of one variable with respect to another variable. Always think of a derivative as a triangle. If one variable ( distance ) is plotted against another variable (time) the differential is the ratio of distance over time. It is the slope of the line. It is rise over run, or the hypotenuse of the right triangle with distance on the y-axis and time on the x-axis
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- Velocity = ds /dt. Your speedometer is making this calculation continuously as you drive down the highway.
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- Always think of “integrals as rectangles“, not triangles. Derivatives are division, Integrals are multiplication. Integrals are the area of a rectangle. To get the area of a rectangle you multiply the height versus the width. Derivatives and Integrals are opposites in a way.
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- Acceleration is the derivative of velocity with time. a = dv / dt. Velocity is the integral of acceleration with time. v = Integral a * dt. “a” is acceleration in meters / second / second. ‘a“ = m/sec^2, “v” = meters / sec^2 * sec = meters / sec.
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- If acceleration is plotted versus time. Acceleration on the y-axis and time on the x-axis, then, velocity is the area under the curve. If the acceleration is constant then the area is easy to find, it is length times width of the rectangle.
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- If the acceleration is increasing at a constant rate then you have a line sloping upward and the area is that of a triangle. Velocity is ½ height * base of the triangle. If the acceleration is constantly changing, like going around a race track, then the velocity is the area under the acceleration curve. Here is where Integration becomes really powerful.
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- You can approximate the area under any curve by making little rectangles, calculating the area of each one and adding them all up to get the total area under the curve. Whenever you have tedious, repetitive additions think Integration.
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- Let’s illustrate with some examples: Suppose a belt was put around the equator of the Earth. Snug tight at a length of 25,000 miles. Suppose you could increase the length of the belt by your height so the belt hovered above the ground. Could you crawl underneath it?
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------------------------- Circumference = 2*pi*radius
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- Do you wonder how we get the differential and integral equations. It is easy. You simply look them up in a Standard Mathematical Tables. I use the C.R.C., Chemical Rubber company book published in 1931. The equations don’t change
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------------------------- The derivative of 2*p*r = 2*pi = 6.28.
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- Derivative means the rate of change of circumference with every increase in radius is 6.28 times. So, if you increase the radius by 1 foot the circumference increases by 6.28 feet. That is your height, 6.28 feet. The belt is 1 foot off the ground. Most people would guess the height of the belt to be a fraction of an inch. Do the math. This illustrates the power of being able to think in derivatives.
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-------------- In Calculus talk, where “C“ is the circumference and “r“ is the radius:
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---------------------------- dC( r ) /dr = a constant. d (2*pi*r) = 2*pi
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- You drop a rock off a bridge and it takes 6 seconds to hit the water. How high is the bridge?
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- Gravity has a constant acceleration on all objects, large and small. The constant acceleration of gravity at Earth’s surface is 32 feet per second per second. Acceleration,
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-------------------------- a (t) = 32 ft/sec^2.
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Velocity is the integral of acceleration with time. v = Integral a * dt.
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------------------------- v = Integral 32 * dt
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------------------------- v = 16 t^2
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------------------------- note that the derivative of 16 t^2 /dt is 32 * dt
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------------------------- the derivative of x^2 is 2x.
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If the rock is falling with a velocity of 16*t^2 in 6 seconds it will have traveled 16*(6)^2 = 16 * 36 = 576 feet. The bridge is 576 feet above the water.
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- Let’s suppose Linda owns a dress shop. She makes a profit of $100 a day on average. Gil being retired can invest her money and make an average of 5% over time. How much money will Gil have accumulated by the time Linda retires in 20 years?
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- Linda works 365 days a year times $100 per day for 20 years = $730,000 total dollars. But, that does not include Gil’s investments. Gil invests the $36,500 the first year and that investment lasts at 5% for 19 years.
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--------------------- 36,500 (1.05)^19 = $92,234
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--------------------- (38,325 + 36,500) (1.05)^18 = $180,075
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--------------------- ( 74,825 +36,500) (1.05) ^17 = $255,159
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---------------------
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--------------------- and so on for 19 years, this becomes a long tedious summation of multiplications. It is better to recognize that long summations are best handled with “integration“. Recognize the compound interest is logarithmic curve. The formula for a logarithmic curve is e^x. “e” is the natural logarithm to the base = 2.718.
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--------------------- The formula for the logarithmic curve we are developing :
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-------------------- Total Dollars = Integral 36,500 dt *e^-.05(20-t)
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-------------------- dt is one year increments.
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-------------------- (20 - t ) means the each year you are compounding interest one less year out of the 20 years.
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-------------------- $36,500 is what Linda makes in profit each year.
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-------------------- The Total Dollars is the area under the logarithmic curve. We calculate the area by adding up (summations) all the individual rectangle areas calculated by multiplying e^-.05(20-t) * dt.
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-------------------- Integral 36,500*dt * Integral e^-.05(20-t) from 0 to 20 years
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------------------- 36,500 Integral ( e^(-1 + .05t))
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-------------------- 36,500 ( e^.05*20 / .05 - e^.05 * 0 /.05)
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-------------------- 36,500 ( e^1 - e^0 / .05) = 36,500 (2.718 - 1)/.05
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-------------------- 36,500 ( 1.718 / .05 )
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-------------------- 36,500 (34.37)
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-------------------- $1,254,345.73
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- Gil was so excited about his $1.2 million nest egg from Linda’s store that he decided to invest $100,000 in store improvements. However, he wanted to be sure Linda was maximizing profits with his investment.
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- The average cost of the dresses that Linda bought on the used market was $30 per dress. After many years of experience Linda estimated her total market even if she gave the dresses away was 20,000. And, if she priced the dresses too high, say $400 each, she would not sell any. The best price was somewhere in between. The number sold, Q, would be 20,000 - 50 * p. Where “p” is the price of each dress. “Q” = quantity sold.
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------------------------ Q = 20,000 - 50*p
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------------------------ Cost = 100,000 - 30 * Q
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------------------------ The rate of change of cost with respect to the quantity sold is the differential of the Cost = 30.
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------------------------ The Revenue with respect to the quantity sold. Revenue = Quantity sold * price. Price = 400 - Q/50.
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------------------------ Revenue (Q) = ( 400 - Q/50 ) * Q = 400 Q - Q^2/ 50
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------------------------ The differential is the slope of the Revenue, dR/dQ = 400 - Q/25
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------------------------ The differential of the Cost is the slope of the cost curve = 30.
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------------------------ When the slope of the Cost curve = the slope of the Revenue curve the profit is maximum. This takes some thought.
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- For example, if the slope of the Cost curve is greater than the slope of the Revenue curve it costs more to sell one more dress then the increase in revenue you get from selling one more dress. The same logic applies to the other side of the curve where the slope of the cost is less than the slope of the Revenue curve. So, we set the two slopes equal to each other in order to find the optimum quantity to sell for maximum profit:
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------------------------ 400 - Q / 50 = 30
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------------------------ Q = 9,250 dresses sold is the optimum quantity
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------------------------ p = 400 - 9,250 / 50 = $215 price for each dress will maximize the profit from the business.
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- Linda’s business is doing great. Gil’s investments are doing great. But, Linda is worried that Gil is spending too much time on the couch reading investment papers. He is not getting enough exercise. She takes his pulse on the couch and it is 75 beats per minute.
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- Linda realizes that Gil has only so many heartbeats in his lifetime. When he gets to the last one he’s done. If Gil would only exercise he would lower his resting heart rate. Of course, when he exercise his heartbeat hastens. There is a trade off here. What is the optimum number of times you should exercise to maximize the number of heartbeats left in your life?
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- How many additional heartbeats would Gil get during a week of exercise? Linda put Gil on the treadmill and got his heartbeat from 70 up to 130 during the exercise. She calculated that the exercise consumed 3,600 heartbeats.
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------------------------- Gil’s resting heartbeat = 25/(Ex +1) + 50
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------------------------- When exercise was zero his heartbeat was 75.
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------------------------- With exercise his heartbeat was 62.5
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------------------------- Ex is the number of exercises
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------------------------- There are 10,000 minutes in a week.
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------------------------ Total heartbeats in a week = 10,000 (25/(Ex+1) + 50) + 3,600* Ex
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--------------------- Take the differential to find the slope of this curve:
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--------------------- 250,000 + 1/(Ex+1)^2 + 3600 = 0
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--------------------- Set this slope equal to zero to find the minimum point on the curve for Total heartbeats and solve for Ex. When a curve passes through a maximum or a minimum the slope always passes through zero.
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---------------------- Ex = 7.5 exercises per week. Now Linda has Gil on the treadmill once a day.
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- If Linda cut back her hours and worked 5 days a week for 50 weeks. And, Gil made 8% interest on her money. Their retirement nest egg would be $2,470,645. Gil would have made more money then Linda by lying on the couch. Of course he was using her money.
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-------------------- 50,000 Integral e^.08t - 1 from 0 to 20 years
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-------------------- 50,000 ( e^.08*20 / .08 - e^.08 * 0 /.08)
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-------------------- 50,000 ( e^1.6 - e^0 / .08)
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-------------------- 50,000 ( 4.95 / .08 - 12.5)
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-------------------- 50,000 (61.9 -12.5)
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-------------------- $2,470,645
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- Hope you re not afraid to try a little calculus. It is very powerful math.
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- November 30, 2019 2514 737
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