What is the need for calculus or differential equations, when we have graphs?

gary36

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Hi,

I am a newbie to calculus and having trouble with understanding need for calculus. My view was that all information can be inferred from a algebraic equation or a function. For example, maxima , minima/increasing/decreasing can be directly inferred from graph. Why derivatives are required in the first place?
Secondly what is the need to express newton's law of cooling and radioactive decay with differential equation. Everything can be inferred from the exponential function. In fact most of the numerical's requires the use of a function and not the derivative. In essence, I am unable to understand the need to model a physical situation using derivatives. Just plot the graph and all we need is contained in this graph. ( I still do not understand the need for instantaneous rate of change, I do not think we use it practically in real life). Any specific example where use of derivatives have an edge over simple algebraic function would be of great help
 
Hi,

I am a newbie to calculus and having trouble with understanding need for calculus. My view was that all information can be inferred from a algebraic equation or a function. For example, maxima , minima/increasing/decreasing can be directly inferred from graph. Why derivatives are required in the first place?
Secondly what is the need to express newton's law of cooling and radioactive decay with differential equation. Everything can be inferred from the exponential function. In fact most of the numerical's requires the use of a function and not the derivative. In essence, I am unable to understand the need to model a physical situation using derivatives. Just plot the graph and all we need is contained in this graph. ( I still do not understand the need for instantaneous rate of change, I do not think we use it practically in real life). Any specific example where use of derivatives have an edge over simple algebraic function would be of great help
How did we (the humans) know:

law of cooling and radioactive decay ......... can be inferred from the exponential function

First the differential equation was set-up from observation.

Differential equation was solved - exponential model prepared and we were able predict future behavior from that.
 
Hi Subhotosh,

I think newtons law of cooling was first experimented, which resulted in exponential decay. Later differential equation was used to form generalized equation. I think it was some kind of curve fitting. So curve came first and derivatives later. So my question is , will not curve suffice?
 
calculus elementary question

Hi

I Think that graph was first plotted and then the differential equation. Not the vice versa
 
Hi

I Think that graph was first plotted and then the differential equation. Not the vice versa
Yes - for one material (I think it was hot water) - after taking hundreds of data points under different starting and ambient temperatures. Then what about hot iron rod? what about any other material? You will again do multitudes of experiments.

This is the beauty of the mathematical model!

From the model - we surmise that you need to know only two data points - and then you can predict its behavior for any time-elapsed?

Similarly, once you know the mathematical-model of a radio tuner, you can tune your radio to any frequency (and you'll know which frequency will be out of range for the particular tuner). Have you taken class in "differential equation" yet?
 
Hi

The point I was trying to make is , with the data points collected (for newtons law of cooling), I know that function is always exponential with two degrees of freedom, viz.. temperature of liquid, solid whatsoever and the surrounding temperature. Now it is easily possible to predict for any material. another example is classical RC circuit. Without modeling , by observation, I can easily establish relation between input and output voltage. Now is it not possible to predict output voltage or current for any kind of input and any values of R and C?

my intention of this question is to get a compelling reason about the need for a differential equation. I think the above case, we can still deal without it.
 
Hi

The point I was trying to make is , with the data points collected (for newtons law of cooling), I know that function is always exponential with two degrees of freedom, viz.. temperature of liquid, solid whatsoever and the surrounding temperature. Now it is easily possible to predict for any material. another example is classical RC circuit. Without modeling , by observation, I can easily establish relation between input and output voltage. Now is it not possible to predict output voltage or current for any kind of input and any values of R and C?

my intention of this question is to get a compelling reason about the need for a differential equation. I think the above case, we can still deal without it.
If you allow graphing (Can you make a graph of the 4-D space-time surface that an object traces while it is embedded in Euclidean 5 space?) then you probably don't need derivatives. However it is often much easier to solve and interpret equations using them.

As a simple example, given a constant acceleration (let's say we're in 1-D) we can derive the equation \(\displaystyle s = s_0 + v_0 t + \dfrac{1}{2}at^2\) much faster using integral Calculus than we can do using only algebraic equations. Yes, you get there eventually but it's only a two line problem using Calculus.

Another example is solving the equation for the speed of a freely falling object with air resistance. You'd have to go through all sorts of algebraic work (I'm not sure even I could do it) but it is reasonably easy to solve for using the differential equation.

-Dan
 
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Hi dan

From your example, I understand that calculus is a short cut to solve few problems. It can still be solved algebraically. See I am unable to see the aaha moments yet. Can anybody share atleast one example where calculus is the real winner and no alternative methods exist.
 
Hi dan

From your example, I understand that calculus is a short cut to solve few problems. It can still be solved algebraically. See I am unable to see the aaha moments yet. Can anybody share atleast one example where calculus is the real winner and no alternative methods exist.
Look, I'm essentially a Mathematical Physicist these days and I can tell you that differential equations make things sooooo much easier.

Yes, you could probably get away with any problem using Algebra. Find the slope of a function at a point on that function using just Algebra. Very very ugly until you define a secant function to estimate the slope and you wind up doing a Calculus problem without realizing it. Yes, I know you are saying that you can just graph it but Algebraic solutions become very "non-trivial" very quickly. In order to solve most differential equations you would have to already know the answer before you could sketch it! Which, for the sake of sanity at least, we use Calculus to help us solve these problems.

I don't mean to assume but it sounds like you have an axe to grind. At a guess you didn't do well in Calculus this semester?

-Dan
 
Hi Dan

In fact I did very well in calculus, embarrassingly good!!. But without knowing the reason of course. Still an example is worth thousands of words.
 
Hi dan

From your example, I understand that calculus is a short cut to solve few problems. It can still be solved algebraically. See I am unable to see the aaha moments yet. Can anybody share atleast one example where calculus is the real winner and no alternative methods exist.
Gary

You may not have noticed that most of the equations in economics are so general that you cannot do much with them algebraically. Frequently all you know about them is the presumed sign of their derivatives. I suppose you could solve problems of optimizing subject to constraints with algebra alone, but why bother when calculus gives a simple method of attack with LaGrangians.

EDIT: Your whole point about graphing is lost if you think about an economy with more than two actors and more than two goods. A simple economic model may well be in eight or nine dimensions.
 
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Hi Jeff

That's exactly my point as well. calculus is needed for complex problems. For simple ones such as population growth or newtons law of cooling , graphing could suffice. Do you agree?
 
Hi Jeff

That's exactly my point as well. calculus is needed for complex problems. For simple ones such as population growth or newtons law of cooling , graphing could suffice. Do you agree?
I agree in this limited sense. Many problems have been reduced to general formulas. If you deal with such problems with enough frequency, just memorize the pertinent formulas, which does indeed usually involve nothing more than basic algebra. If you want to develop a general formula, calculus is frequently the easiest tool to use. For example, if I were trying to determine where the extremum of the general quadratic is, I'd use calculus rather algebra although it can be done by algebra.

Dan, I think is making a different point, which is that new problems in physics, even where they are very specific, are often most easily dealt with by calculus. That sounds plausible because calculus was largely developed to help physicists, but my ignorance of physics is too profound for me to support or contradict him.
 
Dan, I think is making a different point, which is that new problems in physics, even where they are very specific, are often most easily dealt with by calculus. That sounds plausible because calculus was largely developed to help physicists, but my ignorance of physics is too profound for me to support or contradict him.
Yup!

-Dan
 
Hi

After some googling, I got to understand that certain problems (related to motions) cannot be just solved by algebra and requires calculus. And wherever algebra could be used, calculus can become handy. So realistic situation can all be modeled by calculus even though it may be solvable ( in certain cases) by algebra
 
An historical note: Newton and Leibniz, as well as other scientists of that age, were primarily concerned with the orbits of planets and, specifically what force was responsible for the motion of planets. They all knew, of course, "F= ma" and felt that the force should be related to the distance from the sun to the planet. The problem was that at that time "a", the acceleration, had to be calculated as "change in speed over a given time interval divided by the length of that time interval". And speed itself was "distance moved in a given time interval divided by the time interval". So that both require a non-zero time interval but the distance from the sun to the planet is determined at a specific time, not a time interval. To solve this conundrum, it was necessary to define velocity, and acceleration, at a given instant, not over a time interval. That was the basic idea of the "Calculus".
 

  1. If the calculus were unnecessary, or of limited use, it is unlikely it would have been invented or stood the test of time.


  2. If everything were finite, square and straight, and didn’t move or change shape, there would be less use for the calculus.
  3. Learning of ALL of mathematics may tell your brain that there is another way to think. This is important to society.
  4. The very question suggests that you don’t know enough about the calculus. Keep learning and don’t burden yourself with second-guessing the value of the pursuit.

  5. Algebra and geometry tried for a couple millennia to chop things up into small enough pieces to explain everything. The development of the calculus contributed directly to the abandonment of this foolish pursuit. Your question seems to suggest that we go back to wasting our time with things that can’t be done.

  6. Your graph is limited to the precision of your pencil (or monitor). There is so much more to see.


I could go on.

 
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Each new step in math is not (initially) necessary, but makes old things easier. Then it makes new things possible.

The first problems you solved with algebra probably could have been solved just as easily with arithmetic, by trial and error or by experience. For example, you probably didn't see the need to use algebra to solve 2x = 8; you'd just recall your multiplication table. But later algebra problems couldn't be solved without it.

The same is true of calculus. You first learn things that can be done by other means, but once you have mastered that, you can use it for bigger things you couldn't have imagined.

Newton used calculus to come up with his ideas about gravitation, but when he wrote the Principia, he had to use the math everyone else knew, so he restated everything using geometry. So calculus wasn't necessary. But without it, he probably wouldn't have been able to do nearly as much; and most of the progress since then would have been greatly delayed, if not impossible. In particular, calculus allowed scientists to formulate general theories that probably couldn't even be expressed without it. The discovery of radio, for example, started with observations about some differential equations that led to speculation about things that hadn't been observed yet. (I'm oversimplifying everything, of course.)
 
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