High School Calculus/Tangent Lines and Rates of Change
When it comes to mathematics when we say functions we mean a equation that changes it's output value according to the way that it's domain changes. Simply it means how a related quantity changes when another quantity changes. It would be better if you read about graphs and functions before going into this section.
In calculus differentiation is a extremely important concept. All the other portions of calculus depending on differentiation or the immediate rate of the function when x varies. As most people think this is not a hard idea and whole calculus thing is not a hard idea but it's beautiful one. Before studying the derivative of a function, to understand that concept, we need to have a good understanding about the tangent line and rate of change of the function(Which will be explained shortly).
Maybe you have met tangent line the term in your geometry class before, even in calculus when we tan line or tangent line we pretty much contemplate that old meaning. To explain this concept the easiest method to do it is represent the idea graphically. If you have seen the image above, You'd see that we have a curved function and an another line touches the curve, the area that the other one touches the curve is infinitely small. This is the representation of a tangent line. What this mean is the whole idea of differentiation or finding the derivative of a function.
An alternative term to tangent line is slope of a function. The way you find the slope is, Assuming your is linear,
Slope = Change in Y/Change in X = Y2-Y1/X2-X1
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Let's consider a curve like y=x^2 the simplest curve in the world. You'd see this approach is not working for curves. Just try to do this on your own. You'd not get an constant answer. Because the slope or gradient of this function changes all the time.
Mathematical approach of understanding tangent line.
We use the understand you have about normal term 'tangent line' to understand it mathematically. To continue reading in this section you should be familiar with the concept of limit of a function. In the starting section of this book there is chapters for it.
I'll simply explain it anyway. Say we have a function y=f(x), When x approaches a some value but not going to that value, what is y approaching. Let me give you a example so you'd understand it simply. If you are familiar with the function, y=1/x, you'd notice that when x=0 this function's output value is indeterminate, if you come from the right side to 0 simply substituting x values to the function, you'd notice that when x is getting smaller and smaller, y's value is highly increasing and y is approaching +Infinity. If you come from the left side you'd notice that y is approaching -Infinity. So according to the definition of limit if you have learned it, this function is not a valid function because when we approach x from different sides y value does not diverge into a one value it goes not +,- infinities respectively. But i took this function because it's easy to understand limit's basic idea. There is a bunch of techniques you can use to simplify an equation with limits such as L'Hopital's Rule, You would find them in the limit section of this book.
Now you have a basic understanding about a limit of a function. Let's define what a tangent line means. Say we have a function y=f(x), and we have to find that tangent line of the function when x=a assume this function is a curve, Let's think about this a little bit. So according to the understanding that we have about tangent lines from geometry, we need to create a line that just touches the curve, logically in an infinitely small area.
So wouldn't this be obvious if you just take a small, an infinitely small change in x we can call it 'delta x' (we call it delta x because in mathematics delta means change). and take an infinitely small change in y as wall we call it dy. and divide small change in y by small change in x.
Just take a minute and think about it.
Let's do a simple example for this.
Say you have a function (y=x^2), We need to find this function's tangent line when x=5. So we need to find deta x or dx or infinitely small change of x.
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Say x=[5.00000000,5.00000001] (This is not infinitely small but this will do it!) So dx=5.00000001-5.00000000; dx=0.00000001;
To find dy,
Say, dy=Y2-Y1; (Assuming the difference between Y2 and Y1 is infinitely small)
Y2=(5.00000001)^2; (Substituting for x^2=y equation) Y1=(5.00000000)^2;
Y2=25.0000001; (Simplification of above equation) Y1=25; (Simplification of above equation)
So we found dy and dx.
So the slope of this function at x=5 should be approximately equal to,
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Okay!. You have just found the tangent line of the function of y=x^2 when x=5. Hope that's not too much.
Relation between rate of change and tangent line.