Among other things, calculus involves studying analytic geometry (analyzing graphs). The above graph should be familiar to anyone who has studied elementary algebra. The horizontal axis is the 'X' axis and the vertical axis is the 'Y' axis.
The primary concern of
Mathematicians use the Greek letter delta "Δ"
to represent "difference" and so this equation could be written: Using y=3x + 6 (the red line in the graph above), we take the 2 points (x=2, y=12) and (x=-2, y=0) and calculate the slope:
There is an easy method to calculate the slope of linear equations. For equations of the form y = a x + b = 0, the slope equals 'a' (the coefficient of 'x'). This is better than choosing points and calculating differences don't you think ? Just one more quick example - what is the slope of 5y = -3x + 7 ? Since the equation has to be of the form y = ax +b then y = -3/5 x + 7/5 and so the slope = -3/5.
In the previous section, we learned how to calculate the slope of a linear equation (equations
whose exponents = 1).
The graph above is based on a quadratic equation which predicts the distance an
object has fallen (the y-axis) in relation to time (the x-axis). You
The formula states the distance (in feet) = ½ • g • t² (where g = 32 feet/sec²). So, when t = 3
seconds, d= ½ • 32 • 3² = 16 • 9 = 144 feet. But what do we do for choosing a second point? We could try using a value of t= 4 seconds, remembering this is
not the same slope at t=3. So, we get:d= ½• 32 • 4² = 16 • 16 = 256 feet. Now, we have our second set of values: approximate slope yields:Why don't we choose a closer value of x such as 3.1? When that is the case, the distance equals 153.76 feet and our slope is:
How about choosing a value of x that is even closer to 3 than 3.1 ? Besides representing a difference, Δx (called 'delta x') also represents the smallest possible quantity greater than zero. Δx is less than a millionth, less than a trillionth - it's 1 divided by infinity. So, when x _{2} = 3 + Δx , then y_{2} equals
and the slope at x = 3 can be calculated as: differentiation, the
result of these calculations is called the derivative
and this branch of mathematics is called differential calculus.
The derivative is usually represented by dy/dx
or f'(x).
Although the above method does work, it has 2 drawbacks - it is rather cumbersome and it only calculates the slope at one particular point. If we wanted to know the slope at x = 2, we would have to go through all those calculations again. Is there an easier way for determining the slope of an equation at any point? Yes !
Differentiation the Easy Way^{n},
the derivative is equal ton • k • x ^{(n-1)}
Having determined the derivative, we can put it to use by the previous example when we calculated the slope for x=3. When x = 3 (or time = 3 seconds), the slope = 32 • 3 or 96. WOW that's a lot easier huh? What about the slope at 2 seconds ? 32 • 2 equals 64.
And what is the
Just a few more comments. The derivative of a constant (for example the number 7) is always
zero. So, by way of example, the derivative of x² + 7 is 2•x. Also, if an equation
has more than 1 'x' term, simply differentiate
As we learned,
The above graph where velocity = g • T (or v = 32 • T), is based on the
If we calculated the sum of the orange, blue and red areas this would equal the distance
fallen after 3 seconds.
So how were we calculating these areas ? We multiplied the y-axis (which is the quantity
g • T) by the x-axis (time in seconds or 'T') and we multiplied this by ½ . So we
calculated the area by the formula ½ (g • T • T) which equals ½ • g • T², and this is the
precise formula which was used in the previous section !!
Integration the Easy Way
Well congratulations !! If you have read this tutorial carefully, you now have a good understanding of calculus (both differential and integral) !! Granted, this was a very quick, "bare bones" explanation, and it represents a very small tip of an incredibly huge "Calculus Iceberg". However, you now understand the "big picture" of what calculus is all about. As a bonus, the calculator below will help you determine derivatives and integrals.
Derivative and Integral Calculator^{n}
Copyright © 1999 - 1728 Software Systems |