The Calculus Diaries - Jennifer Ouellette [107]
Recall that the steepness of the slope tells us the rate at which those values are changing (the derivative); the steeper the slope, whether trending upward or downward, the faster its value is changing. You can also see this trend in the above formula. If the numerator (top) is larger than the denominator (bottom), then the y’s are changing faster and therefore the line is getting steeper. If the denominator is larger, that means the x’s are changing faster and the line forms a shallow incline, because it is moving more quickly to left or right than it is moving up and down. So it should be clear that the above curve describes a car accelerating, then cruising at a constant speed before decelerating.
Finding the Area. It is an equally simple matter to find the area under this particular curve by breaking it into common geometric shapes: a rectangle bounded by two triangles.
We can find the area of the two triangles by halving the base and multiplying that number by the height, written algebraically as A = ½ bh. (A stands for area in this context, b stands for base, and h stands for height.) We find the area of the rectangle by multiplying the width times the height, written algebraically as A = wh, where w stands for width. Then it’s just a matter of adding those three areas together to find the total area under our simple curve.
For both triangles, h = 5 and b = 3. So we can multiply 2.5 by 3 to get an area of 7.5 for each. The same goes for the rectangle in the center; we multiply 3 by 5 to get 15. Then we add it all together (15 + 7.5 + 7.5) and we end up with a total area of 30. Simple, right?
THINGS GET MESSY
Alas, the real world rarely fits neatly into these sorts of idealized models. In reality, for the above example, our speed and direction would be varying constantly, and we would not be dealing with simple straight lines, but with curves. This is the true value of calculus: It helps us solve more difficult problems dealing with change and motion using known derivatives and integrals for given functions. Once again, the derivative describes rates of change and corresponds to the slope of the tangent line to a particular point on the curve, while the integral corresponds to the area under a curve. It’s just a little trickier to find those values when dealing with irregular geometric shapes.
In chapter 4, we experienced free fall while riding the Tower of Terror and learned that plotting our motion (change in position) as a function of time onto a graph produced a parabolic curve. This parabolic curve represents our position function (height, h, as a function of time, t): Let’s say we want to figure out our instantaneous velocity at a specific point. We need to find the slope of the tangent line for that point, which is equivalent to the derivative of our position function.
So we know our position function, and we also know the value of the constant b, namely our starting height. The Tower of Terror is 199 feet high; we can round up to an even 200 feet to make our calculations easier. So b = 200. Finally, we know the value of a, since falling objects travel at −32 feet per second per second; half of that is −16. (The sign is negative because our height decreases as we fall.) Plug those values into our starting function and we get h(t) = -16t2 + 200. Now we’re ready to start differentiating.
Derivatives: The Hard Way. I won’t lie to you: Things are about to get ugly. But it’s instructive to walk through every painful step just so we can fully appreciate how useful calculus can be when we see the simplified process in the next section. We begin by picking our point (h1,t1). We draw a straight line tangent to that point, and now we want to find the slope, which in turn will give us our instantaneous velocity. The problem is that our chosen tangent line only hits the curve on a single point. We can still pick another nearby point (h2,t2) and use our nifty formula above to calculate the difference between