The Calculus Diaries - Jennifer Ouellette [104]
Where a = −2, we get the exact same parabolic curve, only inverted (falling below the x axis) because the sign is now negative:
Finally, we can add additional variables: f(x) = ax2 + bx + c, also known as y = x2 + bx + c. It’s fun to play with the basic equation and see firsthand how changing each value for the different variables is reflected in the shape of the resulting curve. For instance, this is what you get when you plug in the values a = 3, b = 8, and c = 10:
It’s still the same basic function; the fundamental nature of its “face” hasn’t changed, it’s just expressing different “emotions.”
TOP TEN FUNCTIONS
While it’s useful to practice graphing a few functions by hand, certain functions crop up so frequently that it’s worth committing their “faces” (curves) to memory. The top ten most common functions are listed below. They should already be somewhat familiar, since you’ve encountered all but one (the logarithm) in the text.
For good measure, I’m also including their derivatives and integrals, because it’s important information for any beginning calculus student, and why do the work of crunching those numbers all over again when past generations of mathematicians have done it for you? It will also help you to see the connection between the two in practice, namely, how the derivative undoes the work of the integral, and vice versa.
1. A Constant: f (x) = c
This is the function you’d use for the velocity of a car moving at a constant speed down a straight road, for example, as discussed in chapter 2.
Derivative:
The notation to the left of the equal sign tells us we are taking a derivative of the constant c. The answer is 0 because the derivative measures a rate of change. A constant, by definition, does not change, so the rate (and hence the derivative) is 0. Integral:
∫ cdx = cx
Here, the notation to the left of the equal sign tells us we are taking an integral. Remember that the integral is the flip side of the derivative. If we take a derivative of the velocity to determine the acceleration of a car moving at a constant rate, then we take an integral of the velocity to determine how far we traveled between our starting point (a) and ending point (b). The c tells us that we are dealing with a constant, and the dx tells us we are taking an integral of the derivative of that constant.
If we were taking a definite integral, we would write this differently: = (b - a) c.The a and b variables at the top and bottom of the integral sign simply define the range over which we are taking the integral. On the right side of the equal sign, the notation simply tells us that we are subtracting our starting position (a) from our ending position (b) and multiplying by the constant c to determine how far we traveled.
2. A Straight Line: f (x) = ax + b
This is the function you’d use for the velocity of a car accelerating at a constant rate, for example, also discussed in chapter 2.
Derivative:
Integral:
3. A Parabola: f(x) = ax2
This function pops up all over the place in physics, whether we’re dealing with the trajectory of a cannonball, the acceleration of a falling apple, or our motion (changing position with respect to time) on the Tower of Terror free-fall ride in chapter 4.
Derivative:
Integral:
4. Exponential Growth Curve: f(x) = 10ax
We covered the basics of exponentials earlier. For an exponential function, we fix the base number and let the power to which it is raised be the variable: In this case, the base is 10 and the power is ax. This is the function we would use to describe the almost certain annihilation of the human race by voracious zombies in chapter 6 or the rapid growth rate of the Dutch tulip trade in chapter 5.
Derivative:
Integral: