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MATH 1121.81 Calculus I

One more look at the concept of an antiderivative, and then some formulas for calculating them:

The notion of a slope field*:

If you look at the differential equation, dy/dx = cos(x), it associates a value, dy/dx, to every point on the xy plane.  This is called the slope field of the differential equation, and in this case the value is equal to the cosine of the x-coordinate of the point.  So we can think of every point as having an associated line, through the point, with the given slope.  What we're searching for is a function that is tangent to each of these lines, as x varies through a domain.  So, in the applet below, we see a bunch of red line segments.  These correspond to the slopes of the associated lines, in this case, always equal to the cosine of the x-coordinate.  Notice that these segments trace out what appear to be sine functions.  Click and drag the "initial condition," then slide the x-slider to see the function traced out that has as tangent at each point, the associated line from the slope field.

Applet:

 

So, when we're finding the antiderivative and solving for the constant, we're looking for a function that satisfies the slope field, as well as passing through the point in question.

Ex:  Find the solution to y' = cos(x), y(-1.414) = 2.626 (since that's what's in the picture above.)

y = sin(x) + C (we already saw that the antiderivatives of cos(x)are sin(x) + C).

To find C, we say:

2.626 = sin(-1.414) + C

2.626 = -0.988 + C

C = 3.614

y = sin(x) + 3.614

That's the function being traced out as x slides back and forth.

Here's another (somewhat more sophisticated) slope-field applet, that lets you solve (some) Initial Value Problems (this applet may take a few minutes to load, please be patient):

Applet:

The game is to enter the derivative as f(x) in the "input" box at the bottom (FIRST, Double-Click on the applet itself, to open the actual GeoGebra application.)

f(x) = x - cos(2*x), for example.  (Note, you NEED to include "f(x)=" for it to work!)

Then, slide the blue dot around to the initial condition: y(-1) = 2, for example.

Then slide the slider on the right (for the "constant" in the antiderivative) up and down until the curve passes through the blue dot.  The solution to the IVP shows up in the box.  Warning:  This is easily "breakable."  (All you have to do to fix it is to close the window and re-open it!)  Try several examples and see what happens.  This will also be used in Project #5.

Now:  Rules for calculating the antiderivatives:

Since antiderivatives are, as you'd expect, the opposite of derivatives, every derivative formula read backwards yields some sort of antiderivative formula.  Having said that, it's true that some of them need to be adjusted a bit to be useful, and others have almost no use at all.  As we'll see, there are only a few basic antiderivative rules we'll use.  Much of the rest of the study of integration is learning how to adapt these few rules to many situations:

VERY IMPORTANT:

Linearity

Just as, and because, you can break up derivatives at "+" signs, and "pull through" CONSTANT multiples, you can do the same with antiderivatives:

These are the only actual rules (properties) we have for arbitrary functions.  Everything else is a rule for a specific type of function, and the proof of each of the rules is simply taking the derivative and seeing that it works:

The Big Three:

Basically, these 3 and Linearity are most of all of what we have to work with.  We will learn a version of the backwards chain rule, called u-substitution in a few sections, and how to handle the special case of x-1 in Chapter 5 (we'll invent a special function called the Natural Logarithm to deal with it), and see what happens to the product rule in Calc II (it turns into something called integration by parts), but, by and large, the whole game of solving antiderivatives works down to trying to turn the problem into a linear combination of powers of x, and sines and cosines.

Here are a bunch of pages of examples.  The book has some great exercises in rewriting antiderivatives so that they can be solved.

Now try the homework from Section 4.1; you can also work on Quiz #8.

*Slope Fields (also called "direction fields"):  In fact, you can define a slope field as any function of x and y:

dy/dx = F(x,y). 

Beginning at any initial point, you can then trace an approximate solution by following little line segments of constant "dx" length, and then calculating the slope of the new line segment at the next endpoint.  This is called "Euler's Method" for approximating solutions to Differential Equations.  Google "Slope Field Java Applet" for many, many examples to play with, such as this one:

http://www.math.ust.hk/~amoy/math150/SlopeField.html

or

http://www.dartmouth.edu/~rewn/dfDemo.html

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