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1. Newton's Binomial Theorem Continued Newton's Binomial

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1. Newton's Binomial Theorem Continued Newton's Binomial 1. Newton's binomial theorem continued Newton's binomial theorem is a powerful tool. Given a binomial (x + y)n. We can more quickly expand it using the binomial coefficients. Example 1. (1) (x + y)3 = x3 + 3x2y + 3xy2 + y3. (2) (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5. (3) (x + 3)4 = x4 + 4x3 ⋅ 3 + 6x2 ⋅ 32 + 4x ⋅ 33 + 34 = x4 + 12x3 + 54x2 + 108x + 81. If n is small enough and it is difficult to find the binomial coefficients, we can use Pascal's triangle for aid. 2. Functions We have used the term \function" several times in this course. For example, we have looked at area as a \function" from the plane to the real numbers. In this section we defined functions in a more familiar way. One way to think of a function is as a set of ordered pairs (x; y) such that for a given x, there is at most one y such that (x; y) satisfies a given equation. For example, consider the equation y = x2. A function is the set of pairs (x; y) such that for every x there is at most one y such that (x; y) satisfies y = x2. So for whatever x we choose, there is at most one y that could be its pair. Thus, we can say f(x) = x2 is a function. √ = Consider the equation y√ x. Here, the function \doesn't√ make sense" for all values of x. For example,√ −1 does not exist for us. x only \makes√ sense" for x ≥ 0. However, f(x) = x can still be a function. If we assume that x always yields the positive√ root, and not both the positive and negative, we that√ that for every x√such that x makes sense, there is at most one y such that y = x. So, f(x) = x is another function. Now consider the function y2 = x. This equation seems very similar to our first example. Consider x = 4. There are two y values, y = 2; −2 such that y2 = x. This means that it is not a function. Another way to think of functions is as algorithms. We can say a function is rule that assigns exactly one real number to each number in a particular set of 2 real numbers. For example, f(x) = x√ is a function from all real numbers to the non-negative real numbers. f(x) = x is a function from the non-negative real numbers to the non-negative real numbers. 1 2 A function of the form f(x) = mx + b with m and b real numbers, is called a linear function. In the case that m = 0, we call the function a constant function. Finally, a function of the form f(x) = ax2 + bx + c is called a quadratic function. 3. Graphs Let f(x) be a function. The graph of the function f(x) is the set of points in the cartesian plane (x; y) such that x and y satisfy y = f(x). So if f(x) = 3x + 5, the graph is the set of points (x; y) such that y = 3x+5. We know that this is a line with slope 3 and y-intercept 5. In fact, for all linear functions f(x) = mx + b, the corresponding graph will be a line with slope m and y-intercept b. As you would expect, the graph of a quadratic function will be a parabola, assuming a ≠ 0. √ 2 3 3 For example, f(x) = x , g(x) = x , and h(x) = √x are all functions that \make sense," or are defined on all real numbers. f(x) = x is a function that is defined 1 1 only on the non-negative real numbers. g(x) = x and h(x) = x2 are function that are defined everywhere except for x = 0. See figure 5.25 in your text to see graphs for these functions. Finally, consider the graph of y2 = x. It is shown in figure 5.26 in your text. It looks like a sideways parabola. We've already determined that y2 = x is not a function, but the graph of it gives us a more clear picture of what is going on. Consider all the graphs we've seen. Those that are not functions have two points in the graph (x; y1) and (x; y2) for y1 ≠ y2. Pictorially, if we can draw a vertical line through the graph in such a way that it intersects more than one point, the graph is not a graph of a function. This process is known as the vertical line test and is quite helpful when we know what the graph for an equation looks like. 4. Tangent lines Consider some curve in the cartesian plane and fix some point P on it. See figure 5.10 in your text for reference. Let Q be some other point on the curve, and let PQ be the line between them. Now imagine pushing the point Q along the curve closer to P . As it gets closer and closer, the slope of the line PQ gets closer to that of what is called the tangent line at P . Call the slope of the tangent line at P , mP . Say P = (x; y) and Q = (x′; y′). Let ∆x = x′ − x and ∆y = y′ − y. So the slope of the line PQ is y′ − y ∆y = . x′ − x ∆x 3 Again, imagine we push Q along the curve close to P . This is the same as saying we push x′ close to x, or that we push ∆x close to 0. Recall that as Q gets closer ∆y to P the slope of PQ, ∆x , gets close to the mP . In limit notation, we get ∆y mP = lim . ∆x→0 ∆x The above definition is how Leibniz defined the notion of a tangent line. Notice that he could not simply say \when ∆x = 0..." as this would give us ∆y = 0. So ∆y 0 ∆x = 0 , which is not defined..
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