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Calculus Important notes

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MATH 104
Jacob Levy

Math 104 Section 101 Notes for 2013/9/5 September 5, 2013 Calculus is the study of functions on real numbers. In this course we will look primarily at functions used in business and economics: for example, as a company varies the number of widgets q that it produces, the price p that it can sell them at will change, producing a relationship between p and q. De▯nition 1. a function y = f(x) is a rule that applies to some set of real numbers D, and for each x in the set D, it assigns it a unique real number y. The set D is called the domain of f. Example 2. If f(x) = x , then we can take the domain D to be the set of all real numbers. Example 3. If f(x) = 1=x, then f is de▯ned at every point x 6= 0. p Example 4. If f(x) = x, then f is de▯ned at every x ▯ 0. We can de▯ne a function by its graph, and much of calculus comes from draw- ing connections between the algebraic side (i.e. an equation) and the graphical side (i.e. the graph of a function). For example, here is the graph of f(x) = x : Using the graph, we can determine whether a curve is a function or not. Because a function must assign to each x in the domain a unique value y, each vertical line intersects the graph at only one point (x;y). We can graphically 2 see that the curve of x indeed represents a function. In contrast, the curve below does not represent a function: 1 Note that the dashed vertical line intersects the graph at multiple points, so it cannot assign a unique y to each x. This is called the vertical line test: a graph represents a function if and only if each vertical line intersects it at at most one point. Given the importance of vertical lines in determining whether a graph is a function, it is natural to ask what horizontal lines mean. They have a meaning, but ▯rst, we need a de▯nition: De▯nition 5. A function f is called 1-to-1 if for each y there is at most one x such that y = f(x). Equivalently, we call f invertible. The relationship between x and y can sometimes be switched. In economics, a demand curve relates price and quantity; price can be viewed as a function of quantity produced, but the quantity demanded can also be viewed as a function of the price charged. To see if we can replace the relationship y = f(x) with ▯1 ▯1 x = f (y), we need to check that f would be a function. For that, we require f to be 1-to-1, and this is the case if and only if f satis▯es the horizontal line test: the graph of a function represents a 1-to-1 function if and only if each horizontal line intersects it at at most one point. Example 6. The following function is not 1-to-1, as demonstrated by the hori- zontal line depicted: 2 Example 7. The inverse of a powerpfunction is a root function. The function y = x is 1-to-1 and its inverse is3x, and more in general if a is an odd integer then y = x is invertible and its inverse isax. Remark 8. As can be seen in the example above, the graph of the inverse of a function is ob
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