Class Notes (834,756)
Mathematics (1,058)
MATH 104 (93)
Lecture

# Calculus Important notes

5 Pages
117 Views

Department
Mathematics
Course
MATH 104
Professor
Jacob Levy
Semester
Fall

Description
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
More Less

Related notes for MATH 104
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.