It simply provides an image of what calculus methods

Info icon This preview shows pages 2–4. Sign up to view the full content.

View Full Document Right Arrow Icon
teach you calculus. It simply provides an image of what calculus methods can do, followed by some calculus results that can be applied to economic problems. Figure M.10-1 shows 2 functions: y = 2 x on the left side and y = x 2 on the right. We want to answer two questions about each of these functions over the domain x [0, ). First, what is the slope of the function at any value of x ? Second, what is the area between the function and the horizontal axis and between 0 and the given value of x ? For y = 2 x , the questions are fairly easy to answer. The slope is a constant +2, because y = 2 x is a lin- ear equation. The area required is just the area of a triangle, A = (b x h)/2, with base b = x and height h = y = 2 x . Hence A = (b x h)/2 = ( x y )/2 = ( x )(2 x )/2 = 2 x 2 /2 = x 2 . For x = 1, A = 1 square unit; with x = 2, A = 2 2 = 4 square units; and with x = 3, A = 3 2 = 9 square units. When we turn to Figure M.10-1(b), however, things are not so simple. The slope is not constant, but gets continuously steeper, and we cannot use our triangle formula to calculate the area, because y = x 2 is not a straight line. Calculus has been developed, since its invention in the seventeenth century (virtually simultaneously) by Newton and Liebniz, to deal with precisely these difficulties posed by nonlinear functions. We shall assert (correctly, but without proof) that the slope of the function y = x 2 at any x has the value 2 x , and that the shaded area takes on the value A = x 3 /3. [Any introduc- tory calculus text will provide proofs.] Hence, when x = 1, the slope of the tangent to the function (simply put, the slope of the function) = 2 x = 2 and the area = x 3 /3 = 1/3 square units; when x = 2, the slope = 2 x = 4 and the area = 8/3 square units; and when x = 3, the slope = 6 and the area A = 9 square units. Note that at any value of x , the value of the slope in (b) equals the value for y in (a). This is as we would expect, since in (a), y = 2x, which is the value of the slope of the function y = x 2 in (b). Note also that for x < 3, the shaded area in (a) exceeds that in (b), at x = 3 the two areas are equal (both equal 9 square units), and for x > 3, the shaded area in (b) exceeds that in (a). What does all this mean? We have used the notation m = ∆ y/ x to denote the slope m of a function. Here both ∆ x and ∆ y represent discrete differences between the x values and between the y values at two separate points. Suppose that we shorten the distance between x 1 and x 2 (= x 1 + ∆ x ) until ∆ x becomes infinitesimally small, and x 2 therefore virtually coincides with x 1 , and we do the same for y 1 and y 2 . In the limit, we approach the slope of the tangent to the functions, which we write dy/dx , at x 1 . If y = f ( x ), then we can write the slope (which we also call the derivative of the function) in several different but equivalent ways: dy/dx , y ’, df ( x ) /dx , or f ’( x ). M10-2 MATH MODULE 10: CALCULUS RESULTS FOR THE NON-CALCULUS SPEAKER
Image of page 2

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The derivative of a function of x is also a function of x . In our example, if y = f(x) = x 2 , then dy/dx = df /dx = f (x) = 2 x is also a function of x .
Image of page 3
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern