Math Xb Spring 2004 NameLab 1: Exponential and Logarithmic Functions Section InstructorFebruary 12, 2004 CollaboratorsCertain combinations of the ex ponential functions exand e−xarise so frequently in mathematics and itsapplications that they are given special names. Two of them are the hyperbolic s ine funct ion, denoted sinh x(pronounced “sinch x”), and the hyperbolic cosine function, denoted cosh x (pronounced “cosh x”). Theyare defined as follows.sinh x =ex− e−x2cosh x =ex+ e−x2These functions are similar in some ways to the trigonometric functions we will study later this semester.You need not know any trigonometry to complete this lab.1. Choose one of the two hyperbolic trig functions listed above. Explain how to use the graph of y = exto construct the graph of your chosen function without using a graphing calculator. (Use what youknow about stretching, shrinking, shifting, flipping, adding, and subtracting the graphs o f functions.)Illustrate your explanation with appropriate graphs.2. Simplify cosh2x − sinh2x.3. The graph of y = sinh x passes the vertical line test, and so sinh x is a one-to-one, and hence invertible,function. The inverse of sinh x is the function sinh−1x = ln(x +√x2+ 1). Find and simplify thederivative of sinh−1x.4. It can be shown that if a heavy flexible cable (such as a telephone or power line) is suspended betweentwo points at the same height, then it takes the shape of a curve with equation y = c +a cosh(xa) calleda catenary. (The Latin word catena means “chain.”) Supp ose that a telephone line hangs be tween twopoles 14 meters apart in the shape of the catenary y = 20 cosh(x20) − 15, where x and y are mea suredin meters. Find the slope of this curve where it meets the right pole.-7