Mathematical functions can sometimes be used to generate interesting shapes or patterns. The function f() = sinx/a is generally known as the cardinal sine function (sinc) which has an ocillatory and decaying pattern as illustrated in Figure 1. This function is everywhere continuous except one point, where x = a. Identify this point (value of a) where the function is discontinuous. From what you have learned, despite the discontinuity at x = a, it is possible that the limit ofthis function exists at a. Using the L'Hopital's rule, determine this limit, limx>a f(x). The 2-dimentional version of the cardinal sine function can be used to generate a symmetrical ripple-like surface as illustrated in Figure 2: f (x, y) = (sin /x + y? + c)/Va + y + c , where c is arbitrary constant. With your knowledge in differentiation techniques, determine the expressions of partial derivatives fx and fy. Use c = 7 At this point it is probably clear that f(x) is basically a simplified vertical slice of the surface f(x,y). Taking the slice of f(x,y) where y=0, determine the first non-zero position of x (value of x other than x=0) where this ripple has horizontal gradient (fx=0). The following information in Figure 3 might be useful: 0.8 0.6 0.4- 0.2- -0.2- -0.4 -20 -15 -10 5 10 15 20 Figure 1:1llustration of f(x) Figure 2:1llustration of f(x,y) 12 tan w 10.9041 10 7.7253 8- 4.4934 4 10 12 Figure 3: w = tan(w) has solutions at w = 0, w= 4.4934, w = 7.7253, w = 10.9041, etc.