{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Implicit Differentiation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "implicitplot(\{x^2-y^2=5,4*x^2+9*y^ 2=72\},x=-5..5,y=-5..5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "We s hall show that these two curves are orthogonal; that is to say that th e tangent lines at those intersections are perpendicular. First, we fi nd those intersections first. We substitue y^2=x^2-5 into the second e quation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(4*x^2+9*( x^2-5)-72=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "If x = 3, y^2=4 , which yields the intersections of " }{TEXT 256 5 "(3,2)" }{TEXT -1 5 " and " }{TEXT 257 7 "(3, -2)" }{TEXT -1 53 ". If x=-3, y^2=4, whic h yields the intersections of " }{TEXT 258 6 "(-3,2)" }{TEXT -1 5 " an d " }{TEXT 259 7 "(-3,-2)" }{TEXT -1 89 ". Next, we use the implicit d ifferentiation to find the derivatives for these two curves." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=x^2-y^2=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "implicitdiff(f,y,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g:=4*x^2+9*y^2=72;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "implicitdiff(g,y,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 341 "Let's check the slopes of these two tang ent lines at the intersection (3,2). The slope for the first tangent l ine is '3/2' and the second one is (-4/9)*(3/2) = -2/3. So the tangent lines at (3,2) are perpendicular. It is easy to check that the tangen t lines at other intersections are also perpendicular, so these two cu rves are 'othogonal'." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }