Prove that the curve r(t) = (a cos t, b sin t, c sin t) is...
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Prove that the curve r(t) = (a cos t, b sin t, c sin t) is parameterized by arc length, provided ab + c = 1. 54. Change of variables Consider the parameterized curves r(t) = (f(t), g(t), h(t)) and R(t) = {f(u(t)), g(u(t)), h(u(t))), where f, g, h, and u are continuously differentiable functions and u has an inverse on [a, b]. a. Show that the curve generated by r on the interval a t b is the same as the curve generated by R on u- (a) t u (b) (or u- (b) t u (a)). 1 b. Show that the lengths of the two curves are equal. (Hint: Use the Chain Rule and a change of variables in the arc length integral for the curve generated by R.) 12. Letf(x) = x - 2x- y +1 x -2x + y + 1 - (x, y)(1,0) (along x =1) lim f(x) first.) . lim f(x) does not exist. (Hint: Examine lim f (x) and (x, y)(1,0) (x, y)(1,0) (along y=0) Use the Two-Path Test to show that Prove that the curve r(t) = (a cos t, b sin t, c sin t) is parameterized by arc length, provided ab + c = 1. 54. Change of variables Consider the parameterized curves r(t) = (f(t), g(t), h(t)) and R(t) = {f(u(t)), g(u(t)), h(u(t))), where f, g, h, and u are continuously differentiable functions and u has an inverse on [a, b]. a. Show that the curve generated by r on the interval a t b is the same as the curve generated by R on u- (a) t u (b) (or u- (b) t u (a)). 1 b. Show that the lengths of the two curves are equal. (Hint: Use the Chain Rule and a change of variables in the arc length integral for the curve generated by R.) 12. Letf(x) = x - 2x- y +1 x -2x + y + 1 - (x, y)(1,0) (along x =1) lim f(x) first.) . lim f(x) does not exist. (Hint: Examine lim f (x) and (x, y)(1,0) (x, y)(1,0) (along y=0) Use the Two-Path Test to show that
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