# Worksheet_8 - with the instructor(b Suppose h R → R is a...

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Discussion — Thursday, October 7th Subject: Curves and integration. 1. (a) Sketch the first-octant portion of the sphere x 2 + y 2 + z 2 = 16. Check that P = ( 2,2,2 p 2 ) is on this sphere and add this point to your picture. (b) Find a function f ( x , y ) whose graph is the top-half of the sphere. (c) Imagine an ant walking along the surface of the sphere. It walks down the sphere along the path C that passes through the point P in the direction parallel to the yz - plane. Draw this path in your picture. (d) Use the function from (b) to find a parameterization r of the ant’s path along the portion of the sphere shown in your picture. Specify the domain for r , i.e. the initial time when the ant is at P and the final time when it hits the xy -plane. (e) Adjust your parameterization so that the ant is at P at time 0 and hits the xy -plane at time 1. Hint: See 2(b) below. Check your answer with the instructor. 2. Consider the curve C in R 3 given by r ( t ) = ( e t cos t ) i + 2 j + ( e t sin t ) k (a) Calculate the length of the segment of C between r (0) and r ( t 0 ). Check your answer
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Unformatted text preview: with the instructor. (b) Suppose h : R → R is a function. We can get another parameterization of C by con-sidering the composition f ( s ) = r ( h ( s ) ) This is called a reparameterization . Find a choice of h so that i. f (0) = r (0) ii. The length of the segment of C between f (0) and f ( s ) is s . (This is called param-eterizing by arc length.) Check your answer with the instructor. (c) Without calculating anything, what is | f ( s ) | ? (d) Draw a sketch of C . 3. Consider the curve C given by the parameterization r : R → R 3 where r ( t ) = (sin t ,cos t ,sin 2 t ). (a) Show that C is in the intersection of the surfaces z = x 2 and x 2 + y 2 = 1. (b) Use (a) to help you sketch the curve C . 4. As in 2(b), consider a reparameterization f ( s ) = r ( h ( s ) ) of an arbitrary vector-valued function r : R → R 3 . Use the chain rule to calculate | f ( s ) | in terms of r and h ....
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