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Problem 3 (20 points): Find all values of t where tangent lines for the given parametric equation are horizontal or vertical.

Answer 1

4 Given 3 3t2 and t V는 443. Differentiat i' and 'y' with respect toʻt: 으 dx dt d43.3t2) 3 dy (tf-43) and dt dt dt d och = nix' px 1 Apply power rule dx dx 3t-st 는 - 4t3_132 and dy dt dt dy dt ft3__ 1242 slope = 9 da . d x 3t-st dt dy dx 4tf(t- 32) 3 (-2) dy da 는 4t (t-3) 3Ct-2), The horizontal tangente occur when the derivative equals zero ie dy CO., 4t (t-3 10 3(-2) - - 4t Ct-3) = 0

LEIL 4t=0 and - 3 =0) tao and Therefore t= 0, 3 tangent is horizontal The vertical tangents occur when derivate is infinity on undefined dy dx 는 dy dx 4+(4-3) 3(t-2) ola 3(+-2) = 0 t=2. At tog the tangent is vertical. t = 0, 3 tangent is horizontal t=2 tangent is vertical.
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