53 Worksheet Solutions 8_31

# 53 Worksheet Solutions 8_31 - Math 53: Multivariable...

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Math 53: Multivariable Calculus Solutions for Worksheet 8/31/11: Tangent Lines, Area, and Arclength (Oh my!) Exercise 0.1. (Stewart 10.2.25) Consider the curve x = cos t , y = cos t sin t , 0 t < 2 π . Find all tangent lines to the point (0 , 0) (hap- pening for diﬀerent values of t ). Solution. First we need to ﬁnd which values of t give us the point (0 , 0). This means we wish to simultaneously solve the equations cos t = 0 and sin t cos t = 0 for 0 t < 2 π . The only values in this range for which cos t = 0 are t = π/ 2 and t = 3 π/ 2, so these are the only possible values of t for our point. Since sin t cos t is also 0 for these values of t , we have that t = π/ 2 and t = 3 π/ 2 both give us the point (0 , 0). We now wish to ﬁnd the slope of the tangent line at time t ; as we saw in section and in lecture, this is just g 0 ( t ) f 0 ( t ) (assuming that expression makes sense). Since f ( t ) = cos t and g ( t ) = sin t cos t , we have f 0 ( t ) = - sin t and g 0 ( t ) = cos 2 t - sin 2 t (by the product rule). Thus we have g 0 ( t ) f 0 ( t ) = cos 2 t - sin 2 t - sin t . At time

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## 53 Worksheet Solutions 8_31 - Math 53: Multivariable...

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