1. Find the tangent vector i(t) and unit tangent vector u(t) for each of the following...

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1. Find the tangent vector i(t) and unit tangent vector u(t) for each of the following parametrised curves: (a) r(t)=-(2+12)i- t25+ (6+12)k for 0≤t≤1. (b) x(t)=1-t, y(t)=1+t, z(t) = t² + 1 for t€ R. (c) (t)=te (i-1)+2t3k for t≥0. 2. If a curve C is parametrised by r(t) for t€ [t1, t2], then the total are length of the curve is S= dt. Suppose C is the lower half of a circle of radius A. (a) Using only what you know from elementary geometry, predict what value of S the above integral should give. (b) Now we parametrise the semicircle by 7(t) = Acos (vi) + Asin (vi) j where t goes from 2 to 4x2. Compute the are length of this semicircle using the integral formula above and comment on how it agrees or disagrees with (a). 3. (a) Find the curvature (t) and principal unit normal vector, N(t) for the curve given in Problem 1(b). (b) Show that reaches its maximum value at the point (1,1,1). 1. Find the tangent vector i(t) and unit tangent vector u(t) for each of the following parametrised curves: (a) r(t)=-(2+12)i- t25+ (6+12)k for 0≤t≤1. (b) x(t)=1-t, y(t)=1+t, z(t) = t² + 1 for t€ R. (c) (t)=te (i-1)+2t3k for t≥0. 2. If a curve C is parametrised by r(t) for t€ [t1, t2], then the total are length of the curve is S= dt. Suppose C is the lower half of a circle of radius A. (a) Using only what you know from elementary geometry, predict what value of S the above integral should give. (b) Now we parametrise the semicircle by 7(t) = Acos (vi) + Asin (vi) j where t goes from 2 to 4x2. Compute the are length of this semicircle using the integral formula above and comment on how it agrees or disagrees with (a). 3. (a) Find the curvature (t) and principal unit normal vector, N(t) for the curve given in Problem 1(b). (b) Show that reaches its maximum value at the point (1,1,1).

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