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Problem I: Find the point on the curve t) = (55111 0f + (5 cos t))' + 12th at a distance 2611' units along the curve from the point (0.5.0) in the direction of increasing arc length. (Ans: (0.5.24n)) Problem 2: (Line Integral} Integrate [(1. 31.7.) = x + J; 22 from (0, 0, 0) to (1, l, 1) over the path given by C,:F(t) = 5+6}, 0 s t s 1 _. all\" t|.|.ll c,:r(r)=1'+j+tE. 05 t5 1 s a 5\"\" (A115: :43 + 3) c, 1 cl (LLB) Problemil: Finddiemassofaiinwirelyingalongthecnrve Fm = t! + r] + (4 mi, from point (0, o, 4) to pan: (Ji, 3). The density is 110) = 3r. (Ans: 445 2) Problem 4: Calculate fuel-order partial derivatives, [1,. [3, Use the chain rule if necessary. (a) form) = 4x2 +J'2 (b) may) = '21,, Problem 5: Find all second-order partial derivatives (In. . I\". , f1, and I\") of the following functions. Use the chain rule if necessary. Note. ft, = (5),, = 69 = 5%. I\" = a a a (r, 53%) =$- (a) f(x.y)=x'y+cosy+ysinx a mam-1e)- (we tan-wen... ) Min = (ifn Problem 6: Let f(x,y) = e\". and denote f(x(t),y(t)) = Fa). Evaluatein each case, using the chain rule. (a) x = t2 - 1. y = sin 3: (Ans: col-13mm: sin 3: + 3(c2 - 1) cos 3: (h) x = JHl, y = cost (Ans: em'" (3:72 JHlsin t)) Problem 1: The resistance R of two resistors connected in parallel is 1 1 1 E'E+E (I) Show that the total differential 2 as = (R3): mi +62%) an, (2) Given R1 = 1001] and R; = 4000, will the value of R be more sensitive to variation in 81 or to variation in R2? Give a reason for your answer. (3) Assume It, changes from 20 to 20.! ohms and R, changes from 25 to 24.9 ohms. Approximately how much is the change in R