MATH 3A03 Study Guide - Final Guide: Howaldtswerke-Deutsche Werft, Jhq Rheindahlen, Wiru Language

11, x(t) = Ce": x(0) = 5 12. P=C/t; P = 5 when t 3 13, y v2tfC; the solution curve passes through (1,3) Satishes the equation dP dt 7. Use implicit differentiation to show that za + y2 = r2 is T. 21S 13. a solution to the differential equation dy/day.14. Q 1/(Ct +C);:Q 4 when t 2 Problems 15. Show that y = A + Cekt is a solution to the equation 18. Suppose Q = Cekt satisfies the differential equation dy =k(y-A) dt -0.0302 dt What (if anything) does this tell you about the values of C and k? 16. Find the value(s) of w for which y cos wt satisfies 19. Find the values of k for which y z2 + k is a solution dt2 to the differential equation 17. Show that any function of the form Ci cosh wt + C2 sinh wt 20. For what values of k (if any) does y-5+ 3eka satisfy z the differential equation satisfies the differential equation x,,-w2x = 0. ey=10-2y dx

8. For each of the following iterated integrals, do the following: Sketch the region of integration. Where applicable, give a geometric interpretation of the iterated integral. Evaluate each double integral. Change the order of integration where nec- essary. (a) 104 fets dydr (b)//쁩udy dr (c)//gudy @i jsdady (h)? fsinrardy (i).t{r)lydydz 0 ev 1 2 dzd @j feë´ drdy((o) 2 4x c s(y ) dydr (p)//VI阿drdy a) J sin(3戸) dydx (r) fsin(H) dzdy (s)// dady (t) 9 3 sin (㎡) drdy (u) 4 2ye'dydr (v)//e-va dydz 0 y/2 1 Vy vz 0 z ivITpdrdy@i 2向dady (y)..-75dy dr (w) (1 3)

2. Consider the following function of two variables: z f(x, y) = x/1 + y2 = (a) Find the gradient of the function (b) Find the directional derivative of the function at the point P(5, 3) in the direction of the point Q(8, 4). (c) Suppose that k=e24 andy-2t.|Use the ChainRule to comput4 , and evaluate the expression when t=0. dt'