MATA33H3 Lecture 16: Partial Derivatives & Applications (7.3)

560 Test exercise 7 Write out the solutions carefully. They all quite straightforward. 1 Differentiate the following with respect to x: 38 (b) (Sr +3,- (O cosh, (a) tan 2x sin, 4x er)-3x 2 lfx2 + ,2-2x+2y-23,find/rand얽at the point where x--zy. 4 Ifx-11-cos" and y-3(.-sin") find a and a: in their simple (d) kg.dr.3x-1) sina 田 (x+1 diy when r,y,4ay2以 3 Find an expression for forms Further problems 7 39 1 Differentiate with respect to (b) In(secx+tanx (c) sinxcos'x 2 Find when: a)y-14 cosx if y is a function of x, and x-T+1 show that훑-x(1-x)dr 3 5 Differentiate coshx-1 coshx1 ",.tow. "r-wrmF} o, ,.刍 7 If(x-n'-A(x + y. prove that (2x + y -x+2y. If.e + 2xy + 3,-이.prove that (x + 3 '헒+2tx2 + 2xy + 9 Scanned by CamScanner

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)

1. Evaluate the following using substitution. Do parts (a) through (e) in two ways: i. by writing the integrand in the form f(G(x))g(x) (or y oir t or whatever the variable is) for appropriate functions f, G, and g, with G, = g, and then finding F = Jf; and ii. using differential notation. Do the remaining parts in the way you feel most confident sin(w) Vcos(w) dw sin(vs) b) sin(52) dz m) rlnr nesin(1 e")dx 2v + 1 1 +y2 g) tanxdx 1-w2 ta㎡ (z) sec2(z) dz 1 + 4y'2 i) sec(r/2) tan(x/2) dx 1 - 9w2