MAT135H1 Chapter 3.3: Derivatives of Trigonometric Functions (NOT including limit laws)

Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function. (a) y=ce2x, y, = 2y 2. (c) y = 5 + ce-x-, y, + 2xy = x 2 y = (I + ce-z?/2); (I-ce-x2/2)-1 2y' + x (y2-1) = 0 y ="e + c2x)ex + sinx + x2, y"-2y' + y -2 cos x + x2-4x + 2 y x-1/2(c1 sinx + c2 cosx) + 4x +8; (d) (f) 2 (h) ty+ (+3-2

Integrating Products of sine and cosine Let m and n be integers. Evaluate § sin-x cos" x dr. Strategy The power of cosine is odd n=2k +1 >0 Isin" x cos" x dr-/sin-x(cos x)' cos x dx in-in x) cos.a de Strategy The power of sine is odd ,n = 2k+1>0 jsin" xcos' x ds-(sin' x) cos" rsin.x d (1-cos) cos" xsin a d -10-112)' u" du If the powers of both sine and cosine are odd, both Strategies can be used. Strategy 3-2 The powers of both sine and cosine are even n= 2k 20 and m = 21120 sin" xcos"r dr = (sin, x)', (cos, x)' dc (1-cos 2x ) (1+cos 2x Foil out and use Strategy C-O and C-E

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