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(MATH2351)[2011](f)quiz~=ee822^_73244.pdf
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MATH150 Introduction to ODEs, Fall 2011-12 Hints to Week 01 Revision worksheet: Di.erentiations and Integrations
Name: ID No.: Tutorial Section:
Attempt at least FOUR questions from the following questions! The worksheet MUST be
handed in at the end of the tutorial

(Partial solution of this worksheet will be available at the course website a week after all the tutorials)
1. (Demonstration) Do two/three di.erentiation problems for each of the categories below
(a)
Simply polynomials which may have negative powers;

(b)
simple (polynomials/rational) functions that require product/quotient rules;

(c)
(polynomials/rational) simple functions that require chain rule;

(d)
repeat the above for trigonometic functions when time allowed. Students should be allowed time to do some class problems before turning to integration problems.


2. (Demonstration) Do two/three problems in integration of functions that require either one or a com-bination of
(a)
simple power law;

(b)
substitution;

(c)
trigonometric functions;

(d)
integration by parts.


3. (Class work) Find dy/dx for the following functions:
(a)
y =5x4 + x2 + x: 20x3 +2x + 1;

(b)
y = x3 +1 3 :3x2 . 3x.4;


x
(c) y =(x2 + 1)2(2x + 3): (4x(x2 + 1)(2x + 3) + 2(x2 + 1)2;
x2 +1
(d) y =: .4x(x2 . 1).2;
x2 . 1
x
(e)
y = 1+ x2: ;1+ x2

(f)
y = (1+3x)4: 12(1 + 3x)3;

(g)
y =: ;


sin xx cos x . sin x
xx2
(h) y = (ln x)2: 4. (Class work) Find . f(x)dx for the following functions: 2 ln x x ;
(a) . x 3dx: x4/4 + C;
(b) . 1 x dx: 2 x + C;

. (ax + b)n+1
(c) (ax + b)n dx:+ C;
a(n + 1)
11 1
(d) dx:(ax + b).n+1 + C for n .ln |ax + b| + C for n = 1;
= 1,
(ax + b)n a(1 . n) a
11
(e) dx: ln |ax + b| + C;
(ax + b) a
1
(f)
sin3xdx: . cos 3x + C;

(g)
x cos(1 + x 2)dx: sin(1 + x 2)+ C;

(h)
x sin xdx: sin x . x cos x + C;


3
1
2
5. (Supplementary problems) Find dy/dx for the following functions: 3x . 2
(a)
y = x(3x . 2): +3 x;

2x

(b)
y = (1+2x)5: 10(1 + 2x)4;



3
2
2).
x(1 + x
23
(c) y =1+ x2: ;3
(d)
y = cos 2x: .2 sin 2x;

(e)
y = cos2 x: . sin 2x;

(f)
y = cos x sin x: cos 2x; 1

(g)
y = tan x:;

cos2 x

(h)
y = e2x:2e2x;

(i)
y = ex cos x + e.x sin x:(ex + e.x)(cos x . sin x); 1

(j)
y = ex ln x: ex( +ln x);

(k)
y = ln(xex): 1+ ;


x
1
x
6. (Supplementary problems) Find f(x)dx for the following functions:
4 3

x +3x 3)dx x
3
2
+
x
4 + C;(a) (2 3 4
(b) . x 2 cos x 3dx 1 3 sin x 3 + C;
.
(c) sin(1 + x)dx . cos(1 + x) + C;
(d) . x cos(1 + x 2) dx 1 2 sin(1 + x 2);
(e) . x ln xdx 1 2 x 2 ln x . 1 4 x 2 + C;
.
(f) x 2 e xdx (x2 . 2x + x)ex + C;
(g) . e x sin xdx 1 2 e x(sin x . cos x) + C