Calculus Answers

) An efficiency study of the workers at a factory shows that an average worker who comes to work at 8:00 am will have produced Q(t) = -t^3+9t^2+12t hours later. At what time during the morning is the worker performing most efficiently.

Sketch the graph of the function f defined by f(x) = x^4+ 8x^3 clearly giving all the properties used in it.

Obtain the largest possible domain and range of the function f, defined by f(x) = under root x+1 /x+2. Further, check whether or not lim x tends to a f(x) exists for x= 2, -1

Check the continuity of f(x)={x+1,xis smaller than 1
{ 0, 1greater than equal to x greater than equal to 2
{2-x , x greater than equal to 2
for all points in its domain.

Give an example each with justification, of a function defined on ]1, -1[which is
i) one-one but not onto.
ii) onto but not one-one.

Show that lim x tends to infinity=2/x-3=0

Find the volume and surface area of the solid formed by revolving a right-angled triangle about a side adjacent to the right angle. What is the solid so obtained

8. a) Define two partition P1 and P2 of ] [ such that 2, 5 P1 ⊂ P2 . Find the upper and lower
product sums with respect to f , defined by
⎩
⎨
⎧
− ≥
< = 1 x , x 4
x, x 4
f(x) 2 . Also verify the
relationship between these 4 sums. (4)
b) Derive a reduction formula for ∫( n x) dx, m∈N m l . Hence evaluate ∫( n x) dx 4 l . (4)
c) Find the approximate value of 5/3 ( upto 3 decimal places.

Evaluate
i) (1 | x 3|) dx
4
0
2
∫ + −
ii) ∫
−
+
3
3
[f(x) g(x)]dx , where f and g are odd functions.
iii) cos x dx ∫
iv) ∫sin θsin 2θsin 3θdθ
v) ∫ +
−
dx
1 x
x tan (x )
6
2 1 3

6. a) Find dx
dy , when
i) t x 3cost 2cos t, y 3sin t 2sin 3 3 = − = −
ii) [ \{1} 2 ( ) (sin ) , ] 0, 1 sin π = + ∈ − y nx x x x x l . (3 × 2 = 6)
b) If
⎪⎩
⎪
⎨
⎧
=
≠ = −
0 ; x 0
; x 0
x
1
x tan f(x) 1
, show that f is continuous but not differentiable at
x = 0 .