Calculus Answers

5. a) 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 t Q(t) t 9t 12 3 2 = − + + units t hours
later. At what time during the morning is the worker performing most efficiently. (5)
b) If x y tan−1 = , obtain an equation showing the relationship between n 2 n 1 y , y + + and
n y .

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

3. a) Obtain the largest possible domain and range of the function f , defined by
x 2
x 1 f(x)
+
+ = . Further, check whether or not limf(x) x→a exists for a = −1, 2 . (5)
b) Find dx
df , where ⎥
⎦
⎤

1. Which of the following statements are true or false? Give reasons for your answers. (2 × 5 =10)
i) The function f : R → R , given by f(x) n | x 1 x | 2 = l + + is neither even nor odd.
ii) ∫ = −
0
x
2 2 sin(t )dt sin x
dx
d .
iii) The area enclosed by the x-axis and the curve y = cosx over the interval ⎥
⎦
⎤ ⎢
⎣
⎡ π π − 2
3
,
2
is 0 .
iv) If f and g are functions over R such that f + g is continuous, then f must be
continuous.
v) x − y + 2 = 0 is a tangent to the curve 3 2 (x + y) = (x − y + 2) at (−1, 1).

Verify that the Pfaffian differential equation yz dx + (x^2y- zx) dy + (x^2z-xy) dz =0 is integrable and hence find its integral.

The differential equation of a damped vibrating system under the action of an external periodic force is: d^2x/dt^2 +2 m0 dx/dt +n^2x = a cospt Show that, if n>m0>0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.

Solve the following DEs (I) (dy/dx-1)^2(d^2y/dx^2 +1)^2 y = sin^2(x/2)+x. (2) 2x^2y(d^2y/dx^2)+4y^2= x^2(dy/dx)^2 + 2xy(dy/dx).

Solve : x×dy/dx+y lny = xye^x.

Find the charge on the capacitor in an RLC circuit at t= 0.01sec. when L= 0.05 Henry, R= 2 ohms, C= 0.01 Farad. E(t) =0, q(0)= 5 Columbus and I(0)=0.

Solve : d^2y/dx^2 -2tanx dy/dx +5y = e^x. sec x