Mathematics I (Calculus) 2075
Attempt any three questions:
1. (a) A function is defined by f(x) = |x| , calculate f(-3), f(4), and sketch the graph.
Given,
Now,
Now, for sketching graph calculating y = f(x) for different values of x
Graph:
(b) Prove that the 
does not exist
Given
Now,
Here,
Hence, 
doesn’t exist.
2. (a) Find the domain and sketch the graph of the function f(x) = x2 - 6x .
Given
For domain, for all real
values of x, f(x) exist. So, domain is set of all real number i.e. domain
is 
For Graph, calculating the values of y = f(x) for different values of x;
Plotting these points on graph we get:
(b) Estimate the area between the curve y = x2 and the lines y = 1 and y = 2.
Given
And lines:
y =1, y = 2
The given curve is the parabola and the sketch of the given curve is
Required area 
3. (a) Find the Maclaurin series for cos x and prove that it represents cos x for all x.
We need to find derivatives of f(x) = cos x, so
Therefore, Maclaurin series for cos x is
Since the cosine function and all the derivatives of cosine function have absolute value less than or equal to 1. So, by Taylor’s inequality
Now,
i.e.
for all values of x.
This implies that the series converges to cosx for every value
of x.
(b) Define initial value problem. Solve that initial value problem of y' + 2y = 3, y(0) = 1.
The problem of finding a function y of x when we know
its derivative and its value y0 at a particular point x0
is called an initial value problem.
Given,
Comparing given equation with 
we have
P =2 and Q=3
Now,
Applying the initial condition y(0)=1
Applying this value, we have:
(c) Find the volume of a sphere of a radius a .
The sphere of radius a can be obtained rotating the half circle graph (semi-circle) of the function
about the x-axis.
The volume V is obtained as follows:
by the symmetry about the y-axis,
4. (a) If 
does 
exist? Justify.
Here
As 
we get 
form.
So, set 
where m is some constant value then,
Along 
we observe 
and we get
And at m =1,
Thus,
So, the limit does not exist.
4(b) Calculate 
for f(x, y) = 100 - 6x2y and 
Attempt any ten Questions:
5. If f(x) = 
and g(x) = 
, find fog and fof.
Given,
Now,
6. Define continuity on an interval. Show that the function 
is continuous on the interval [ -1,1] .
A function f is continuous from the right at a
number a if 
and f is continuous from the left at a if 
.
A function f is continuous on an interval if it is continuous at every number in the interval. If f is defined only one side of an end point of the interval, we understand continuous at the end point to mean continuous from the right or continuous from the left.
Given,
Let 
then
Which shows that f(x) is continuous at 
.
For the end points i.e. x=-1
Which shows that f(x) is continuous at the left end point x = -1
Similarly for the end point x=1
Which shows that f(x) is continuous at the right end point x = 1.
Hence f(x) is continuous at [-1,1]
7. Verify Mean value theorem of f(x) = x3 - 3x + 2 for [-1, 2].
Given,
f(x) = x3-3x+2
Since, f(x) = x3-3x+2 is continuous on [-1, 2] and f’(x) = 3x2-3
so, differentiable on (-1, 2).
Thus f(x) = x3-3x+2
satisfy the both conditions for mean value theorem. So, there exist 
such that
Clearly, 
Hence, mean value theorem satisfied.
8. Stating with x1 = 2, find the third approximation x3 to the root of the equation x3 - 2x - 5 = 0.
Given,
By Newton’s method we have
When x1=2
Then
9. Evaluate 
Here
Take,
Put 
Then 
So that
Thus, form (i)
10. Find the volume of the resulting solid which is enclosed by the curve y = x and y = x2 is rotated about the x-axis.
11. Find the solution of y'' + 4y' + 4 = 0.
Given,
The characteristics equation of given differential equation is
Here the roots are real and equal.
The general solution is
12. Determine whether the series 
converges or diverges.
Given,
Here,
Now;
Therefore, by nth term test for divergence, the given
series is divergent.
13. If a = (4, 0, 3) and b = (-2, 1, 5) find |a|, the vector a - b and 2a + b.
Given
a = (4 , 0, 3)
b = (-2, 1, 5)
Now,
14. Find 
and 
if z is defined as a function of x and y by the equation x3 + y3 + z3 + 6xyz = 1.
Given,
Now,
Differentiating w.r.to x
Again,
Differentiating w.r.to y
15. Find the extreme values of the function f(x, y) = x2 + 2y2 on the circle x2 + y2 = 1.
Given,
And, let
By method of Lagrange’s multiplier, for some scalar 
This implies,
This gives
From (ii) we have x = 0 or λ = 1. If x = 0,
then (i) gives y = ±1. If λ = 1, then y = 0 from (iii), so then (i) gives x =
±1. Therefore, f has possible extreme values at the points (0, 1), (0, −1) (1,
0), and (−1, 0). Evaluating f at these four points, we find that
f(0, 1) = 2
f(0, −1) = 2
f(1, 0) = 1
f(−1, 0) = 1
Therefore, the maximum value of f on the circle x 2
+ y 2 = 1 is f(0, ±1) = 2 and the minimum value is f(±1, 0) = 1.