# BSc.-Calculus-Mumbai-April 2019

## Semester: 2

[Total Time: 2 ½ Hours]
[Total Marks: 75]
N.B. 1) All questions are compulsory.
2) Figures to the right indicate marks.
3) Illustrations, in-depth answers, and diagrams will be appreciated.
4) Mixing of sub-questions is not allowed.
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Q. 1 Attempt All (Each of 5Marks) (15M)

A) Select correct answer from the following:

1) In which of the following method, we approximate the curve of the solution by
the tangent in each interval.
a) Simpson’s Method
b) Euler’s method
c) Newton’s method
d) None of the above

2)ʃ 1/(9x2 + 25) dx =
a) (3/5) tan-1(3x/5) + c
b) (1/9) tan-1(3x/5) + c
c) (3/5) tan-1(5x/3) + c
d) (1/15) tan-1(3x/5) + c

3) A function is said to be invertible if and only if it is______________
a)Bijective          b) injective             c) Inflexion              d) Surjective

4) \lim_{x\rightarrow\infty}\;\frac7{2x}=
a)1                     b)infinite                 c) zero                    d) None

5) If f(x, y)= x3y3+ y3+1 then fx(x, y) is
a)3x2                  b) 3xy                      c) y3x                      d) None

B) Fill in the blanks:
(????????????????????????????????????????, ∞, (4i+5j)/41, (4i+5j)/31, -∞, ex, derivative, x – 3 log|x + 3| + c)

1) \lim_{{}\rightarrow\infty}\left(5-2x\right)=

2) The derivative of ex is __________________.

3) Unit vector of 4i+5j is______________.

4) ʃ x/(x +3) dx = ____________

5) The rate of change of one variable with respect to another is called _____________.

C) Answer the following in one line

1) Define Tangent Plane

2) Define Critical Point

3) Define the term Definite Integral

4) Evaluate \int_\frac\pi3^{2\pi}\sin\;x\;dx

5) Linearization of a function

Q. 2) Attempt the following (Any THREE) (15M)

a) Show that   \lim_{x\rightarrow1}2x^2+\;3x\;-\;1=

b) Discuss the continuity of the function f\left(x\right)=\sqrt{4-x^2}

c) Show that the function f(x) = ????3– 9????2 +30x + 7 is always increasing.

d) Find the relative extrema of f(x) = 4xy-x4-y4 using both first and second derivative test.

e) Using Newton’s method find the approximate root for the equation
f(x) = x- cosx

f) Divide 100 into two parts such that sum of their square is minimum.

Q. 3) Attempt the following (Any THREE) (15M)

a) Evaluate \int\sin\;^{-1}\;\sqrt x\;dx

b) Evaluate \int_\frac\pi6^\frac\pi3\frac1{\left(1+cot\;x\right)}dx

c) Estimate \int_0^4x^2dx  using simpson’s rule and n = 4.

d) Solve the differential equation
Sec2x tan y dx + sec2y tan x dy = 0

e) Solve dy/dx = 1 – y; y(0) = 0, find y(0.1) and y(0.3) using Euler’s method. Taking
h = 0.1.

f) Solve the differential equation \left(x+1\right)\frac{dy}{dx}-y\;=e^x\left(x+1\right)^2

Q. 4) Attempt the following (Any THREE) (15M)

a) Show that f(x, y) = 2x2 +3xy is continuous at (2, 3)

b) Find the second order derivatives of f(x,y)=????2????3 + ????4????

c) If z=x2y, x=t2 and y=t 3Use chain rule to find \frac{dz}{dt}

d) Find the directional derivative of f(x, y)=x3 +2xy2 at the point (-2, -3) in the direction of the vector a= ???? + ????

e) Find the gradient vector of f(x, y) if f(x, y) = 10 − 8????2 − 2????2. Evaluate it at (2, 3)

f) Find the equation for the tangent plane and parametric equations for
normal line to the surface z=x2y at the point (2, 1, 4)

Q. 5) Attempt the following (Any THREE) (15M)

a) Locate all relative extrema and saddle points of
f(x, y)= 3x2 – 2xy +y2 – 8y

b) Solve the differential equation
\frac{dy}{dx}\left(4+\;y\;+1\right)2^2

c) Draw the graph of y= 4 – 3x2 + x3 and find the intervals on which the function y is
increasing and decreasing(draw the graph on the answer sheet itself)

d) Find the asymptotes of the function y=\;\frac x{\left(x+1\right)\left(x+2\right)^2

e) Solve the differential equation
dy/dx = (4x + y + 1)2
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