LinkedIn Insight BSc.-Calculus-Mumbai-November 2017 - Grad Plus

BSc.-Calculus-Mumbai-November 2017


Subject: Calculus

Semester: 2

[Time: 2 ¼ Hours ]
[Total Marks: 75]
Please check whether you have got the right question paper.
NB. 1) All questions are compulsory.
2) Figures to the right indicate marks.

Q.1) Answer the following questions. (15M)

A) Choose the best choice for the following questions :

1) Let be defined on an interval, x1 and x2 be the points on the interval, then f is said to be a a constant if
p) f(x1)=(fx2)=0
q) f(x1)=f(x2)=1
r) f(x1)=f(x2)= k
s) all of these

2) if” (a) exists and f has an infection point at x=a, then
p) f” (a)>0
q) f” (a)<0
r)  f” (a)=0
s) none of these.

3) If a function f is continuous on an interval [a,b], then which of the following is true:
p) is integrable on [a,b]
q) f is differntiable on [a,b]
r) Either (p) or (q)
s) None of these

4) Graph of a function of two variables is a surface in
p) 1- space
q) 2- space
r) 3- space
s) None of these

5) which of the following is true about the function f\;(x,y)=\frac{xy}{1+x^{2+y^2}}?
p) Continuous everywhere
q) Continuous except where 1+ x2 + y2 =0
r) Either (p) or (q)
s) Nither (p) nor (q)


B) Fill in the blanks for the following questions.

1) Two non-negative numbers, x and y, have a um gueto 10. The latest possible product of the two numbers is obtained by maximizing f (x) = ___________________ for x in the interval

2) If y=f(x) is a smooth curve on the interval [a, b] then the arc length of this curve over[a,b] defined  as ____________.

3) A solution of a differential equation \frac{dy}{dx}-y=0 is given by ___________

4) If f(x,y)=\sqrt{y+1\;\log}\;(x2-y), the value of f(e,0) is given by ___________

5) The value of lim (x,y) →(3, 2) x cos (π y) = _____________


C) State true false for the following questions:

1) Newtons Method uses the tangent line to y=f(x) at x=xn to compute xn+1

2) The differential equation \frac{d^2y}{dx^2}=\frac{dy}{dx\;} has a solution which is constant.

3) and are functions of two variables such that is and f+g  and f g are both continuous themselves continuous.

4) If (x,y)→L as (x, y)→(x0,y0), then f (x,y),→L as(x,y) → (x0,y0) along any smooth curve.

5) A function of two variables is said to have an absolute maximum at a point (x0,y0) if f(x0,y0) ≤ f (x, y) for all points (x, y) in the domain of f.


Q. 2) ANSWER Any three of the following questions. (15M)

a) Find the intervals on which f(x)=x2– 4x+3 is increasing and the interval on which is decreasing.

b) Use first and second derivative tests to show that f(x) =x3– 3x+3 has a relative at x=1 and a relative maximum at x=1

c) Locate the critical points at f(x) = 3x4+12 x .

d) Find the absolute maximum and minimum values or fix)=4x2-12x+10 in [1, 2] a)

e) A firm determines that x units of its product can be sold daily at p Rupees per unit, where x =1000p  The cost of producing units per day 3is C(x) =3000+20x.
1) Find the revenue function R(x)
2) Find the profit function p (x)
3) Assuming that the production capacity is at most 500 units per day, determine how many units the company must produce and sell each day to maximize the profit
4 Find the maximum profit

f) The equation x3-x- 1= 0 has one real solution. Approximate it by  Newtons Method


Q. 3) Answer any THREE of the following questions. (15M)

a) Find the area under the curve \;=3\sqrt xy over the interval(1.4)

b) Find the area of the region enclosed by x= y2 and y=x- 2 integrating with respect to y 15

c) Find the approximate value of \int_1^2\frac1x\;dx using Simpson’s rule with n=14

d) Solve differential equation \frac{dy}{dx}=2\;(1+\;y^2)\;x

e) Use Euler’s Method with a step size of 0.5 to find an approximate solution of the initial-value problem \frac{dy}{dx}=y^\frac13 y(x) = 1 over 0≤ x ≤ 4.

f)  Solve the differential equation \frac{dy}{dx}-y\;=e^x by the method of integrating factors:


Q. 4) Answer any THREE of the following question: (15M)

a) Find \lim_{(x,\;y)\;\rightarrow(0,0)}\frac{xy}{x^2+y^2} along the x-axis and (2) along the parabola y=x2 

b) Determine whether the limit exists. If so, find its value.


c) Find f(2, 1) and fy (1, 2) for the function f(x, y)= 10x2y4 – 6xy2 + 10x2

d)  Find the directional derfratwe of (x,\;y,\;z)\;=\frac{z-x}{z+y} at the point, ( 1, 0, -3 ) in the direction of the vector a= 6i+ 3j – 2k

e) Find parametric  equations of the tangent line to the curve of intersection fo the paraboloid z = x2 +y2 and the ellipsoid 3x2 + 2y2 +z2 =9 at the point (1, 1, 2) 

f) Find all relative extrema and saddle points of f (x, y) = 1- x2– y2


Q. 5) Answer THREE of the following questions: (15M)

a) Let f(x)=x2 +px + q . Find the value of p and q such that f(1) = 3 is an extreme value of f on [0, 2] . is this value a maximum or minimum?

b) Show that y=xe^\frac{-x^2}2 satisfies the equation xy = ( 1- x2) y

c) Find the area of the region below the interval [2, 1] and above the curve y=x3

d) Solve differential equation \frac{dy}{dx}y\;=e^x

e) Evaluate \lim_{(x,y)\rightarrow(0,0)}\frac{x^2y^2}{\sqrt{(x^2+y^2})} by converting to polar coordinates.



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