BSc.-Calculus-Mumbai-November 2017

MUMBAI UNIVERSITY

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.
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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=x_n to compute x_n+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_x  (2, 1) and f_y (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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