**[Time: 21/2 Hours]**

**[ Marks:75]**

**Please check whether you have got the right question paper.**

**N.B: 1. All questions are compulsory.**

** 2. Figures to the right indicate marks.**

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

1) Let f be defined on an interval, and let x1 and x2 be points on the interval, then f is said to be decreasing if

p) f (x_{1}) > f(x2) whenever x_{1} < x_{2}

q) f(x_{1}) > f(x2) whenever x_{1} > x_{2}

r) f(x_{1})= f(x2) whenever x_{1} < x_{2}

s) None of these

2) If a function f is concave up on (a,b) then which of the following is true on (a,b)

p) f’ > 0

q) f’< 0

r) f’ = 0

s) None of these

3) If f is integrable on [a, b] and f(x)≥ 0∀x ∈ [a,b], then

p) \int_a^bf\left(x\right)>0

q) \int_a^bf\left(x\right)≥0

r) \int_a^bf\left(x\right)=0

s) None of these

4) A rule that assigns a unique real number f (x,y) to each point (x,y) in some set D in the xy-plane is called

p) a function of one variable

q) a function of two variable

r) a function of three variables

s) None of these

5) which of the following is true about the function f(x,y)=3x^{2}y^{5}?

p) Discontinuous at (0,0)

q) Discontinuous at (1,1)

r) Continuous everywhere

s) None of these

B) Fill in the blanks for the following questions:

1) A function f has a relative maximum at x0 if there is an open interval containing x0 on which f (x) is —- f (x0) for every x in the interval.

2) The points on the curve y=f(x) where the rate of change of y with respect to x changes from increasing to decreasing, or vice versa is known as——-

3) The integral \int_0^\pi\sqrt{\left(1+\cos\;x\right)}^2 is the arc length of y=———– from x=0 to x=????.

4) If \left(x,y\right)=\;\frac{x-y}{x+y+1}, the value of f (y+1,y) is given by ———-

5) The value of \lim_{(x,y)\rightarrow\left(0,1\right)}e^{xy^2} _______

C) State true or false for the following questions: **(15M)**

1) If f (x)=0 has a root, then Newtons Method starting at x=x_{1} will approximate the root nearest x_{1}.

2) The order of the differential equation \left(\frac{dy}{dx}\right)^2=\;\frac{dy}{dx} is one.

3) If f (x,y)→L as (x,y) approaches (0,0) along the x-axis ,and if f(x,y) →L as (x,y) approaches (0,0) along the \lim_{(x,y)\rightarrow\left(0,0\right)}f(x,y)\;=\;L

4) If a function f is continuous at every point in an open set D, then f is continuous on D.

5) A function f of two variables is said to have a relative maximum at a point (x0,y0) if there is a disk

centered at (x_{0},y_{0}) such that f(x_{0},y_{0})≤ f (x,y) for all points (x,y) that lie inside the disk.

Q. 2) Answer any THREE of the following questions: **(15M)**

a) Find the intervals on which f(x) = x^{2} – 3x+8 is increasing and the intervals on which it is decreasing.

b) Use first and second derivative tests to show that f(x) =3x^{2} -6x + 1 has a relative minimum at x=1.

c) Sketch the graph of the equation y=x^{3} -3x +2 and identify the locations of the intercepts (draw the graph on the answer sheet itself).

d) Find the absolute maximum and minimum values of f(x)= (x-2)^{2} in [1,4].

e) A garden is to be laid out in a rectangular area and protected by a chicken wire fence. What is the largest possible area of the garden if only 100 running feet of chicken wire is available for the fence?

f) The equation x^{3} – 2x – 2 = 0 has one real solution. Approximate it by Newton’s Method.

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

a) Find the area under the curve y = x^{3} over the interval [2,3].

b) Find the area of the region bounded above by y = x+6, bounded below by y = x^{2} and bounded on the sides by the lines x = 0 and x = 2.

c) Find the approximate value of \int_1^2\frac12dx using Simpson’s rule with n=10.

d) Solve the differential equation \frac{dy}{dx}=\frac yx

e) Use Euler’s Method with a step size of 0.25 to find approximate solution of the initial-value problem \frac{dy}{dx}=\;x-\;y^{2,} y(x) =1 over 0≤ x ≤ 1.

f) Solve the differential equation

\frac{dy}{dx}+3y\;=e^{-2x} by the method of integrating factors.

Q. 4) Answer any three of the following questions: **(15M)**

a) If f \left(x,y\right)=\frac{xy}{x^2+y^{2'}} Find the limit of (x,y)→(0,0)

1) along x-axis and

2) along the line y=x.

b) Evaluate \lim_{\left(x,y\right)\rightarrow\left(0,0\right)}\sqrt{x^2+y^2}.\;\log\;(x^2+y^2),by converting to polar coordinates.

c) Find ????_{????} (????, ????)and ????????(x,y) for f(x,y)=2????^{3}????^{2} + 2???? + 4????, and use those partial derivatives to compute ????????(1,3) and ???????? (1,3).

d) Find the directional derivative of f(x,y)=????^{????????} at (2,0) in the direction of unit vector that makes an angle of ^{π}/_{3} with the positive x-axis.

e) Find an equation of the tangent plane to the surface ????^{2} + 4????^{2} + ????^{2} = 18 at the point (1.,2,1). Also find the parametric equation of the line that is normal to the surface at the point (1,2,1).

f) Find all relative extrema and saddle points of f (x,y) = 3????^{2} − 2???????? + ????^{2} − 8????.

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

a) Find the absolute maximum and minimum values of f(x)\;=\frac{x-2}{x+2} on [-1,5]

b) Show that for any constants A and B, the function y= ????????^{2????}+????????^{−4????} satisfies the equation y”+2y-8y=0.

c) Find the area of the region under the curve y=x^{2}+1 and over the interval [0,3].

d) Solve differential equation \frac{dy}{dx}+2xy=x

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

\lim_{\left(x,y\right)\rightarrow\left(0,0\right)}\frac{x^4-16^4}{x^2+4y^2}

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