B.Sc. (Computer Science) Sem 2 Calculus Oct 2023 Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Solve the following.(Any 4) [Marks : 20] =x74+5, x >2,at x =2 , then find f is differentiable or not? 2.Find the asymptote of the function f(x) = 3.If f(x) = 69 — 24 x —9 2 x3, find the intervals on which the function f(x) is increasing and 4 Divide the number 100 into two parts such that their product is maximum 5.For what values of x the curve y = 3 x*- 2 x? is concave downwards and when it is concave 6.Determine the absolute extrema for the function f(x) = 4 x in the interval [0 , 2 ]
2. Q2 Solve the following.(Any 4) [Marks : 20 ] 2.Find the area bounded by y = x? ,x 4 Estimate x? dx using Simpson’s rule and n = 4 6.Find the length of the curve in one period of the cycloid x =t— sint , y= 1 t. The values of t Run from 0 to
3. Q3 Solve the following.(Any 4) [Marks : 1 Find the linearization of function f(x,y ) = x3+ xy + y? at the point (1,2) 2.Find the directional derivative of f(x,y) =3x+4yatu=(2,3) along
4. Q3 If f(x,y + find Vf at the point the maximum and minimum rates of change in f at u if and u =( 1,2,3) Also state their direction 5.Prove that the function f(x,y)= 2 x*+ 3 - has neither a local maximum nor a local 6.Find the equation of the tangent plane and normal line for function 45 at (1,2,-3)
5. Q4 Solve the following.(Any 5) [Marks : 1.Find the derivative of f(x) = 2.Show that the function f(x) = x3+ 30 x +7 is always increasing implicit differentiation of function att =(2,3) 6.Find for the function f( x, y )=y*+ - x?y? and Check whether = or not 7 Find the derivative of f(x,y) = y?,x = = 4t+ 1 using Chain Rule Date: 07-10-2023 Marks: 75 Time: 02.00 pm to 04.30 pm 1.Solve the following.(Any 4) [Marks : 20]
6. Q1 If f(x) — 24 x —9 2 x3, find the intervals on which the function f(x) is increasing and
7. Q2 For what values of x the curve y x*- 2 x3 is concave downwards and when it is concave 3.Find the asymptotes of the function y = 4.Find the intervals on which function f is increasing or decreasing, where f(x) = 27x +5 5.Divide the number 25 into two parts such that the square of one part is multiplied with the cube of the other part is maximum 6.Determine the absolute extrema for the function f(x) = sin x —x cos x ]
8. Q2 Solve Evaluate the following.(Any 4 ) [Marks : 20]
9. Q2 Evaluate dx 3.Find area between the graph of cos x and sin x on [0 , 77/4 4 Find the arc length of the curve y = [0,4 ] 5.Solve by using substitution y = v x 6.Evaluate dx for n = 6 by Simpson’s rule
10. Q3 Solve the following.(Any 4) [Marks : 20]
11. Q1 Find the directional derivative of f(x,y) =3 y atu along
12. Q2 Find the maximum and minimum rates of change in f at u if f( and u = ( 1,2,3) Also state their direction 3.Find the linearization of function f(x )= +x y+ y? at the point (1,2) 4.Find the equation of tangent and normal for the function x? + x*y —2 x y?+ 25 =0 at the
13. Q5 The function f (x ,y )=x? Find all local maximum and local minimum
14. Q6 For the function f (x , y ) = e*” x sin y
15. Q4 Solve the following.(Any 5 ) 15] 1.Show that, the function f(x) = 10 + 12x +3 x3 is always decreasing 2.Divide the number 100 into two parts such that their product is maximum 5.For the function y ) = e*+ cos 2y.Hence find V f ( 0,5) 6.For the function f(x x*+ 7 5 Check whether f = fyx or not 7.Find implicit differentiation of the function f(x , y )= y e* +x sin y—2 at (0 ,2)

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