BSc Data Science SEM II 2021 2022 May 2022 CALCULUS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Attempt the following 40 marks
   • 1) Given f(x) = x? — 12x? + 45x + 8 at which point does f (x)has its minimum value
   • 2) Which is correct formula for finding the area enclosed the curves f(x) and y2 = g(x) is given by
   • 3) Function f should on [a,b] according to Rolle’s theorem
   • 4) Which formula is applicable for revolution about x-axis
   • 5) The equation of the normal to the curve y =sinx at (0 is
   • 6) ) How would you split the rational fraction f(x) into partial fraction f(x) =
   • 7) The constant of integration c does not appear in the fina! value of integral hence it is called as
   • a) Definite integral Indefinite integral May be indefinite integral d) None of these
   • 8) ) is greater one then it is
   • a) Hyperbola b) Parabola c) Ellipse d) None of these find eccentricity with the help of given equation r 10) In this equation dN/dT is the growth rate of the population in a given instant where N is
   • a) Population Size Population rate c) per Capita d) Number of terms
   • 11) = then
   • 12) Local minimum value of f if f(c) < f(x) where xis near c
   • 13) Find the equation of tangent to the circlex? + y? = 25 at (3 4)
   • a) dy/dx =-x/y b) 3x+4y =25 d) 4x+3y =25
   • 14) Find the velocity at time t where t=2 if f(t) = — 6t? +
   • 15) If f'(x) is more than zero on an interval then f is -------- on that interval
   • 16) The line y=x+1 is a tangent to the curve y* = 4x at the point
   • 17) Find the average value of f(x) = x? [1,2]
   • 18) ) Find the directrix of the parabola y? = 10x
   • 19) Evaluate — 2x + 1)dx
   • 20) Evaluate the Foci on X-axis =1
   • A) following (Solve any one ) 4
   • i) The position of the partical is given by the equation s =f(t) — 6t? + 9t where t is measured in seconds and s in meters i) Find velocity at time t ii) what is velocity after 2s and 4s iii) When partical is moving forward iv) When partical is at rest ‘
   • ii) If f(x) = xe*find f(x) find nth derivative f"(x) and also differentiate y = x?sinx
   • B) Attempt the following (Solve any one ) 3
   • i) =25 find an equation of tangent to the circle x* + y* = 25
   • ii) Find the point on the curve y = x* — 6x* + 4 where tangent line is horizantal
2. Q3 A) Attempt the following (Solve any one ) 4 marks
   • i) Evaluate logx dx by using integral by parts
   • ii) Evaluate x sinx*dx by using substitution method
   • B) Attempt the following (Solve any one ) 3
   • i) Evaluate dx
   • ii) Finding Area of region between two curves if f(x) =x+4 and =3 -x/2 belong to [1,4]
3. Q4 ) A) Attempt the following (Solve any one ) 4 marks
   • i) Use the mean value theorem to solve Suppose f(0)=-3 (x) =x5 Vx How large can f(2) possibly be
   • ii) The marginal cost manufacturing x shoes is 6 the cost producing a pair of shoes is Rs 12 Find total and average cost function
   • B) Attempt the following (Solve any one ) 3
   • i) Evaluate average value of continuous function if f(x) = x? — 5x 6 cos(mx) :
   • ii) Calculate consumer surplus if demand function p =50-2x and x= 20
4. Q5 Attempt the following (Solve any one ) 4 marks
   • i) Define Continous function of two variable at (a,b) and Show that ie 2) does not exit also find extreme of f(x ,y ) = — x?
   • ii) Define linear approximation of (a,b) Show that f(x ,y) = xe* is dif ferentiable at (1,0 ) and find its linearization.Use it to approximate f(1.1 ,-0.1)
   • B) Attempt the following (Solve any one ) 3
   • ii) Arectangular box without a lid is to be made from 12m? of card board find maximum volume of such box
5. Q6 A) Attempt the following (Solve any one ) 4 marks
   • i) Eliminate the parameter to find a cartesian equation of the curve
   • B) Attempt the following (Solve any one ) 3
   • i) Find the polar equation of the conic with focus at the origin and given directrix X=4

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