BSc Mathematics SEM II 2016 17 2016-17 Calculus Question Paper - Mumbai University | munotes

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1. Q1 following questions. 15 marks
2. Q1 Let f be defined on an interval, and let x1 and x2 be points on the interval, then f is said to be 5 marks
   • p) f (x1) > f(x2) whenever xi < X2
   • q) f(x1) > f(x2) whenever xi > X2
   • r) f(x2) whenever < X2
   • s) None of these
3. Q2 If a function f is concave up on (a,b) then which of the following is true on (a,b)
   • s) None of these
4. Q3 If f is integrable on [a, b] and [a,b], then
   • s) these
5. Q4 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 function of one variable
   • q) a function of two variable
   • r) a function of three variables
   • s) None of these
6. Q5 which of the following is true about the function
   • p) Discontinuous at (0,0)
   • q) Discontinuous at (1,1)
   • s) None of these Fill in the blanks for the following questions: 05
7. Q1 A function f has a relative maximum at there is an open interval containing Xo on which f (x) is f (xo) for every.x in the interval
8. Q2 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
9. Q3 The integral J cosx) dx is the arc length of y=----------- from x=0 to f (x,y) the value of f (y+1,y) is given by
10. Q5 The value of
    • Q.P. Code :05600
    • c) State true or false for the following questions: If f (x)=0 has a root, then Newtons Method starting at will approximate the root nearest x1
11. Q2 The order of the differential equation (2) = is one
12. Q3 If as (x,y) approaches (0,0) along the x-axis ,and if f(x,y) (x,y) approaches (0,0) along the y-axis then f (x,y)=L
13. Q4 function f is continuous at every point in an open then f is continuous on D
14. Q5 A function f of two variables is said to have a relative maximum at a point (Xo,Yo) if there is.a disk centered at (Xo,Yo) such that f (x,y) for all points (x,y) that lie inside the disk
15. Q2 any THREE of the following questions: 15 marks
    • a) Find the intervals on which f(x) = x? — 3x+8 is increasing and the intervals on which it is decreasing
    • b) Use first and second derivative tests to show that f(x) relative minimum at x=1
    • c) Sketch the graph of the equation y=x? -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)? in [1,4]
    • e) 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? - 2x 2 = 0 has one real solution. Approximate it by Newtons Method any THREE of the following questions: 15
    • a) Find the area under the curve y = x? over the interval [2,3]
    • b) Find the area of the region bounded above by y = x+6, bounded below by y = x? and bounded on the sides by the = 0 and
    • c) Find the approximate value of dx using Simpson’s rule with n=10
    • e) Use Euler’s Method with size of 0.25 to find approximate solution of the initial-value problem
    • f) Solve the differential equation + 3y = by the method of integrating factors
16. Q4 THEREE of the following questions: 15 marks
    • a) If find the limit of f(x,y) as 1) along x-axis and 2) along the line y=x
    • b) Evaluate x2 + y2.log(x? + y2),by converting to polar coordinates Find for + 2y + 4x, and use those partial derivatives to compute
    • d) Find the directional derivative of f(x,y)=e*” at (2,0) in the direction of unit vector that makes an angle with the positive x-axis
    • e) Find an equation of the tangent plane to the surface x” + 4y* + = 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 — 2xy + — By
    • Q.P. Code :05600
17. Q5 Answer any THREE of the following questions: 15 marks
    • a) Find the absolute maximum and minimum values of f (x) on [-1,5]
    • b) Show that for any constants A and B, the function y= Ae2* + satisfies the equation y’+2y-8y=0
    • c) Find the area of the region under the curve and over the interval [0,3]
    • e) Determine whether the following limit exists. If so, find its value
    • P. Code :

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