Find the Maclaurin series for cosx and prove that it represents cosx for all x.Define initial value problem. Solve that initial value problem of y′+2y=3, y(0)=1.Find the volume of a sphere of radius r.[4+4+2]
Applications of Derivatives
1.
Verify Mean value theorem of f(x)=x3−3x+2 for [-1,2].[5]
2.
Find the volume of the resulting solid which is enclosed by the curve y=x and y=x2, is rotated about the x-axis.[5]
3.
Find the extreme values of the function f(x,y)=x2+2y2 on the circle x2+y2=1.[5]
Function of One Variable
1.
A function is defined by
f(x)=∣x∣
Calculate f(-3), f(4), and sketch the graph.Prove that the limit does not exist.
x→2limx−2∣x−2∣
[5+0+5]
2.
Find the domain and sketch the graph of the function
f(x)=x2−6x
Estimate the area between the curve y=x2 and the line y = 1 and y = 2.[3+2]
3.
If f(x)=2−x and g(x)=x, find fof and fog.[5]
4.
Starting with x1=2, find the third approximation x3 to the root of the equation x3−2x−5=0.[5]
Infinite Sequence and Series
1.
If f(x,y)=xylim(x,y)→(0,0)xf(x,y) does not exist, justify.Calculate ∬Rf(x,y)dA, for f(x,y)=100−6x2y, and R:0≤x≤2,−1≤y≤1.[5+5]
2.
Determine whether the series converges or diverges
n=1∑∞5n2+4n2
[5]
Limits and Continuity
1.
Define continuity on an interval. Show that the function is continous on the interval [1,-1].
f(x)=1−1−x2
[5]
Ordinary Differential Equations
1.
Find the solution of y′′+4y′+4=0.[5]
Partial Derivatives and Multiple Integrals
1.
Find ∂x∂z and ∂y∂z if z is defined as a function of x and y by the equation x3+y3+z3+6xyz=1.[5]
Plane and Space Vectors
1.
If a=(4,0,3) and b=(−2,1,5), find ∣a∣, the vector a−b and 2a+b.[5]