Find the Maclaurin series for ex and prove that it represents ex for all x.Define initial value problem. Solve that initial value problem of y′+5y=1, y(0)=2.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+3 for [-1,2].[5]
2.
Sketch the curve y=x3+x.[5]
3.
Find the length f the arc of the semicubical y2=x2 between the points (1,1) and (4,8).[5]
4.
Find the extreme values of f(x,y)=y2−x2.[5]
Derivatives
1.
Find the derivative of
f(x)=x
State the domain of f.Estimate the area between the curve and the line x=0 and x=2 where curve is
y2=x
[3+2+5]
Function of One Variable
1.
A function is defined by
f(x)={x+2,1−x,x<0x>0
Calculate f(-1), f(3), and sketch the graph.Prove that the limit does not exist.
x→0limx∣x∣
[5+0+5]
2.
If f(x)=x and g(x)=3−x, find gof and fog.[5]
3.
Define limit of a function.
x→∞lim(x−x)
[5]
Infinite Sequence and Series
1.
For what values of x does the series converge?
n=1∑∞x(x−3)n
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 integral is convergent or divergent.
∫1∞x1dx
[5]
3.
Test the convergence of the series
n=1∑∞n!nn
[5]
Limits and Continuity
1.
Use continuity to evaluate the limit,
x→4lim5+x5+x
[5]
Ordinary Differential Equations
1.
Find the solution of y′′+6y′+9=0, y(0)=2, y′(0)=1.[5]
Plane and Space Vectors
1.
Define cross product of two vectors. If a=i^+3j^+4k^ and b=2i^+7j^−5k^, find the vector a×b and b×a.[5]