Find the Maclaurin series expansion of f(x)=sinx for all x.[5]
Applications of Derivatives
1.
Show that every member of the family of function y=1−cet1+cet is a solution of the differential equation y′=21(y2−1).[5]
2.
Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under the curve y=x for 0 to 1.[5]
Derivatives
1.
Find the unit normal and binormal vectors for the circular helix r(t)=costi^+sintj^+tk^.[5]
2.
The position vector of an object moving in a plane is given by r(t)=t2i^+t2j^. Find its velocity, speed, and acceleration when t=1 and illustrate geometrically.[5]
Function of One Variable
1.
Sketch the graph of f(x)=x2. Find its domain and range.Evaluate
x→1−limsin−1(1−x1−x)
[5+5]
2.
Where the function f(x)=∣x∣ is differentiable? Discuss.A farmer has 1200 m. of fencing and wants to fence off a rectangular field that borders a straight river. He needs to fence along the river. What are the dimensions of the field that has the largest area?[5+5]
Infinite Sequence and Series
1.
Determine whether the sequence an=(−1)n is convergent or divergent.[5]
Ordinary Differential Equations
1.
Find the solution of the initial value problem x2y′+xy=1, y(1)=2, x>0.Find the area enclosed by the line y=x−1 and the parabola y2=2x+6.[5+5]
Partial Derivatives and Multiple Integrals
1.
If f(x,y)=x2+y2xy, does lim(x,y)→(0,0)f(x,y) exist? Justify.[5]
2.
If f(x,y)=2x3+x2y2−y4, find fx(1,−2), fy(1,−1) and fyx(1,−1).[5]