Define arc-length of the curve.Find the length of the curve y=(2x)32 from x=0 to x=2.[1+4]
Application of Derivative
1.
Define the concavity of the function.The graph of the function is then f(x)=x4−4x3+10. Find the intervals on which f is increasing and on which f is decreasing.Find where the graph of f is concave up and where it is concave down.Find the local maximum or local minimum value of function if exist.[2+3+3+2]
Differentiation
1.
Find dxdy of the following. y2−x2=cos(xy),x2=y29.[5]
First Order Differential Equations
1.
What is a first order linear differential equation?Solve the initial value problem: tdtdy+2y=t3, t>0, y(2)=1.[1+4]
Functions and their graphs
1.
If a function is defined by
f(x)={1+xx2if x≤−1if x>−1
Evaluate f(−3), f(−1), and f(0) and sketch the graph.Define different types of discontinuity at a point. At what points the function becomes continuous of the function f(x)=x2−7x+10x−2[5+5]
2.
Find the domain and range of the function f(x)=5x+10.Draw the graph of the function y=x2 shifted up by 1 unit, down by 2 units, also shift 3 units to left, and 2 units right with new position of function.[2+3]
Infinite Sequence and Series
1.
Determine whether the following series are convergence or divergence ∑n=1∞5n−15, ∑n=0∞n!1.[5]
Integration
1.
Find the area of the region between the x-axis and the graph of f(x)=x3−x2−2x, 1≤x≤2.[5]
2.
Evaluate the following integral. ∫04π1+cosxdx,∫3x3x2−7x+1,dx[5]
Limits and continuity
1.
Define horizontal and vertical asymptotes.Find the appropriate asymptotes to the function: f(x)=x−x2+16.[2+3]
Partial Derivatives
1.
Define Gradient vector and directional derivative.Find the direction in which f(x,y)=2x2+2y2 increases and decreases most rapidly at the point (1,1).What is the direction of zero change in f at (1,1)?Derivative of f(x,y) at the point (1,1) in the direction v=3i−4j.[5+2+3]
2.
What is chain rule for function w=f(x,y)?To use this rule, find the derivative of w=xy w.r.t. t along the path x=cost, y=sint. Also, find derivative of w at t=2π.[1+4]