Find the Maclaurin series expansion of f(x)=ex at x=0.[5]
Applications of Derivatives
1.
As dry air moves upward, it expands and cools. If the ground temperature is 20∘C and the temperature at height of 1 km is 10∘C, express the temperature T (in ∘C) as a function of the height h (in kilometer), assuming that linear model is appropriate.(a) Draw a graph of the function in part (b). What does the slope represent? (c) What is the temperature at a height of 2.5 km?[5+5]
2.
Find where the function f(x)=3x4−4x3−12x2+5 is increasing and where it is decreasing.[5]
3.
Sketch the graph and find the domain and range of the function f(x)=2x−1.[5]
4.
The area of the parabola y=x2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulting surface.Find the solution of the equation y2dy=x2dx that satisfies the initial condition y(0)=2.[5+5]
Derivatives
1.
Find y' if x3+y3=6xy.[5]
Infinite Sequence and Series
1.
Determine whether the series converges or diverges
n=1∑∞5n2+4n2
[5]
Limits and Continuity
1.
Show that the function f(x)=x2+7−x is continuous at x=4.[5]
Ordinary Differential Equations
1.
Show that y=x−x1 is a solution of the differential equation xy′+y=2x.[5]
Partial Derivatives and Multiple Integrals
1.
If f(x,y)=x3+x2y3−2y2, find fx(2,1) and fy(2,1).[5]
Plane and Space Vectors
1.
If a=(4,0,3) and b=(−2,1,5), find ∣a∣, 3b, a+b and 2a+5b.Estimate the value of