1. Compute the following limits, if they exist. Else, argue why the limit does not exist.
e2x + x3 + lnx
( b) lim
2. The function f(x) is defined on the interval [0,2] and is between 4 − x and x2 + 2 for all x in this interval. Does it have to be continuous at x = 1? Explain why or why not. (5)
3. Show that the equation x7−3x−1 = 0 has at least one solution in the interval [−1,1].
(5)
4. (a) Show that
d 1 arctanx = .
dx 1 + x2
(b) Consider the function
f(x) = 2 arctanx − x.
Find its domain, horizontal and vertical asymptotes, local minima, local maxima, and inflection points of f. Identify the regions where the graph of f is concave upward or concave downward. Finally, sketch the graph of the function.
(5+10)
5. An airplane is flying towards a radar station at a constant height of 6km above theground. The distance s between the airplane and the radar station is decreasing at a rate of 400km/h when s = 10km. What is the horizontal speed of the plane? (10)
1
6. Compute the following definite or indefinite integrals.
(10+10+5) 7. Find the derivative of the function
(5)
