Question 1:
(1). Compute
limx→0 (3x2 − 5)(x − 3)
(2). Compute
limx→−2
x2 + 7x + 10
x+2
(3). Compute
limx→0
sin 2x
5x
(4). Compute
x2
limx→∞
(x − 5)(3 − x)
1
2
Question 2:
The following function is not defined at a certain point. How should it be defined
in order to make it continuous at that point?
(a)
f (x) =
sin x
3x
(b)
f (x) =
x4 + 2x2 − 3
x+1
3
Question 3:
Find the derivative of the following functions:
(a).
2x2 − 1
f (x) =
x+7
(b).
f (x) =
sin x
tan x
(c)
f (x) = (x2 + 1)(1 − 3x − 4x3 )
4
Question 4:
Find the equation of the tangent line to the function y = (x2 + 1)−1 at the point
(1, 21 ).
5
Question 5:
Aircraft A is moving north with constant speed 10/second. After 4 seconds,
Aircraft B is sent to catch Aircraft A. The distance function of Aircraft B is 3t2
since then. Will Aircraft B catch Aircraft A within 6 seconds? Why?
(Hint: let f (t) be the distance function of A after B is sent and let g(t) be the
distance function of B. B catches A when g(t) = f (t).)