Math 1210-007
Fall 2013
Practice Midterm 1
Name :
UID :
Question
1
2
3
4
5
6
Total
Points
Out of
10
10
10
10
10
10
60
1. Compute the following limits
(a)
x−4
lim √
x→4
x−2
(b)
lim
x→5
x2
sin x
− 7x + 10
(c)
lim
x→0
tan 4x
x
2. (a) Give definition for continuity of a function f (x) at a point a.
(b) Give definition for differentiability of a function f (x) at a point a.
(c) Show that the function
f (x) =
1
if x = 0
x sin x1 if x 6= 0
is continuous but not differentiable at 0.
3. (a) Show that
x5 + x3 − 1 = 0
has a real solution between 0 and 1.
(b) Find the vertical and horizontal asymptotes for the function
√
.
x
x2 − 4
4. Find the derivatives for the following functions (no need to simplify your answer):
(a)
f (x) = sin(t2 + 1)
(b)
g(x) =
5x2 + x − 1
x−3
5. Take first, second and third derivative of the following function
(a)
h(x) = 2x3 − 5x + 10
(b)
ξ(x) = cos 5x
6. Find the equation of the tangent line to the graph
y 2 = x3 − x
√
at (2, 6).