Practice Test
(Midterm 1)
1. Evaluate the following limits (if exist) and also find the horizontal asymptotes for the last two problems
(if exist).
5x4 − 4x3 + 2
.
x→+∞ 1 − 3x2 − x4
x2 − 1
(d) lim
.
x→−∞
x
(x2 − 2x)(x2 − 4)
x→2
(x − 2)2
2
3x + 1
(b) lim 3
x→−1 x + x2
(c)
(a) lim
lim
2. Find the points of discontinuity of the following functions:
 3
x
x+1
2

if x < −1
 3 − 4x + 1
(b) g(x) = 4
(x
− 4)
(a) f (x) = 2x2 − 1
if − 1 ≤ x ≤ 0

 3
3x + x2 − x − 1 if x > 0
3. If lim f (x) = 1, then find lim [f (x) + x2 − 1].
x→4
x→4
5
, then what is the value of f (c)?
3
5. Find the slope and the instantaneous rate of change of the following functions at any point x and the
specified points:
4
2x − 1
1
(a) y =
).
, at (1, 16
x2 + x
√
3
2x − 1
(b) f (x) =
, at (0, −1).
2x + 1
6. Find the slope of the horizontal tangents of the following curves:
4. Let f be a function which is continuous at x = c, if lim f (x) =
x→c
(a) f (x) = x6 − 6x4 + 8.
p
(b) y = x2 − x + 4.
(c) for a function f (x) for which f 0 (x) =
x(x − 1)
.
x2 + 1
7. Find
(a) y (4) , if y = x6 − 15x3 .
√
d2 y
(b) y (5) , if
= 3 3x + 2.
2
dx
x2
(3)
(c) f (x), if f 0 (x) = 2
.
x +1
√
8. If f (x) = x + x, find the rate of change of f (4) at x = 1.
9. Suppose a particle is moving along a straight line, at time t seconds its distance from the starting point
is s meter. If s is related to t by the following formula:
s = 100 + 10t + 0.01t3
Find it’s velocity and acceleration at time t = 2 seconds.
10. Solve the problems 27 and 28 from Page No. 675 of your text book.
11. Solve the problems 18, 20, 27, 30, 33 − 36 from Page No. 703 of your text book.