Practice Test
(Midterm 1)
1. Evaluate the following limits (if exist) and also find the horizontal asymptotes for the last two problems
(if exist).
2x)(x2
x!2
(x 2)2
2
3x + 1
(b) lim 3
x! 1 x + x2
(a) lim
(x2
5x4 4x3 + 2
.
x!+1 1
3x2 x4
x2 1
(d) lim
.
x! 1
x
4)
(c)
lim
2. Find the points of discontinuity of the following functions:
8 3
x
x+1
>
4x2 + 1
if x < 1
<3
(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
x!4
1].
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
p
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 .
p
d2 y
(b) y (5) , if
= 3 3x + 2.
2
dx
x2
(3)
(c) f (x), if f 0 (x) = 2
.
x +1
p
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.
Practice Test
(Midterm 2)
1. Sketch the graph of the following functions.
1
(a) y = x +
x
x
(b) y = 2
x
1
x2
(c) y = 2
x +1
(d) y = x3
2. Find the derivative of the following functions.
◆1
t2 + 3 4
⇥
1 t
p
(b) y = ln(x3 x2 + 1)
p
(c) y = ln(3x + 1)
(a) y = ln
✓
(d) y = log2 (1 x
1
(e) y =
ln x
1 + ln x
(f) y =
x2
x2
(g) y = x
x2 )
(h) y = e3 + eln x
(i) y =
e2x
2
+
2x
e
2
2
(j) y = e2x ln(4x)
(k) y =
ex e
ex + e
(l) y = 4(x
2
x
x
+1)
(m) y = (e3x + 4)10
(n) y = 4(ex )3
4ex
2
dy
using the method of Implicit Di↵erentiation.
dx
(a) x4 + 2x3 y 2 = x y 3
p
(b) 2x + 2y = x2 + y 2
3. Find
(c) x + 2xy = 2. Also find the equation of tangent line at x = 1.
(d) x ln y + 2xy = 2. Also find the slope of the tangent at (1, 1).
(e) x2 y = ex+y
(f) yex = y 2 + x
2
2
(g) x + 4y
tangents.
4x
2. Also find the slope of tangents at x = 0.
4 = 0. Also find the points where the curve has horizontal tangents and vertical
4. Problem No. 8, 10, 11, 13, 16, on Page No. 795
5. Evaluate the following indefinite integrals.
Z
(a)
x(x 1)2 dx
Z
2
(b)
xe(1+x ) dx
Z
7x3
⇥
(c)
dx
1 x4
796 of the text book. Problem No. 8 on Page No. 805.
(d)
(e)
(f)
Z
3x
x2
Z 
Z
x
x(x2
1
dx
1
(x + 1)2
1)10 dx
dx
x3
3x + 1
dx
x 1
(k)
(5x3 + 1)2 · 45x2
dx
(5x3 + 1)3
(h)
(x3
e3x ) dx
(l)
e(4x
(i)
y 2 ey dy
(j)
p
3
(g)
3
(m)
x2
dx
x3 4
(n)
2
3)2
x5
e(2 3x6 )
· 2(4x2
3) · 8x dx
dx
1600e0.4x dx
6. Suppose the rate of growth of the population of a city is predicted to be
dp
= 200t1.04
dt
where p is the population and t is the number of years past 2005. If the population in the year 2005 is
50, 000, what is the predicted population in the year 2015?
7. Suppose that the marginal cost for x units of a product is M C = 4x + 50, the marginal revenue is
M R = 500, and the cost of the production and sale of 10 units is $1000. What is the profit function for
this product?
Page 2
Practice test 3
1. Evaluate the following definite integral.
! 3
3
dy
(a)
2
1 y
! 4
√
(b)
4x + 9 dx
(c)
(d)
(e)
(f)
!
!
!
!
(g)
(h)
0
2
√
3
0
2x3 − 8 x2 dx
(i)
2
−1
(x3 − 3x2 )3 (x2 − 2x) dx
(j)
2
4
3
(x − 5x + 2x) dx
0
(k)
0
(x + 2)13 dx
(l)
−1
!
!
!
!
!
!
3
(2x − x2 )4 (1 − x) dx
0
2
1
5
dz
z3
2
e4x dx
0
e
3y −1 dy
1
2
3
8x2 e−x dx
0
1
0
3x3
4x4 + 9
2. Find the following area.
(a) The curve y = x3 + 3x + 2 bounded by the x-axis and the vertical lines x = 1, x = 2.
2
(b) The curve y = xex bounded by the x-axis and the vertical lines x = 1, x = 3.
(c) The curve y = e−x bounded by the x-axis and the vertical lines x = −1, x = 1.
3. Problems from 43 − 48 in your textbook, Page 888, Section 13.2.
4. Problem No. 13, 15, 19, 21, 23, 24, 25, 30, 32, Page 899, Section 13.3.
5. Problem No. 1, 5, 6, 10 − 12, 17 − 20, 21 − 22, 23 − 26, Page 930, Section 13.7.
6. Problem No. 4 − 7, 19, 20, 22, Page 956, Section 14.1.
7. Problem No. 2, 5, 7, 8, 13 − 15, 27, 29, 31, 33, 36, 39, 41, 42, 44, Page 966, Section 14.2.