Midlands State University
Department of Mathematics
HMT101 Calculus I
Lecturer: W D Govere
Dept of Applied Education
AEM102 Calculus
Module Outline
a)
b)
c)
d)
e)
f)
g)
h)
i)
Number Systems: natural, integral, rational and irrational numbers
Mathematical induction
The real number system: decimal and geometric representation
Inequalities and their solutions
Sequences
Functions: exponential, logarithmic, trigonometric and hyperbolic and
their inverses
Limits and continuity
Differentiation: derivatives of a single variable
Integration: definite and indefinite integrals, techniques of integration:
method of substitution, integration by parts and reduction formulae,
fundamental theorem of calculus
Recommended reading
L Hostetler- Calculus
E L Swokowski- Calculus
Spiegel- Advanced Calculus
L M Mudehwe- Calculus 1 (ZOU Module)
Any other text on Calculus
1
Tutorial Worksheet 1 Number Systems / Induction / Inequalities
1. Prove that (i)3, (ii) 5, cannot be rational.
2. Show that 2+3 cannot be rational.
3. Show that 7-3 is irrational.
4. Assuming that 5 is irrational show that (2+5) is irrational.
m
r
5. Assuming that 2 is irrational, show that if
and are rational numbers with
n
s
r
m
r
 0 then
 2 is irrational.
s
n
s
6. Prove that between any two rational numbers there is a rational number.
p
7. Express 0.mnmnmn... where m and n are distinct integers, in the form
where p
q
and q are integers.
8. Does the decimal 0.123456789101112131415161718… whose digits are natural
numbers strung end-to-end represent a rational or irrational number. Give a reason for
your answer.
9 Write down the following terminating decimals as recurring decimals: 0.23, 0.325,
0.108
10 Express the following recurring decimals as terminating decimals:
.
.
.
0.74 9,  8.003 9, 316.319
Prove the following by PMI:
n
n(n  1)
n(n  1)( 2n  1)
11. (a)  r 
(b)  r 2 
6
2
r 1
r 1
12. (1  x) n  1  nx , n  2,3,... x  1, x  0
n
 n(n  1) 
(c)  r  

 2 
r 1
n
2
3
13. n  2 n  n  N . Hint: Note that 2k = k + k.
14. 4 n  3n 2  1  n  2
15. n! n 2  n  4
16 a  ar  ar 2  ...  ar n 1 
a(r n  1)
, r 1
r 1
d 2n
17 Deduce a formula for
(sin x) in terms of sin x and prove your formula for all
dx 2 n
n using PMI.
n
n(n  1)( n  2)
18 Using the results in 11(a) & (b) show that  r (r  1) 
and prove
3
r 1
this result by PMI.
1
1
1
19. Deduce a formula for the sum
and prove it by PMI.

 ... 
1 2 2  3
n(n  1)
Solve the following inequalities:
20. x 2  3x  2  10  2 x
21. | x  1 |  2 x  1
2
22. | 3x  5 |  9
23. | x  1 |  3 | x  2 |
| x | 2
4
24.
x2
25 Solve the inequality
5x  9
 2 x  3 and indicate your solution on the real number
x 1
line.
26 Show that x, y, z  R, x 2  y 2  z 2  xy  yz  xz
3
Tutorial Worksheet 2 Sequences
1.Write down the 1st five of each of the following sequences
1
 1  n 


 n 1
 (1) n 1 
 cos n 
n
(a) 
(e) 1    (f) (1  n) n 
 (b) n (c) 
 (d) 
2 
 n  
 (n  1) 
 n! 
 n 1 


(g) S1  1 , S n1  3S n  1
2. Write down the 6th and 7th terms of each of the following sequences and give a
formula for the nth term.
 
1 3 5
7 9
,  , ,  , , ...
5 8 11 14 17
(b) 1,0,1,0,1,…
1
1
1
(c) , 0, , 0, , ...
2
3
4
(d) 5,5,5,5,5,…
2 3
4
(e) , 0, , 0, , ...
3 4
5
3. Consider the following sequences {S n } , where
(a)
1
2n  3
2n
2n 2  1
(b) S n 
(c) S n 
(d) S n 
2
4n
3n  2
n 1
n
For each of the above sequences,
(i)
find the number l to which the sequence converges
(ii)
Generally, if  > 0, provide an expression for N in terms of  such that
| S n  l |   whenever n  N
(iii)
find the range of values for the index n such that | S n  l |  0.01
(a) S n  3 
4. State whether each of the following sequences {S n } tends to a limit, and if so, what
the limit is. Use the definition in terms of  and N to prove your answers correct.
(a) S n  n
(f) S n 
1
(b) S n  2
n 1
2
2n  1
 1
(d) S n    (e) S n 
2n
 2 
n
(c) S n  (1)
n
n
n 1
5. For {S n } given by the following formulae, establish either the convergence or
divergence of {S n } .
 n(1) n 
 n2 
 2n 2  3 
 nn 
1  (1) n 
(a) 
(e) 
 (b) 
 (c)  2
 (d) 

n 1 
n
 n 1 
 n 1
 n 1 
 (n  1) 


(f) ln n  ln( n  1)
6. Find the limit of each of the following sequences
 en 
 cos 2 n 
 5n 
 cos n 
(a)  2 n 
(b)  4 
(c) e  n ln n (d)  n  (e) 

e 
 n 
 3 
n 
2

 3n  5n  4 

(f) 

2
n

7




4
7 Draw a graph for each of the following sequences and describe whether the
sequence is monotone, convergent or divergent, etc.
1 1 1 1 1
(a) ,  , ,  , , ...
3 5 7 9 11
1
(b) {S n } defined recursively by S1  2 , S n 1  2 
Sn
n
(1) n
(d) S n  1 
n 1
n
8 Evaluate
(c) S n 
 n 3
(i) lim 

n  n  5


2 n 5
 n 1 
(iv) lim 

n  n  2


 2n 2   

(ii) lim  2
n  2n  5 


3n  2
 2n  4 
(v) lim 

n  2 n  3


3n 2 5
 n8
(iii) lim 

n  n  2


2n
n 2  n 1
n
1
 1
9. (a) Show that lim 1   
n 
e
 n
(b) Investigate the behaviour (by sketching a graph) of the following sequence
 n
 n  1 when n is odd
sn  
1  1 when n is even
 n
10. Find the limits of the following sequences
2
n
n
 (1) n 
 n  1
1  
1  

 1  

(a)  2    (b) 1    (c) 
(d)
(e)
1








2
n  

 n  
 n  
n n 
n2
11. (a) Suppose {s n } is a sequence converging to 0, and {t n } another sequence such
that 0  t n  s n n . Show that {t n } converges to 0.
1
(b) Investigate the behaviour of the sequence {n 2 n } , i.e find out whether it is
monotone increasing / decreasing, bounded below / above, convergent / divergent. [A
graph may be useful].

12. Show that lim 2 n  3n
n 

1
n
3
5
Tutorial Worksheet 3 Functions, Limits & Continuity
Q1 Sketch the graphs of the functions (i) tan x (ii) cot x (iii) sec x (iv) csc x .
Q2 Sketch the graphs of the functions (i) cos 1 x (ii) tan 1 x (iii) sec 1 x (iv) csc 1 x .
Q3 Sketch the graphs of the functions (i) tanh x (ii) coth x (iii) sec hx (iv) cos echx .
Q4 Sketch the graphs of the functions (i) tanh 1 x (ii) coth 1 x (iii) sec h 1 x (iv)
cos ech 1 x .
Q5 Verify each of the following identities
1  cosh 2 x
(a) cosh 2 x  cosh 2 x  sinh 2 x (b) cosh 2 x 
(c) tanh 2 x  sec h 2 x  1
2
Q6 Show that


(a) cosh 1 x  ln x  x 2  1 , x  [1, )
1 1 x 
(b) tanh 1 x  ln 
 , 1  x  1
2 1 x 
1 1 x2 
 , 0  x 1
(c) sec h 1 x  ln 


x


Q7 Find the domain of each of the following functions of real numbers.
(3x  5)( x  4)
(a) f ( x) 
(b) f ( x)  4  x ln( x  3) (c) f ( x)  sin 5 x
x 3  16 x
Q8 Find the range of each of the following functions
1
(a) f ( x)  x 2  9 (b) f ( x)  x  | x | (c) f ( x) 
(d) f ( x)  ln( x  3)
( x  3) 2
Q9 Use the definition of a limit to prove that
(a) lim (2 x  3)  5 (b) lim (5 x  7)  2 (c) lim (3 x  7)  10
x 1
x  1
x  1
(d) lim (3x  7 x  1)  49
2
x3
(e) lim ( x  6)  5 (f) lim (2 x 2  5x  6)  4
3
x1
x2
 2 , x 1
Q10 Let f ( x)  
2
1  x , x  1
(a) Draw the graph of f
(b) Determine lim f ( x) and lim f ( x)
x 1
x 1
(c) Does lim f ( x ) exist? Give a reason for your answer.
x1
Q11 Suppose that lim f ( x)  l1 and lim g ( x)  l 2 , then prove that
x  x0
(a) lim ( f ( x)  g ( x))  l1  l 2
x x0
x  x0
(b) lim cf ( x)  cl1
x x0
Q12 Determine the following limits
2 x  2 x
x3  1
(a) lim
(b) lim
x 0
x 1  x 2  x  2
x
x 1
3x  1
(d) lim
(e) lim
2
x  | 2  x |
x 1
x  24 x  5
3
5x  2
x 1
(g) lim 2
(h) lim
x  8 x  3
x 0 x  1
x4 1
(c) lim
x  1 x  1
|x|
(f) lim
x   | x | 1
(i) lim
x 
x
x 1
2
6
(j) lim
x 

(m) lim

x 1  x

3x 4  8 x 3  16
x 2 x 3  3 x 2  4
(k) lim
1  cos 2 x
(n) lim

x 
x
2
1  cos x
(p) lim
x 0
x2
x

x4 x

(l) lim
x 
x
(o) lim
x 0
2
x 2  3x  2  x
sin 3 x
x
2
(q) lim (1  cot x)
x
Q13 Find lim f ( x) if
x 
tan x
(r) lim (cos x) x

2
x 0
2
2x 2
2x 2  5
.

f
(
x
)

x2 1
x2
Q14 Use the squeeze theorem to determine lim
x  
sin x
.
x
Q15 Evaluate each of the following limits
 x8
(a) lim 

x  x  2


(e) lim (1  3x)
x 0
2x
2
x
 2x  1 
(b) lim 

x  2 x  3


(f) lim (1  2 x)
x
3
x
x 0
 x 1 
(c) lim 

x  x  2


3x2
 x  2
(d) lim 

x  x  1


3x2
 2x 2   
ln( 1  x)

(g) lim
(h) lim  2
x  2 x  5 
x 0
x


3 x 2 5
4  x 2 , x  1
Q16 Let f ( x)   2
3 x
, x 1
(a)Draw the graph of f
(b)Determine lim f ( x) and lim f ( x)
x 1
x 1
(c) What value, if any, must be assigned to f (1) to make f continuous at x =1?
Give a reason for your answer.
 x 2  2 x  15
, x3

Q17 Let g ( x)  
x3

k
, x3

What value, if any, must k be so that g is continuous at x = 3? Give a reason for
your answer.
Q18 The function f : R  R is defined by
1

 x sin , x  0
f ( x)  
x
 0 , x  0
Prove that f is continuous at x = 0.
7
Tutorial Worksheet 4 Differentiation, Integration
Q1 Show that
 2 1
 x sin   , x  0
(a) f ( x)  
is differentiable at x = 0.
 x
 0
, x0


1
 x sin   , x  0
(b) f ( x)  
is not differentiable at x = 0.
 x
 0
, x0

Q2 (a) Use the definitions of cosh x and sinh x to show that
d
d
cosh x  sinh x and
sinh x  cosh x
(i)
(ii)
dx
dx
(b) Use the results in (a) to show that
d
d
tanh x  sec h 2 x
coth x   cos ech 2 x
(i)
(ii)
dx
dx
d
d
sec hx   sec hx tanh x (iv)
cos echx   cos echx coth x
(iii)
dx
dx
Q3 Compute the derivative of each of the following functions
1
( x  1) 2
(a) x 3 sin x
(b)
(c)
(d) sin 3 (6 x)
2
5
2
(3x  2 x  1)
3x
(e) x 6 ln( 3x 2  5 x  7)
(f) ln( x  1  x 2 )
Q4 Compute the derivative of each of the following functions
(a) sin 1 x
(b) sinh 1 x (c) tan 1 x
(d) tanh 1 x (e) cos ech 1 x
(f) x sin x
(g) x cos x
dy
1 x 
Q5 If y  cos 1 
.
 , find
dx
1 x 
d
1
Q6 Let cosh 1 x  ln( x  x 2  1) . Show that
.
cosh 1 x 
2
dx
x 1
Q7 Evaluate the following integrals
3
2x  3
6 x 3  13x 2  7 x  10
dx (ii) 
(i)  2
(iii) x 2 e ( x 6) dx
dx
2
x  4x  5
3x  2 x  5
1 1
x
dx
dx
(iv)  ( x 2  2 x  2) cos( x 3  3x 2  6 x  7)dx (v) 
(vi)  2
2
x (ln x)
x 1
(vii)
1 1
 x (ln x) dx
(viii)
cos x  sin x
 cos x  sin x dx (ix)
tan 1 x
 1  x 2 dx (x)
 2x
7  4x
dx
 7x  5
2
3x 3  11x 2  3x  2
dx
(xi) 
x( x  1) 3
Q8 Evaluate (i)  ln xdx
(ii)  tan 1 xdx
(iii)  sin 1 xdx
Q9 Find the reduction formula for S n   sin n xdx
8

 
Q10 If I n   2 x n sin xdx prove that I n  n 
0
2
evaluate I 4 .
n 1
 n(n  1) I n  2 . Hence or otherwise

Q11 Let I n   e  x sin n xdx . Show that (1  n 2 ) I n  n(n  1) I n 2 n  2 . Hence or
0
otherwise evaluate I 4 .
9