The University of Papua New Guinea
Department of Mathematics, Statistics & Computer Science
1.20801 Calculus I
Assignment 1 – due date 29th March, 2019
TOTAL MARKS = 80
Marks are awarded for correct reasoning as well as for correct answers. Please clearly display all steps in each
solution.
1.
(a) For the function whose graph is shown, state the
following.
d) lim
h
lim f ( x)
i)

4
1  tanh
sinh  cosh
x 7
lim f ( x)
ii)
x 3
lim f ( x)
iii)
e)
x0
iv) lim f ( x)
lim
x 6
 
 x   3x 
23 x 3 x 7
x  5
v) lim f ( x)
x 6
8
x
vi) The equations of the asymptotes.
f)
 sin x 
lim 
x 0
 x 

g) Explain informally
why
1 1 
lim   2   
x 0  x
x 
(b) Sketch the graph of an example of a function f that satisfies all
of the given conditions.
lim f ( x)  2, lim f ( x)  2, f (1)  2.
x 1
h) Verify the limit in
x 1
part (g)
algebraically.
2.
Calculate the limit of the following;
4.
 4 1


lim
a)
 1   3  1 


5.
b) lim
x 1
2 x x  1
x 1
a)
Find the points of
discontinuity, if any
for the given
function. Why?
f ( x)  4 
8
x x
4
1
c)
sec x

x  1  tan x
lim
2
3. i) Use First Principles differentiation formula to find
the limits of;
f ( x) 
a)
1
x3
b) f ( x)  625 x
c) Find a value of the
constant c, if possible
that will make the
function continuous
everywhere.
c
 , ___ x  0
f ( x)   x 2
9  x 2 __ x  0

Give reasons for your answer.
ii) Find the first derivative using a differentiation formula.
1
1 

7
a) y  21 3x  2   4  2 x 2 


b)


1
 x 1 
f ( x)  2 x  1 x 2  7 x 

 x 1

 t 
c) y  1  tan 4   
 12  

 
d) r  1  
 7

3
5 .a) Find an equation of
the normal to the curve
y  Cosx at the point
0,1
b) Illustrate part (a) by
graphing the curve and
the lines on the same
graph.
7
2