QUESTION

1.
Find a value fo the constant k, if possible, that will make the function continuous everywhere.
(a) f (x) =
(b) f (x) =
(
9  x
2
, x  3
k/x
(
9  x
2
, x < 3
2
, x  0
k/x
2
, x < 0
2.
Use the deﬁnition
f
0
(x) = lim
h
to derive the derivative of
h!0
f (x) =
p
f (x + h)  f (x)
x for x > 0.
CRICOS No. 00213J 1
MAB121/MAN121
Calculus and Differential Equations
MAB126 Mathematics for Engineering 1
3.
Sketch the graph of the derivative of the function whose graph is shown.
y
(a)
y
(d)
x
x
(b)
30

y
(e)
y
45

x

SOLUTION

1.    (a) We are given

We have to find the value of k that will make the function continuous everywhere.
For,

The function being a polynomial is continuous.
For

The function being rational, is continuous.
We shall take the case of

If the function is continuous at x=3, then we have

Therefore,

2.    (b) We are given

We have to find the value of k that will make the function continuous everywhere.
For,

The function being a polynomial is continuous.
For

The function being rational is continuous.
We shall take the case of

Therefore, for any real number k,

Therefore, there exists no value of k such that the function is continuous at x=0.
2.
We are given,
…… (1)             for x>0
Therefore,
…… (2)
Subtracting (1) from (2), we get

Dividing both sides by h, we get

Taking   on both sides, we get

We know that by Binomial theorem,

Therefore, we get

Therefore, from the definition, we get

3.    (a) The graph of the function is

The graph of the derivative of the function is shown below.

(b) The graph of the function is

The function is

The graph of the derivative of the function is shown below.

(c) The graph of the function is

The function is

The graph of the derivative of the function is shown below.

(d) The graph of the function is

The function is

The graph of the derivative of the function is shown below.

(e) The graph of the function is

The function is

The graph of the derivative of the function is shown below.

(f) The graph of the function is

The function is

The graph of the derivative of the function is shown below.

JC90

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