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
Problem Solving Task 2
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
“The presented piece of writing is a good example how the academic paper should be written. However, the text can’t be used as a part of your own and submitted to your professor – it will be considered as plagiarism.
But you can order it from our service and receive complete high-quality custom paper. Our service offers “Maths” essay sample that was written by professional writer. If you like one, you have an opportunity to buy a similar paper. Any of the academic papers will be written from scratch, according to all customers’ specifications, expectations and highest standards.”
