BTEC HNC - Analytical Methods - Calculus

CalculusAnalytical Methods for Engineers

By Brendan Burr

Brendan Burr BTEC Higher National Certificate in ElectronicsCalculus

Table of Contents

TABLE OF CONTENTS - 2 -

TASK 1 - 7 -

1.1 Differentiate the following algebraic and trigonometric functions using basic rules : - 7 -

(a) (b) - 7 -Solution (a):- - 7 -Check (a):- - 7 -Solution (b):- - 8 -Check (b):- - 8 -

1.2 Differentiate the following using the product rule : - 9 -

- 9 -Solution:- - 9 -Check:- - 10 -

1.3 Differentiate the following using the quotient rule : - 11 -

- 11 -

Solution:- - 11 -Check:- - 12 -

1.4 Differentiate the following using the function of a function rule : - 13 -

- 13 -Solution:- - 13 -Check:- - 14 -

2


TASK 2 - 15 -

2.1 Successive Differentiation - 15 -

If find : - 15 -

(a) (b) - 15 -

Solution (a):- - 15 -Solution (b):- - 15 -Check (a & b):- - 15 -

2.2 Logarithmic differentiation - 16 -

Use logarithmic differentiation to differentiate the following :- - 16 -

- 16 -


2.3 Differentiation of Inverse Trigonometric and Hyperbolic functions - 17 -

Differentiate the following with respect to the variable :- - 17 -

(a) (b) - 17 -

Solution (a):- - 17 -Check:- - 17 -Solution (b):- - 18 -Check:- - 18 -

3


TASK 3 - 19 -

3.1 Integrate the following indefinite integral using the basic rules :- - 19 -

- 19 -


3.2 Integrate the following definite integral correct to 2 decimal places:- - 21 -

- 21 -


3.3 Integrate the following function using algebraic substitutions:- - 22 -

- 22 -Solution:- - 22 -Check:- - 23 -

3.4 Integrate the following function using Partial Fractions:- - 24 -

- 24 -


3.5 Integrate the following using integration by parts:- - 26 -

- 26 -Solution:- - 26 -Check:- - 26 -

4


TASK 4 - 27 -

4.1 A lidless box with square ends is to be made from a thin sheet of metal. Determine the least area of the metal for which the volume of the box is 3.5 cu. m.

- 27 -Solution:- - 27 -

4.2 The distance x moved by a body in t seconds is given by :- - 29 -

- 29 -

Determine : - 29 -

(a) The velocity and acceleration at the start. - 29 -Solution:- - 29 -

(b) The velocity and acceleration when t = 3 sec. - 30 -Solution:- - 30 -

(c) The values of time t when the body is at rest. - 30 -Solution:- - 30 -Check:- - 30 -

(d) The distance travelled in the third second of movement. - 31 -Solution:- - 31 -

4.3 An alternating current is given by , where f is the frequency in Hz and t is time in secs. - 32 -

Determine the rate of change of current when t = 20 mS given that f = 50 Hz. - 32 -Solution:- - 32 -Check:- - 33 -

4.4 In an electrical circuit an alternating voltage is given by volts.- 34 -

Determine the following each correct to 2 decimal places over the range t = 0 to t = 10 mS. - 34 -

(a) Mean value (b) R.m.s value - 34 -Solution (a):- - 34 -Check:- - 34 -Solution (b):- - 35 -Check:- - 36 -

5


4.5 The speed of a car v in m/s is related to time t sec by the following equation :- - 37 -

- 37 -

Determine the maximum speed of the car in km/h - 37 -Solution:- - 37 -

TASK 5 - 38 -

5.1 Use Maclaurin's series to find a power series for as far as the term in - 38 -


5.2 Show using Maclaurin's series that the first 4 terms of the power series for is given by :- - 40 -

- 40 -

Solution:- - 40 -

5.3 Find the first 4 terms of the series for by applying Maclaurin's Theorem. - 41 -


5.4 Determine the following limiting values using L’Hopitals Rule: - 42 -

- 42 -

Solution:- - 42 -

EVALUATION - 43 -

CONCLUSION - 44 -

BIBLIOGRAPHY - 44 -

Books - 44 -

Catalogues - 44 -

Websites - 44 -

6


TASK 1

1.1 Differentiate the following algebraic and trigonometric functions using basic rules :

(a) (b)

Solution (a):-

Check (a):-

The Blue Line is: y=6*x ' deriv. of y=3x^2

7


Solution (b):-

Check (b):-

The Blue Line is: y=8*sin (2*x) ' deriv. of y=-4*cos(2x)

8


1.2 Differentiate the following using the product rule :

Solution:-

9


Check:-

The Blue Line is: y= (0.5x^-0.5)*(2+ln(2x)) ' deriv.of y=ln(2x)*(sqrt(x))

10


1.3 Differentiate the following using the quotient rule :

Solution:-

11


Check:-

The Dark Blue Line is: y=(3*(sin (4*x))*e^(3*x) - 4*e^(3*x)*cos (4*x))/(sin (4*x)^2) ' deriv. of y=(e^(3x))/(sin(4x))

12


1.4 Differentiate the following using the function of a function rule :

Solution:-

13


Check:-

The Blue Line is: y=5*(2*x^3 - 5*x)^4*(2*3*x^2 - 5) ' deriv. of y=((2(x^3))-(5x))^5

14


TASK 2

2.1 Successive Differentiation

If find :

(a) (b)

Solution (a):-

Solution (b):-

Check (a & b):-

The Green Line is: y=3x^4 + 2x^3 - 3x + 2The Blue Line is: y=12*x^3 + 6*x^2 - 3 ' deriv. of

y=3x^4 + 2x^3 - 3x + 2The Red Line is: y=36*x^2 + 12*x ' deriv. of y=12*x^3 + 6*x^2 - 3 '

deriv. of y=3x^4 + 2x^3 - 3x + 2

15


2.2 Logarithmic differentiation

Use logarithmic differentiation to differentiate the following :-

Solution:-

Check:-

The Blue Line is: y=((x-1)*(3+x)*(1+(x-2+x))-(x-2)*(1+x)*(3+(x-1+x))) /(((x-1)*(3+x))^2) ' deriv. of y=((x-2)(x+1))/((x-1)(x+3))

16


2.3 Differentiation of Inverse Trigonometric and Hyperbolic functions

Differentiate the following with respect to the variable :-

(a) (b)

Solution (a):-

Check:-

The Blue Line is: y=1/sqr (1 - (x/2)^2)*1/2 ' deriv. of sin(y)=x/2

17


Solution (b):-

Check:-

The Blue Line is: x=3*cosh y ' deriv. of 3*sinh(y)=x

18


TASK 3

3.1 Integrate the following indefinite integral using the basic rules :-

Solution:-

19


Check:-

The Blue Line is: y=(x^2)-4x+3The Red Line is: y=((1/3)*(x^3))-(2x^2)+3x

The Pink Line is: y=3 + (3*1/3*x^2 - 4*x) ' deriv. of y=((1/3)*(x^3))-(2x^2)+3x

The Red Line is: y=((1/3)*(x^3))-(2x^2)+3x

20


3.2 Integrate the following definite integral correct to 2 decimal places:-

Solution:-

UNITS2

Check:-

21


3.3 Integrate the following function using algebraic substitutions:-

Solution:-

Let:

Differentiate

22


Check:-

The Red Line is: y=2sin(4x+9)The Blue Line is: y=-0.5cos(4x+9)

The Pink Line is: y=2*sin (9 + 4*x) ' deriv. of y=-0.5cos(4x+9)The Blue Line is: y=-0.5cos(4x+9)

23


3.4 Integrate the following function using Partial Fractions:-

Solution:-

Expand :

Split the function into partial fractions, using linear factors:

(So values for A and B are correct)

24


Check:-

The Red Line is: y=(2ln(x-3))-(2ln(x+3))The Blue Line is: y=12/((x^2)-9)

The Pink Line is: y=2*1/(x - 3) - 2*1/(3 + x) ' deriv. of y=(2ln(x-3))-(2ln(x+3))The Red Line is: y=(2ln(x-3))-(2ln(x+3))

25


3.5 Integrate the following using integration by parts:-

Solution:-

Check:-

TASK 4

26


4.1 A lidless box with square ends is to be made from a thin sheet of metal. Determine the least area of the metal for which the volume of the box is 3.5 cu. m.

Solution:-

2 Ends Area = 2 Sides Area = 1 Bottom Area =

Total Area =

Volume, V= Height x Length x Breadth

27


(For the minimum area)

So the minimum area of material required is:

28


4.2 The distance x moved by a body in t seconds is given by :-

Determine :

(a) The velocity and acceleration at the start.

Solution:-

for velocity in m/s

for acceleration in m/s^2

So if t = 0 seconds then:

29


(b) The velocity and acceleration when t = 3 sec.

Solution:-

(c) The values of time t when the body is at rest.

Solution:-

Check:-

30


(d) The distance travelled in the third second of movement.

Solution:-

At t = 3:

At t = 2:

31


4.3 An alternating current is given by , where f is the frequency in Hz and t is time in secs.

Determine the rate of change of current when t = 20 mS given that f = 50 Hz.

Solution:-

kA

32


Check:-

The Blue Line is: y=10*sin(2*pi*50x)The Red Line is: y=1000*(cos (2*p*50*x))*p ' deriv. of

y=10*sin(2*pi*50x)

33


4.4 In an electrical circuit an alternating voltage is given by volts.

Determine the following each correct to 2 decimal places over the range t = 0 to t = 10 mS.

(a) Mean value (b) R.m.s value

Solution (a):-

Check:-

34


Solution (b):-

From Trigonometry:

35


Check:-

36


4.5 The speed of a car v in m/s is related to time t sec by the following equation :-

Determine the maximum speed of the car in km/h

Solution:-

Maximum speed is when the =0

(Maximum Speed is when time=2 seconds.

37


TASK 5

5.1 Use Maclaurin's series to find a power series for as far as the term in

Solution:-

Using Maclaurin’s Theorem we get:

38


Therefore multiplying these two answers together gives:

Check:-

Because this answer is only as far as , there isn’t a feasible check. The only way to do it would be to keep expanding to , then the two equations would equal an identical figure and proof of correctness can be achieved.

39


5.2 Show using Maclaurin's series that the first 4 terms of the power series for is given by :-

Solution:-

40


5.3 Find the first 4 terms of the series for by applying Maclaurin's Theorem.

Solution:-

Substituting into Maclaurin’s Theorem we get:

Check:-

Check by multiplying out:

41


5.4 Determine the following limiting values using L’Hopitals Rule:

Solution:-

Firstly, let x=1

Differentiate the Numerator and Denominator, leaving:

Let x=1

42


EvaluationFor Task 1 I had to differentiate algebraic and trigonometric functions using the product, quotient and function of a function rules. By using the examples that we had covered in class, I was able to get an answer for each of these Sub-Tasks.After working out the answers for Task 1, I used Graphmatica to check my answers. This was fairly straight forward as it simply required me to enter the task equation to produce a waveform, you can then find the derivative of that curve, which creates another formula. This formula should be the same as my answer, which it was on all occasions.Moving onto Task 2, I had to determine higher order derivatives for multiple functions. I used the same process as before to work out the answers and then check them. When checking Task 2.2, I noticed that the formula that was produced from the check was different to my answer. I decided to enter my answer into the equation bar and see if my waveform fitted over the top of the derivative waveform. Fortunately for me it fitted precisely which meant my answer was correct, even though there were two different formulas.I had a similar problem with Task 2.3, where the formula given to me from Graphmatica was different from my answer, however entering my answer the two equations presented an identical waveform.I also had a problem with the check for Task 2.3 and 2.4 because of the inverse function on the formula. It was difficult to enter the equation into Graphmatica, I found out how to rearrange the formula to get the output I required for Task 2.3. However for Task 2.4, I couldn’t work out how to enter a hyperbolic sin into the equation. I gave up checking these two answers in the end due to the amount of time that was being wasted, so they are kind of irrelevant but I have left them in this assignment anyway.I then had to integrate functions using basic rules. I found these straight forward and ran through this Task quite quickly. I checked Task 3.1 by comparing two graphs, one with the question equation and the other with my answer, both the waveforms are identical and therefore proves that my answer is correct.I worked out the answer to Task 3.2 and then checked it on Graphmatica by finding the area under the curve between 0 and 1. The check worked out to be slightly different from my answer, but close enough to be considered correct.I used the same principles for the check for Tasks 3.3 and 3.4 as I did in 3.1, the answers compared to the other graph appeared identical and therefore my answers seem to be correct.My check for Task 3.5 was simply by differentiating my answer to get the question equation again. This shows that unless I have integrated and differentiated incorrectly, then the answer is correct.Task 4 required me to make use of my knowledge of calculus and apply it to solve engineering problems. Using the examples we had in class, I found this very straight forward, however I did find that it was difficult to check my answers, theoretically. Task 5 asked me to use power series methods to determine engineering variables. I checked my answers for this and all seems correct.

43


ConclusionI was fortunate to realise that this assignment was going to take a long time, so began it early in March. I used the notes gathered in class to do each Task after we had covered it every weekend, this helped significantly a couple of months down the line as there were a number of assignments due in then.I felt this assignment is an example of the effort I put into all of my assignments, with regards to my presentation and accuracy.This unit has taken a lot of effort to get right. There is so much detail required in the answers that it is very easy to make a simple mistake and result in having an incorrect answer. Even so, I feel that I have put in the extra hours to ensure that I am correct rather than handing an assignment in and hoping it is correct.

Bibliography

Through guidance from my lecturer, the following text books, catalogues and websites I was able to complete this assignment:

Books

BTEC National Engineering (Mike Tooley & Lloyd Dingle) ISBN: 978-0-7506-8521-4Success in Electronics (Tom Duncan & John Murray)ISBN: 0-7195-4015-1Higher Engineering Mathematics (John Bird) ISBN: 0-7506-8152-7

Catalogues

N/A

Websites

N/A

44

Documents

BTEC HNC - Analytical Methods - Calculus