Steve Goddard

Contents

Topic PageDifferentiation of Common functions 1Differentiation of a Product 1Differentiation of a quotient 1Function of a Function 2Successive Differentiation 3Logarithmic Differentiation 4Differentiation of Inverse Trigonometry and Hyperbolic Functions

5

Integration of Common Functions 5Integration Using Algebraic Substitutions 6Integration Using Partial Fractions 7Integration by Parts 8Analyse engineering Situations and solve Engineering Problems Using Calculus

9

Maclaurin’s Series 16

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Steve Goddard

Analytical Methods – Assignment 1

Calculus

Differentiation of Common Functions

1.

If

Since , a = 3 and n = 2 thus

2.

Differentiation of a Product

3.

Differentiation of a Quotient

4.

Using the quotient rule:

Let U =

And

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U=

V=

Steve Goddard

Let V =

Putting these values into the equation:

=

Function of a Function

5.

Using the function of a function rule:

Rewriting U as gives:

Successive Differentiation

6. Find:

6.1

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6.2

Logarithmic Differentiation

7. Use logarithmic differentiation to differentiate the following:

First of all I took logs from each side:

Differentiation of Inverse Trigonometric and Hyperbolic Functions

Differentiate the following with respect to the variable:

8.

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9.

Integration of Common Functions

Determine the following indefinite integral:

10.

+c

Evaluate the following definite integrals correct to 4 significant figures:

11.

12.

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Integration Using Algebraic Substitutions

Integrate with respect to the variable:

13.

14.

Integration Using Partial Fractions

Integrate with respect to x:

15.

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Next I multiplied the numerators by the main denominator and cancelled out the relevant values

Next I will substitute a strategic value to make one side of the equation = 0. Firstly I will make x = -3

Therefore:

Doing the same again but for the other side I will use x = +3

From this I now know that: A= 2 and B = -2

Now that I now A and B I can put these into the original equation

To integrate this I split it into two parts

Therefore:

Integration by Parts

Determine the following integrals using integration by parts:

16.

Let u = Let du =

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Putting this into the by parts formula:

17.

From the integration by parts formula

Let from which i.e.

And let from which

Expressions for u, du and v are now substituted into the by parts formula

Analyse engineering Situations and Solve Engineering Problems Using Calculus

18. Find the turning points of:

And distinguish between them, showing your calculations and deductions

Given that

I determined that

Let Now I will solve the values for

If then:

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Using the quadratic formula,

if

Steve Goddard

Putting these values into the Y equation:

19. In an electrical circuit an alternating voltage is given by Volts.

Determine to 2 decimal places over the range t=0 to t = 10ms:

19.1 The mean and

From excel I have determined that the equation produces a sine wave:

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Because it is a sine wave I can use the equation

From the graph I can see the maximum value is 25 so:

19.2 The r.m.s.

This is very similar to the above equation

I already know that the maximum value is 25 so:

20. 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

Area x =

Area y =

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-30

-20

-10

0

10

20

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Series1

Series2

XY

Steve Goddard

I already know that V = so:

To work out what is, using the total area equation:

Next I differentiate this answer:

Putting the value for x back into the original equation for total area:

21. The distance, x, moved by a body in t seconds is given by:

Distance =

Therefore:

Velocity = v =

Acceleration

Find:

21.1 The velocity and acceleration at the start

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Steve Goddard

Velocity =

Acceleration =

21.2 The velocity and acceleration when t = 3s

Velocity =

Acceleration =

21.3 The values of t when the body is at rest

This is a quadratic equation:

21.4 The values of t when the acceleration is

21.5 The distance travelled in the third second

For 3 seconds:

Distance =

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So if t = 3 then:

For 2 seconds:

So if t = 3 then:

99.5 - 24.33 = 75.17 metres

22. An alternating current i amps is given by:

Where: f is the frequency in Hz t is time in seconds

Determine the rate of change of current when t = 20ms, given that f=50Hz

23. The speed of a car, v, in metres per seconds is related to time, t, in seconds by the following:

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Determine the maximum speed of the car in kilometres per hour

Max Speed =

24. Determine the area enclosed by:

The lines on the graph to the right represent the equations below:

From these I will work out the x values at which the two lines intersect.

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-2

-1

0

1

2

3

4

5

1 2 3 4 5 6 7 8 9 10 11

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Maclaurin’s Series

25. Use Maclaurin’s series to find a power series for:

As far as the term

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26. Show, using Maclaurin’s series, that the first 4 terms of the power series for

is given by:

By definition;

Now use the known series for (which is ):

=

Note the even power terms cancel out and the odd powers appear twice:

=

(All summations go from n=0 to infinity).

So the series goes:

27. Find the first 4 terms of the series for by applying Maclaurin’s Theorem

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So the first four terms are:

28. Determine the following limiting values:

If you substitute x=1 directly into the expression, you obtain 0/0, which is undefined

Using l'hopital's rule:

Differentiate both the numerator and denominator with respect to x.

Thus:

When x=1 substituted into the above equation is definable (i.e doesn’t = 0/0), l’Hopitals rule doesn’t need to be used again. Therefore this expression is correct.

Then substitute the value x=1 into this new expression,

=1/9

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Steve Goddard

Bibliography

Higher Engineering Mathematics 5th Edition – John birdIn-class notes – Roger MaceyCourse Hand outs – Roger Macey

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