Steve Goddard

Contents

Topic PageAlgebra and Partial Functions 2Logarithms, Exponentials and Hyperbolic Functions

8

Arithmetic and Geometric Progressions and the binomial series

13

Page 1 of 16

Steve Goddard

Analytical Methods – Assignment 1

Algebraic Methods

Algebra and Partial Functions

1. Solve the following polynomial division

So = with a remainder of 8

Check answer using remainder theorem:

Page 2 of 16

Steve Goddard

2. Solve the following equation using the factor theorem:

Page 3 of 16

Steve Goddard

3. Use the remainder theorem to find the remainder for the following:

The remainder theorem states that the remainder, , of a polynomial, , divided by a linear divisor, , is equal to .

So to work out the remainder of the above equation I will use x = 3.

Checking my answer through long division:

4. Find the remainder when the following expression is divided by (x+1)

Page 4 of 16

I checked this using the remainder theorem:

When x = -1

Steve Goddard

5. Resolve the following into partial fractions:

5.1

Equate co-efficients:

By rearranging equation 3:

Substituting into 2:

Solve simultaneously:

Page 5 of 16

Steve Goddard

5.2

I can multiply this equation by the first denominator

Simplified this gives me:

Equate co-efficients

I checked my calculations by using a partial fraction calculator from the internet

Page 6 of 16

Steve Goddard

5.3

Equating the Co-efficients

Multiply equation 1 by 3:

Subtract Equation 2

Using this I will solve equation 1

Check:

Logarithms, Exponentials and Hyperbolic Functions

Page 7 of 16

Equation 2

Equation 1

Steve Goddard

6. Evaluate to 3 significant figures:

Using e as the approximate value of 2.7183

7. Solve the following equations correct to 3 significant figures:

7.1

Check:

7.2

7.3

Log each side

Page 8 of 16

Steve Goddard

Page 9 of 16

Steve Goddard

8. The voltage across a capacitor at time T is given by:

Where C = 10μF and R = 20KΩ. Determine:

8.1 The time for the voltage to reach 5v

8.2 Voltage after 1ms

-t = 0.01

So:

9. Evaluate the following to 4 significant figures

9.1 cosh 2.47

9.2 sinh 1.385

10. A telegraph wire hangs so that its shape is described by:

Page 10 of 16

Steve Goddard

Evaluate correct to 3 significant figures the value of y when x is 10.

First I will work out cosh 0.5

Putting this into the original equation will give me

To 3 significant figures

11. If find values for A and B

Equating the coefficients gives:

And

So:

Adding the two equations together gives me:

Substituting this into the first equation gives me:

12. If find values for P and Q

Page 11 of 16

Steve Goddard

Equate Coefficients:

13. Solve the equation 3.52 Cosh x + 8.42 Sinh x = 5.32 correct to 2 decimal places

Hence = 1.22 or = -0.33

So x = ln 1.22 or x = ln (-0.33) which has no real solution. Hence x = 0.20 rounded up correct to 2 decimal places.

Page 12 of 16

Steve Goddard

Arithmetic and Geometric Progressions and the Binomial Series

14. Determine the 15th term of the series: 12, 17, 22, 27…

First of all I noticed that the pattern in these numbers were that it was increasing every time by 5.

Therefore the 15th term in the series is: 82

I checked this using excel: -------------------------------------------->

15. The sum of 10 terms of an arithmetic progression is 200 and the common difference is 4. Find the first term of the series. For this I worked out some rough minimum and maximum values and put the first values into excel. I then filled the values down by 4 and also filled across to get values for numbers increasing by 1 each time.From this screen I managed to work out the first value of the sequence that equated to 200.

The answer was 2

16. An oil company drills a hole 10Km deep. Estimate the cost of drill if the cost is £20 for drilling the first metre with an increase of £3 per metre for each succeeding metre.

I worked out the cost using excel, I put in 20 and then filled the numbers down 10000 times going up in stages of 3. I then took the sum of all these numbers to give me an answer.

= 150215020

17. Determine the 10th term in the series: 2, 6, 18, 54

I worked out that the pattern in these numbers was that it was multiplied by 3 each time. I continued the trend until I had the 10th term which was:

= 39366

Page 13 of 16

Steve Goddard

18. Find the sum of the first 12 terms of the series: 1, 4, 16, 64…

The pattern in the sequence is that it is being multiplied by 4 each time.

I got these values and calculated the combined total of the numbers as shown on the right.

19. Find the sum to infinity of the series: 4, 2, 1, ½, ¼…..

20. Use the Binomial Series to expand:

To do this I used the formula for binomial expansion:

When a = 1 and n = 6:

I will now simplify the equation:

21. Expand the following ascending powers of x as far as the term in the using the binomial series:

State the limits for which the expansion is valid.

Page 14 of 16

Steve Goddard

Using the binomial formula:

This is valid for

Page 15 of 16

Steve Goddard

Bibliography

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/partial.html

Higher Engineering Mathematics 5th Edition – John bird

www.Wikipeida.org

Page 16 of 16