Analytical Methods - Algebraic Methods
-
Upload
steven-goddard -
Category
Documents
-
view
1.715 -
download
5
description
Transcript of Analytical Methods - Algebraic Methods
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