Category Archives: MATH 270

MATH 270 Week 3 iLab Recent

MATH 270 Week 3 iLab Recent

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MATH 270 Week 3 iLab Recent

1. Given the following geometric series write out the first 5 partial sums. Explain how to do this.

2. Do these partial sums appear to be approaching a finite sum for the entire series? If so, what does that sum appear to be?

3. How do we know for sure that the above infinite geometric series has a sum? What is the formula that we can use to find the sum of an infinite geometric series?

4. Identify the following and use them to determine the sum of the infinite geometric series above.

5. Can we find the sum of these infinite geometric series? In other words, do these series converge? Explain. If you can find the sum, do so.

MATH 270 Week 4 iLab Recent

MATH 270 Week 4 iLab Recent

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MATH 270 Week 4 iLab Recent

Directions: Determine the Taylor Series for the following function: for n = 4 where a = 1

1. Determine the first four derivatives of

2. Evaluate and the derivatives above at x = a = 1

3. The formula for the Taylor Series is . Plug the values from question 2 into the formula. Then, simplify the fractions and write the Taylor Series approximation for .

4. Enter and the series approximation from step 3 into a graphing calculator. View the graphs and the table of values. Does the Taylor series approximation appear to converge on for all values of x? If so, what does this mean about when I can use the Taylor series to approximate ? If not, for what values of x can I use the series to approximate f(x)?

5. Under what circumstances would I be better off using a Taylor series rather than a Maclaurin series to approximate a function?

6. Determine a Taylor Series approximation for where n = 4 and a = 3

Part II: Fourier series

Write the Fourier series for the following function: f(x) =

1. Find using the formula

2. Find using the formula .

Note: We can use the integration formula . To do this we will set u = kx, which makes du = kdx. To complete the substitution we need to have a k attached to both the f(x) and dx portion of the integral. Thus, we are multiplying in two k’s. To balance this we must also divide by 2 k’s outside the integration. Thus, the constant outside the integration will look like .

3. Find . using the formula .

Note: We can use the integration formula . To do this we will set u = kx, which makes du = kdx. To complete the substitution we need to have a k attached to both the f(x) and dx portion of the integral. Thus, we are multiplying in two k’s. To balance this we must also divide by 2 k’s outside the integration. Thus, the constant outside the integration will look like .

4. The Fourier series definition is . Replace with the answers you got in questions 1, 2, and 3 to write the Fourier series. Note: If you replaced k with 2k or 2k + 1 when creating then you must do the same with the k in the cos(kx) and/or sin(kx) in the Fourier series definition above.

5. Under what circumstances can you assume = 0? What about when ? How can you determine this from the graph of the given function? How can you determine this algebraically?

MATH 270 Week 5 iLab Recent

MATH 270 Week 5 iLab Recent

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MATH 270 Week 5 iLab Recent

Part I: Separable Differential Equations

1. What is meant by separable? What should you look for in a differential equation to decide if it is separable or not?

2. Is separable? If so, how can it be separated? Hint: Remember that .

3. Integrate both the left and right sides of the result of question 2.

5. Can the result of question 3 be easily solved for y? If so, solve for y. If not, how should we write the final answer? Under what circumstances would the integration result in an equation that should not be solved for y?

7. Take the result of question 6 and find the particular solution given that y(0) = ½ . How is the initial condition used to find a particular solution? What are we solving for when we find a particular solution?

MATH 270 Week 6 iLab Recent

MATH 270 Week 6 iLab Recent

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MATH 270 Week 6 iLab Recent

Part I: Second Order Homogeneous Differential Equations

1.What is meant by homogeneous? What should you look for to determine the order of a differential equation?

2.Solve the characteristic equation above for the roots . Hint: you may need to solve by factoring or by using the quadratic formula.

4. Write the solution of the differential equation based on the roots found in question 3.

5. Solve the following homogeneous differential equation by repeating the process above.

7. Take the result of question 6 and find the particular solution given that y(0) = 1 and . Why do we need two initial conditions for second order differential equations? How are the initial conditions used to find a particular solution? What must be done in order to use the second initial condition?

Part II: Second Order Non-Homogeneous Differential Equations

1. Follow the six steps below to solve the following:

Remember that the final solution form is

Step 1: Find . This is the homogeneous solution. Use the process outlined in part I above to find it.

Step 5: Solve for the unknown variables A, B, etc. to determine .Hint 1: match like terms on the left and right side. For example, if you had 2A + 3Ax – 5B = 2x + 4, you would set 3Ax = 2x and 2A – 5B = 4. You may have to distribute to clear some parentheses first. Hint 2: You will have to solve a system of equations to find A, B, etc. What are some ways to solve a system of equations?

Step 6: Combine yp and yh to obtain the complete solution