Description

1. Please specify the significant figures for the following number (10pts):

a. 0.1210

b. 100.0E+3

c. 2.12

d. 0.212

e. 10000.0

2. Based on Taylor Series, please prove sin(Ã°Ââ€˜Â¥) = Ã°Ââ€˜Â¥ Ã¢Ë†â€™

Ã°Ââ€˜Â¥

3

3!

+

Ã°Ââ€˜Â¥

5

5!

Ã¢Ë†â€™

Ã°Ââ€˜Â¥

7

7!

+Ã¢â‚¬Â¦. (Hint: evaluate sin(x) at

the expansion point of 0) (30 pts)

3. Use a forward difference and a centered difference approximation to estimate the first derivative of

the function: f(x) = 10x

4

– 4x

3 + 3x

2 + 6x. Evaluate the derivative at x = 1 using a step size of h = 0.1

(that is, xi+1 = 1.1, xi = 1, and xi-1 = 0.9). Compare your results with the true value of the derivative

and compute the true percent relative error Ã¯ÂÂ¥t. (40 pts)

4. Evaluate and interpret the condition numbers for

a. Ã°Ââ€˜â€œ(Ã°Ââ€˜Â¥) = Ã°Ââ€˜â€™

Ã¢Ë†â€™Ã°Ââ€˜Â¥

2

+ Ã°Ââ€˜Â¥

2

for x = 10 (10 pts)

b. Ã°Ââ€˜â€œ(Ã°Ââ€˜Â¥) =

Ã¢Ë†Å¡Ã°Ââ€˜Â¥

3+1

4

Ã¢Ë†â€™ 2Ã°Ââ€˜Â¥ for x = 300 (10 pts)

EMCH 201: Introduction to Application of Numerical Methods

Instructor: Lang Yuan,

Department of Mechanical Engineering, university of South Carolina

Assignment 1

Assignment due date: 11:59 PM, Monday January 25th, 2021

Point Possible: 100 (20+30+40+20)

Submission requirements:

a. Detailed problem-solving process needs to be provided for full points

b. Upload readable electronic copy to blackboard in a single file

c. Late submission will not be graded

1. Please specify the significant figures for the following number (10pts):

a. 0.1210

b. 100.0E+3

c. 2.12

d. 0.212

e. 10000.0

2. Based on Taylor Series, please prove sin(Ã°Ââ€˜Â¥)

=Ã°Ââ€˜Â¥Ã¢Ë†â€™

Ã°Ââ€˜Â¥3

3!

+

Ã°Ââ€˜Â¥5

5!

Ã¢Ë†â€™

Ã°Ââ€˜Â¥7

7!

+Ã¢â‚¬Â¦. (Hint: evaluate sin(x) at

the expansion point of 0) (30 pts)

3. Use a forward difference and a centered difference approximation to estimate the first derivative of

the function: f(x) = 10×4 – 4×3 + 3×2 + 6x. Evaluate the derivative at x = 1 using a step size of h = 0.1

(that is, xi+1 = 1.1, xi = 1, and xi-1 = 0.9). Compare your results with the true value of the derivative

and compute the true percent relative error Ã¯ÂÂ¥t. (40 pts)

4. Evaluate and interpret the condition numbers for

2

a. Ã°Ââ€˜â€œ(Ã°Ââ€˜Â¥) = Ã°Ââ€˜â€™ Ã¢Ë†â€™Ã°Ââ€˜Â¥ + Ã°Ââ€˜Â¥ 2 for x = 10 (10 pts)

b. Ã°Ââ€˜â€œ(Ã°Ââ€˜Â¥) =

Ã¢Ë†Å¡Ã°Ââ€˜Â¥ 3 +1

Ã¢Ë†â€™

4

2Ã°Ââ€˜Â¥ for x = 300 (10 pts)

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