EMCH 201 University of South Carolina Application of Numerical Methods Exercises

  

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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(𝑥) = 𝑥 −
𝑥
3
3!
+
𝑥
5
5!
−
𝑥
7
7!
+…. (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 t. (40 pts)

4. Evaluate and interpret the condition numbers for
a. 𝑓(𝑥) = 𝑒
−𝑥
2
+ 𝑥
2
for x = 10 (10 pts)
b. 𝑓(𝑥) =
√𝑥
3+1
4
− 2𝑥 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(𝑥)
=𝑥−
𝑥3
3!
+
𝑥5
5!
−
𝑥7
7!
+…. (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 t. (40 pts)
4. Evaluate and interpret the condition numbers for
2
a. 𝑓(𝑥) = 𝑒 −𝑥 + 𝑥 2 for x = 10 (10 pts)
b. 𝑓(𝑥) =
√𝑥 3 +1
−
4
2𝑥 for x = 300 (10 pts)
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