MEEN 357-504
Engineering Analysis for Mechanical Engineers
Fall 2025
Exam 1
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• Formulae are provided on the last page.
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• You have 75 minutes to complete the exam.
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Section 1
Section 2
Section 3
Total
70 pts
35 pts
95 pts
200 pts
1
MEEN 357-504
Engineering Analysis for Mechanical Engineers
Section 1: (70 pts) Multiple Choice
Indicate the correct answer for each. (5 pts each)
(1)
(2)
(3)
(4)
Iterative algorithms will only stop when a solution is found.
True
False
Computer-numerical calculations are always accurate.
True
False
Which of the following is an example of an avoidable source of error?
Round-off error
Truncation error
Coding errors
Floating point limitations
When the true value is unknown and an iterative algorithm is used, the most appropriate error
measure is:
(5)
Absolute error
True relative error
Approximate relative error
Intrinsic error
In bracketing methods, valid stopping criteria can include (select all that apply):
When the function value is "close enough" to zero
When the bracket size is "small enough"
When the error is "small enough"
When exceeding a maximum number of iterations
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Fall 2025
MEEN 357-504
(6)
Engineering Analysis for Mechanical Engineers
Bracketing methods are generally slower than open methods but guarantee convergence if the
initial bracket contains a root.
(7)
True
False
The Newton-Raphson method finds where the slope of the function at the current point
intersects the x-axis.
(8)
(9)
True
False
Simpson's 1/3 Rule requires how many equidistant points?
Two
Three
Four
None of the above
Simpson's 3/8 Rule integrates which of the following polynomials exactly (select the highest order
option that is true):
(10)
Linear
Quadratic
Cubic
Quartic
For a dataset with six (6) points and equal spacing, which integration rule(s) can be applied to
estimate the integral?
Multiple application Simpson's 1/3 rule
Multiple application Simpson's 3/8 rule
A combination of Simpson's 1/3 and 3/8 rules
Only the multiple application Trapezoidal rule
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Fall 2025
MEEN 357-504
(11)
Engineering Analysis for Mechanical Engineers
When integrating a cubic polynomial, the truncation error of using the Simpson’s 1/3 rule is
_______ that of the Simpson’s 3/8 rule.
(12)
(13)
Less than
Equal to
Greater than
Reducing the step size in the finite difference method always reduces true relative error.
True
False
The truncation error of using the central finite difference is typically _______ that of the forward
finite difference.
(14)
Less than
Equal to
Greater than
Given the error trend below for some numerical method, what is the error order?
𝑂(ℎ)
𝑂(ℎ2 )
𝑂(ℎ3 )
None of the above
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Fall 2025
MEEN 357-504
Engineering Analysis for Mechanical Engineers
Fall 2025
Section 2: (35 pts) Python
Answer each of the following questions succinctly. Place your answers in the boxes provided when applicable.
(15)
Based on the Python code below, what is the value of x when the script ends?
(10pts)
(16)
Based on the Python code below, what is the value of c when the script ends?
(10pts)
(17)
Write a Python function called factorial that takes an integer n and returns its factorial using a
loop.
(15pts)
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MEEN 357-504
Engineering Analysis for Mechanical Engineers
Fall 2025
Section 3: (95 pts) Short Answer
Answer each of the following questions and show your work. Place your answers in the boxes provided when
applicable.
(18)
Using the centered finite difference method, approximate the derivative of the following
equation at 𝑥 = 1.5. Assume a step size of 0.5.
(15pts)
𝑓(𝑥) = −4𝑥 3 + 2𝑥 2 + 11
(19)
Let 𝑓(𝑥) = 𝑥 4 . Estimate 𝑓(2) using a second-order Taylor Series approximation around the point
𝑥 = 1. Show all of your work.
(15pts)
(20)
What is the relative approximation error (in percent) for the first step in the bisection method
when the initial bracket is defined by 𝑥𝑙 = 1 and 𝑥𝑢 = 4?
(15pts)
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MEEN 357-504
Engineering Analysis for Mechanical Engineers
Fall 2025
1 1
(21)
Consider the function 𝑓(𝑥) = −3𝑥 2 + 5𝑥 + 1 on the interval 𝑥 ∈ [− , ]. What is your
2 2
approximation of the root after one iteration of the false-position method? Use the interval
bounds as your initial bracket.
(15pts)
(22)
Using the same function in Question (21), what is your approximation of the root after one
iteration of the Newton-Raphson method starting at 𝑥 = 0?
(15pts)
(23)
Given the following data, estimate the integral of 𝑓(𝑥) on the interval 𝑥 ∈ [0, 0.8] using an
appropriate combination of Simpson’s 1/3 rule and trapezoidal rule. You must use each rule at
least once.
Note: the majority of points on this problem will be awarded for the proper setup.
(20pts)
𝑥
0
0.16
0.32
0.48
0.64
0.80
𝑓(𝑥)
0.2
1.3
1.7
3.2
3.2
0.2
7
MEEN 357-504
Engineering Analysis for Mechanical Engineers
SCRATCH PAPER
8
Fall 2025
MEEN 357-504
Engineering Analysis for Mechanical Engineers
SCRATCH PAPER
9
Fall 2025
MEEN 357-504
Engineering Analysis for Mechanical Engineers
Fall 2025
Formulae
𝜀𝑡 = |
true value − approximation
| × 100
true value
𝑁
𝑓(𝑥) = ∑
𝑛=0
𝑥𝑟 =
𝑥𝑢 + 𝑥𝑙
2
𝑓′(𝑥𝑖 ) =
𝐼=ℎ
𝜀𝑎 = |
𝑥 new − 𝑥 old
| × 100
𝑥 new
𝑓 (𝑛) (𝑎)
𝑓 ′′ (𝑎)
(𝑥 − 𝑎)𝑛 = 𝑓(𝑎) + 𝑓 ′ (𝑎)(𝑥 − 𝑎) +
(𝑥 − 𝑎)2 + ⋯
𝑛!
2!
𝑥𝑟 = 𝑥𝑢 −
𝑓(𝑥𝑖 ) − 𝑓(𝑥𝑖−1 )
ℎ
𝑓(𝑥0 ) + 𝑓(𝑥1 )
2
𝑓(𝑥𝑢 )(𝑥𝑙 − 𝑥𝑢 )
𝑓(𝑥𝑙 ) − 𝑓(𝑥𝑢 )
𝑓′(𝑥𝑖 ) =
𝐸𝑡 = −
𝑥𝑖+1 = 𝑥𝑖 −
𝑓(𝑥𝑖+1 ) − 𝑓(𝑥)
ℎ
𝑓(𝑥𝑖 )
𝑓′(𝑥𝑖 )
𝑥𝑖+1 = 𝑥𝑖 −
𝑓′(𝑥𝑖 ) =
𝑓(𝑥𝑖+1 ) − 𝑓(𝑥𝑖−1 )
2ℎ
1 ′′
1
𝑓 (𝜉)(𝑏 − 𝑎)3 ≈ − 𝑓 ′′ (𝑏 − 𝑎)3
12
12
𝐼=
ℎ
𝑓(𝑥0 ) + 4𝑓(𝑥1 ) + 𝑓(𝑥2 )
[𝑓(𝑥0 ) + 4𝑓(𝑥1 ) + 𝑓(𝑥2 )] = (𝑏 − 𝑎)
3
6
𝐼=
3ℎ
𝑓(𝑥0 ) + 3𝑓(𝑥1 ) + 3𝑓(𝑥2 ) + 𝑓(𝑥3 )
[𝑓(𝑥0 ) + 3𝑓(𝑥1 ) + 3𝑓(𝑥2 ) + 𝑓(𝑥3 )] = (𝑏 − 𝑎)
8
8
10
𝑓(𝑥𝑖 )(𝑥𝑖−1 − 𝑥𝑖 )
𝑓(𝑥𝑖−1 ) − 𝑓(𝑥𝑖 )
0
