NUMERICAL SOLUTIONS (LAB 1)
EVALUATING COMPLEX FUNCTIONS USING TAYLOR SERIES EXPANSION
OBJECTIVE:
To know how Taylor series can be used as an approximation technique for functions that may be
difficult to evaluate directly and comprehend accuracy of approximations through error
determination as more terms would be used in expansion.
INTENDED LEARNING OUTCOME:
The conduct of Taylor Series Expansion laboratory in numerical analysis ensures comprehensive
understanding of Taylor series and its applications in numerical methods. At the end the students
are expected to:
 Develop a clear understanding of what Taylor series are, how they are derived, and the
mathematical principles behind them.
 Derive Taylor series expansions for various types of functions, including power series,
Maclaurin series (centered at 0), and general Taylor series (centered at arbitrary points).
 Analyze and estimate the errors associated with Taylor series approximations, including
truncation error and round-off error.
 Apply Taylor series expansions to solve a variety of mathematical, scientific, and
engineering problems, including but not limited to differentiation, integration, solving
differential equations, and root-finding.
 Develop critical thinking skills to determine when and how to apply Taylor series expansion
techniques to solve real-world problems effectively.
DISCUSSION:
(1 PARAGRAPH – ATLEAST 4-6 SENTENCES)
FORMULA:
(IF APPLICABLE)
PROBLEM 01:
1. Use zero through fourth order Taylor’s expansion to approximate the value of the function,
defined as 𝑓(𝑥) = −0.1𝑥 4 − −0.15𝑥 3 − 0.5𝑥 2 − 0.25𝑥 + 1.2 at x =1. Use a base point
x=0, then calculate the error associated with each function.
SOLUTIONS:
(ONE COLUMN CALCULATION)
RESULTS, INTERPRETATION AND CONCLUSION:
(SUMMARY OF ANSWERS- IN TABLE)
PROBLEM 02:
2. To calculate a planet’s space coordinates, we have to solve the function
𝑓(𝑥) = 𝑥 − 1 − 0.5𝑠𝑖𝑛𝑥
Let the base point be a = x = π/2 on the interval [0, π]. Determine the highest-order Taylor
series expansion resulting in a maximum error of 0.015 on the specified interval. The error is
equal to the absolute value of the difference between the given function and the specific
Taylor series expansion.
i
SOLUTIONS:
(ONE COLUMN CALCULATION)
RESULTS, INTERPRETATION AND CONCLUSION:
(SUMMARY OF ANSWERS- IN TABLE)
PROBLEM 03:
3. Use Taylor’s Theorem to find an estimate for 𝑓(𝑥) = 𝑒 −𝑥 at x=1 with base point of 0 .
Employ the zero-, first-,second-, and third-order versions and compute the truncation error
for each case.
SOLUTIONS:
(ONE COLUMN CALCULATION)
RESULTS, INTERPRETATION AND CONCLUSION:
(SUMMARY OF ANSWERS- IN TABLE)
PROBLEM 04:
4. The following infinite series can be used to approximate e :
x
𝑥2 𝑥3
𝑥𝑛
𝑒 =1+𝑥+
+
……….+
21
3!
4!
𝑥
(a) Prove that this Maclaurin series expansion is a special case of the Taylor series expansion
with xi = 0 and h = x.
(b) Use the Taylor series to estimate the above expansion for 𝑥𝑖 + 1 = 2 and 𝑥𝑖 = 0.75. Employ
the zero-, to tenth versions and compute the |ε | for each case.
t
SOLUTIONS:
(ONE COLUMN CALCULATION)
RESULTS, INTERPRETATION AND CONCLUSION:
(SUMMARY OF ANSWERS- IN TABLE)
PROBLEM 05:
5. Derive the MacLaurin series for the following functions:
𝒇𝟏 (𝒙)=
𝟏
𝟏+𝒙
𝒇𝟐 (𝒙)=
𝟏
𝟏−𝒙
𝒇𝟑 (𝒙)=
𝟏
𝟏+𝒙𝟐
SOLUTIONS:
(ONE COLUMN CALCULATION)
RESULTS, INTERPRETATION AND CONCLUSION:
(SUMMARY OF ANSWERS- IN TABLE)
GENERALIZATION:
2 paragraphs (at least 4-6 sentences each paragraph)
Note: strictly follow the format
Deadline: Sept. 15, 2023 @ Engineering Faculty
Place your submissions in a brown expandable envelope