Numerical solutions to Ce problems

1. TAYLOR SERIES
EXPANSION

2. INTRODUCTION
• TAYLOR SERIES IS A
FUNDAMENTAL TOOL IN
NUMERICAL ANALYSIS.
• USED TO APPROXIMATE
FUNCTIONS, DERIVATIVES,
AND INTEGRALS.
• IMPORTANT IN SOLVING CIVIL
ENGINEERING PROBLEMS
WHERE EXACT SOLUTIONS ARE
COMPLEX.

3. TAYLOR SERIES EXPANSION
TAYLOR SERIES EXPANSION
• A SERIES EXPANSION OF A FUNCTION ABOUT A POINT. IF A FUNCTION 𝑓(𝑥) HAS CONTINUOUS
DERIVATIVES UP TO (N + 1)TH ORDER, THEN THIS FUNCTION CAN BE EXPANDED IN THE
FOLLOWING WAY:
• 𝑓 𝑥 = σ𝑛=0
∞
𝑓 𝑛 𝑎
𝑥−𝑎 𝑛
𝑛!
= 𝑓 𝑎 + 𝑓′ 𝑎 𝑥 − 𝑎 +
𝑓′′(𝑎)(𝑥−𝑎)2
2!
+ ⋯+
𝑓 𝑛 𝑎 𝑥−𝑎 𝑛
𝑛!
+ 𝑅𝑛,
• TAYLOR SERIES PLAYS A VITAL ROLE IN NUMERICAL METHODS BECAUSE IT CAN BE USED TO
PREDICT THE VALUE OF A FUNCTION AT ONE POINT IN TERMS OF THE VALUE OF THE
FUNCTION AND ITS DERIVATIVE AT ANOTHER POINT. GENERALLY, ACCORDING TO THIS
THEOREM, ANY SMOOTH FUNCTION BE APPROXIMATED AS A POLYNOMIAL.

4. TAYLOR SERIES EXPANSION
TAYLOR SERIES EXPANSION
• IF TWO POINTS ARE VERY CLOSE TO EACH OTHER, THE VALUE OF THE
FUNCTION IF THE POINTS ARE TO BE SUBSTITUTED WOULD BE APPROXIMATELY
THE SAME. THIS IS THE ZERO-ORDER APPROXIMATION OF THE TAYLOR SERIES
EXPANSION. MATHEMATICALLY,
• 𝑓 𝑥𝑖+1 = 𝑓(𝑥𝑖)
THIS EQUATION IS APPLICABLE ONLY IF THE FUNCTION IS CONSTANT. FOR A
LINEAR FUNCTION, FIRST ORDER-APPROXIMATION IS NEEDED TO PREDICT
THE NEXT VALUE OF A FUNCTION AT DIFFERENT POINT.
𝑓 𝑥𝑖+1 = 𝑓 𝑥𝑖 + 𝑓′(𝑥𝑖)(𝑥𝑖+1 − 𝑥𝑖)
• AS OBSERVED, THE EQUATION NOW HAS FIRST ORDER DERIVATIVE TERM
WHICH IS EQUAL TO THE SLOPE OF A FUNCTION.

5. TAYLOR SERIES EXPANSION
TAYLOR SERIES EXPANSION
• FOR CURVATURE EQUATION, AN ADDITIONAL SECOND-ORDER DERIVATIVE IS ADDED TO
THE EQUATION.
• 𝑓 𝑥𝑖+1 = 𝑓 𝑥𝑖 + 𝑓′ 𝑥𝑖 𝑥𝑖+1 − 𝑥𝑖 + 𝑓′′(𝑥𝑖)
𝑥𝑖+1−𝑥𝑖
2
2!
• SIMILARLY, ADDITIONAL DERIVATIVE TERMS MAY BE ADDED TO THE EQUATION TO
COMPLETE THE TAYLOR SERIES EXPANSION.
• 𝑓 𝑥𝑖+1 = 𝑓 𝑥𝑖 + 𝑓′ 𝑥𝑖 𝑥𝑖+1 − 𝑥𝑖 + 𝑓′′ 𝑥𝑖
𝑥𝑖+1−𝑥𝑖
2
2!
+ ⋯ + 𝑓𝑛 𝑥𝑖
𝑥𝑖+1−𝑥𝑖
𝑛
𝑛!
+ 𝑅𝑛.
THE TERM 𝑥𝑖+1 − 𝑥𝑖 IS ACTUALLY THE DISTANCE BETWEEN TWO POINTS. THUS, THE
EXPANSION MAY BE WRITTEN AS
𝑓 𝑥𝑖+1 = 𝑓 𝑥𝑖 + 𝑓′ 𝑥𝑖 ℎ + 𝑓′′ 𝑥𝑖
ℎ2
2!
+ ⋯ + 𝑓𝑛 𝑥𝑖
ℎ𝑛
𝑛!
+ 𝑅𝑛
• AND 𝑅𝑛 IS THE REMAINDER TERM.

6. TAYLOR
SERIES
FORMULA
&
CONCEPT
• Taylor Series represents a
function as an infinite sum
of derivatives.
• Formula: f(x) = f(a) +
f'(a)(x-a) + (f''(a)/2!)(x-a)^2
+ (f'''(a)/3!)(x-a)^3 + ...
• The more terms used, the
more accurate the
approximation.

7. NUMERICAL
APPROXIMATION
• Taylor Series is
used to
approximate
derivatives:
- f'(x) ≈ (f(x+h) -
f(x)) / h (First-
order
approximation)
- Higher-order
derivatives
improve
accuracy.
• Also used in
integration and
solving
differential
equations.

8. APPLICATIONS
IN CIVIL
ENGINEERING
Structural Analysis:
Predicting beam
deflections and stress
distributions.
Fluid Mechanics:
Approximating velocity
and pressure
distributions.
Geotechnical
Engineering: Modeling
soil settlement and
foundation behavior.

9. SAMPLE PROBLEM 1:
APPROXIMATE SIN(X)
USE ZERO- THROUGH SECOND-ORDER TAYLOR SERIES EXPANSIONS TO
APPROXIMATE THE FUNCTION
𝑓 𝑥 = 3𝑥2 − 6𝑥 + 5
• FROM 𝑥𝑖 = 0 WITH ℎ = 1. THAT IS, PREDICT THE FUNCTION’S VALUE AT 𝑥𝑖+1 = 1.

10. USE ZERO- THROUGH SECOND-ORDER TAYLOR SERIES
EXPANSIONS TO APPROXIMATE THE FUNCTION
𝑓 𝑥 = 3𝑥2 − 6𝑥 + 5
FROM 𝑥𝑖 = 0 WITH ℎ = 1. THAT IS, PREDICT THE
FUNCTION’S VALUE AT 𝑥𝑖+1 = 1.
SOLUTION:
@𝑛 = 0
𝑓 𝑥𝑖 = 𝑓 0 =
3 0 2
− 6 0 +
5 = 5
𝑓 𝑥𝑖+1 = 𝑓(𝑥𝑖)
𝒇 𝟏 = 𝟓
@𝑛 = 1
𝑓 𝑥𝑖 = 𝑓 0 = 3 0 2 − 6 0 + 5 = 5
𝑓′ 𝑥𝑖 = 6𝑥 − 6 = 6 0 − 6 = −6
𝑓 𝑥𝑖+1 = 𝑓 𝑥𝑖 + 𝑓′(𝑥𝑖)(𝑥𝑖+1 − 𝑥𝑖)
𝒇 𝟏 = 𝟓 − 𝟔 𝟏 = −𝟏
@𝑛 = 2
𝑓 𝑥𝑖 = 𝑓 0 = 3 0 2 − 6 0 + 5 = 5
𝑓′ 𝑥𝑖 = 6𝑥 − 6 = 6 0 − 6 = −6
𝑓′′ 𝑥𝑖 = 6
𝑓 𝑥𝑖+1 = 𝑓 𝑥𝑖 + 𝑓′ 𝑥𝑖 𝑥𝑖+1 − 𝑥𝑖 +
𝑓′′(𝑥𝑖)
𝑥𝑖+1−𝑥𝑖
2
2!
• 𝒇 𝟏 = 𝟓 − 𝟔 𝟏 + 𝟔
𝟏 𝟐
𝟐!
= 𝟐

11. SAMPLE PROBLEM 1:
APPROXIMATE SIN(X)
• APPROXIMATE SIN(0.5) USING THE FIRST THREE TERMS OF TAYLOR SERIES AT X=0.
• SOLUTION:
• SIN(X) ≈ X - X³/3! + X⁵/5!
• SIN(0.5) ≈ 0.5 - (0.5³/6) + (0.5⁵/120)
• ≈ 0.5 - 0.0208 + 0.0021
• ≈ 0.4813 (APPROXIMATION)

12. SAMPLE PROBLEM 2: BEAM
DEFLECTION APPROXIMATION
• A BEAM DEFLECTION FUNCTION IS GIVEN AS Y(X) = 2X³ + 3X².
• FIND Y(1.1) USING TAYLOR SERIES EXPANSION AT X=1 UP TO SECOND ORDER.
• SOLUTION:
• Y(1) = 2(1)³ + 3(1)² = 5
• Y'(X) = 6X² + 6X → Y'(1) = 6(1)² + 6(1) = 12
• Y''(X) = 12X + 6 → Y''(1) = 12(1) + 6 = 18
• USING TAYLOR EXPANSION: Y(1.1) ≈ Y(1) + Y'(1)(0.1) + (Y''(1)/2)(0.1)²
• ≈ 5 + 12(0.1) + (18/2)(0.01)
• ≈ 5 + 1.2 + 0.09 = 6.29 (APPROXIMATION)

13. CONCLUSION
• Taylor Series is a powerful tool for numerical
solutions.
• Helps in approximating functions, derivatives,
and integrals.
• Widely used in civil engineering applications
for accurate predictions.
• Numerical methods ensure practical solutions
when exact methods are difficult.