Engineering Mathematics For Btech Student

Main Uses of Taylor’s and Maclaurin’s Theorem for a Single Variable
Taylor’s and Maclaurin’s Theorems for a single variable function are mainly used for:

🎯 Approximating Complex Functions:
When a function is difficult to calculate directly (like exponential, logarithmic, or trigonometric functions), it can be approximated using a polynomial form near a point.
Example:
sin
⁡
x
≈
x
−
x
3
3
!
sinx≈x−
3!
x
3

​
for small
x
x.

🧮 Error Estimation in Calculations:
In numerical analysis, Taylor’s series helps measure how accurate an approximation is by providing a remainder term.

⚙️ Solving Differential Equations:
Used to find approximate or series solutions to differential equations when exact solutions are hard to obtain.

📈 Analyzing Function Behavior:
Helps study the local behavior of a function—whether it increases, decreases, or has maxima/minima near a specific point.

💡 In Engineering and Physics:

Used in control systems, circuit analysis, and mechanics.

To linearize nonlinear systems (small signal approximation).
Example:
e
x
≈
1
+
x
e
x
≈1+x for small
x
x.

🧠 In Computer Science & Machine Learning:

For optimization algorithms (e.g., gradient descent).

To approximate cost or activation functions near minima.

👉 In short:
Taylor’s theorem for a single variable is mainly used to approximate, simplify, and analyze nonlinear functions using polynomial expansions — making complex problems easier to compute and understand.

1. Taylor’ s and Maclaurin Theorem For Function of Two Variables Here is where our presentation begins by Sumit Kumar & Team SARDAR BEANT SINGH STATE UNIVERSITY, GURDASPUR 143530 B.TECH (CSE) ‘E’ ENGINEERING MATHEMATICS - 1

2. Table of contents 01 04 02 05 03 Q/A Overview of Single Variable Theorem’s for two Variable Applications Conclusion

3. Q1. What is a Taylor series ? Taylor series is a function of an infinite sum of terms in increasing order of degree. Taylor series of polynomial functions is a polynomial. Q2. What is a Maclaurin series? A Taylor series is called Maclaurin series when the function is centered at zero point. Q3. What are Uses of these ? These theorems are widely used to approximate values of functions, study their behavior near a point, and simplify real-world problems in engineering, physics, and computer science.

4. Taylor’s theorem  expands a function f(x) about any point x = a , OverView/ Summary of TAYLOR’S and MACLAURIN’S THEOREM for a SINGLE variable function

5. while  Maclaurin’s theorem  is a special case of Taylor’s series where a=0. The summary of Taylor's and Maclaurin's Theorem for a single variable focuses on approximating a function with a polynomial . OverView/ Summary of TAYLOR’S and MACLAURIN’S THEOREM for a SINGLE variable function

6. Taylor’ s and Maclaurin Theorem For Function of Two Variables We know that Taylor’s theorem for a function f(x) for a single variable x is Now let, f( x,y ) be independent variable x and . If y is kept constant then by Taylor’s theorem for a function of a single variable x, we have ---1 Now keeping x constant and applying taylor theorem for a function single variable y, we have ---2 Using 2 we can write 1 as  

7. Taylor’ s and Maclaurin Theorem For Function of Two Variables = Cor 1: Put x=a, y=b; we have Cor 2: Put a+h =x, b+k =y in Cor 1 Cor 3: Put a=0 and b=0 in Cor 2  

8. Taylor’ s and Maclaurin Theorem For Function of Two Variables

9. Relevance to real-world applications

10. Relevance to real-world applications

11. Relevance to real-world applications

12. “Every complex curve hides a simple truth — Taylor’s theorem reveals it, one derivative at a time.” “In engineering, perfection is impossible — but through Taylor’s theorem, approximation becomes art.” — Sumit

13. A picture is worth a thousand words

14. Thanks! Do you have any questions? Email: [email protected] [phone] LinkedIn Profile: www.linkedin.com/in/sumitkumar2007 Please keep this slide for attribution

15. Internet Books – NP Bali pdf by Manpreet Mam Reference