This document discusses the concepts of continuity, derivatives, and Taylor series, focusing on the function log(1+x) around points x=5 and x=10. It provides derivative calculations and Taylor series expansions, emphasizing the function's behavior and applications. The project also facilitated knowledge gain and improved teamwork through sharing ideas and perspectives.

1. Represented By : Arijit Dhali

5. • Continuity:
Continuity is the formulation of the intuitive concept of a function that varies with no abrupt
breaks or jumps.
• Derivative:
Derivative of a function of a real variable measures the sensitivity to change of the function
value with respect to a change in its argument.
• Taylor Series:
Taylor series of a function is an infinite sum of terms that are expressed in terms of the
function's derivatives at a single point.
• Lagrange Remainder:
Lagrange remainder is: where M is the maximum of the absolute value of the (n + 1)th
derivative of function on the interval from x to c. The error is bounded by this remainder.

9. • Infinite Series for the function Log(1+x) around x=5
Here the function f(x) = log(1+x), -1<x<∞ is continuous and derivable at each point of (-1, ∞)
Now,
f(x) = log(1-x)
f’(x) =
𝟏
𝟏+𝐱
f’’(x) =
−𝟏
(𝟏+𝐱)𝟐
f’’’(x) =
𝟐
(𝟏+𝐱)𝟑 and so on……..
Now using Taylor’s formula for the function f(x) about x=5 is,
f(x) = f(5) + (x-5) f’(5) +
(𝒙−𝟓)𝟐
𝟐!
f’’(5) +
(𝒙−𝟓)𝟑
𝟑!
f’’’(5) + …+
(𝒙−𝟓)𝒏
𝒏!
𝒇𝒏 𝟓 + θ 𝐧 − 𝟓 , 0 < θ < 1
Since, f(5) = log(6), f’(5) =
𝟏
𝟔
, f’’(5) = −
𝟏
𝟑𝟔
, f’’’(5) =
𝟐
𝟐𝟏𝟔
…………..and so on.
Therefore, f(x) = log 6 +
𝟏
𝟔
(x-5) +
(𝒙−𝟓)𝟐
𝟐!
(
−𝟏
𝟑𝟔
) +
(𝒙−𝟓)𝟑
𝟑!
(
𝟐
𝟐𝟏𝟔
) +……+
(𝒙−𝟓)𝒏
𝒏!
(−𝟏)𝒏−𝟏 𝒏−𝟏 !
𝟔𝒏
Log(11) = 1 +
𝟏
𝟔
(x-10) -
(𝒙−𝟏𝟎)𝟐
𝟕𝟐
+
(𝒙−𝟏𝟎)𝟑
𝟔𝟒𝟖
-……+
(𝒙−𝟏𝟎)𝒏
𝒏
(−𝟏)𝒏−𝟏
𝟔𝒏

10. • Infinite Series for the function Log(1+x) around x=10
Here the function f(x) = log(1+x), -1<x<∞ is continuous and derivable at each point of (-1, ∞)
Now,
f(x) = log(1-x)
f’(x) =
𝟏
𝟏+𝐱
f’’(x) =
−𝟏
(𝟏+𝐱)𝟐
f’’’(x) =
𝟐
(𝟏+𝐱)𝟑 and so on……..
Now using Taylor’s formula for the function f(x) about x=5 is,
f(x) = f(10) + (x-10) f’(10) +
(𝒙−𝟓)𝟐
𝟐!
f’’(10) +
(𝒙−𝟏𝟎)𝟑
𝟑!
f’’’(10) + …+
(𝒙−𝟏𝟎)𝒏
𝒏!
𝒇𝒏 𝟏𝟎 + θ 𝐧 − 𝟏𝟎 , 0 < θ < 1
Since, f(10) = log(10), f’(10) =
𝟏
𝟏𝟏
, f’’(10) = −
𝟏
𝟏𝟐𝟏
, f’’’(10) =
𝟐
𝟏𝟑𝟑𝟏
…………..and so on.
Therefore, f(x) = 1 +
𝟏
𝟏𝟏
(x-10) +
(𝒙−𝟏𝟎)𝟐
𝟐!
(
−𝟏
𝟏𝟐𝟏
) +
(𝒙−𝟏𝟎)𝟑
𝟑!
(
𝟐
𝟏𝟑𝟑𝟏
) +……+
(𝒙−𝟏𝟎)𝒏
𝒏!
(−𝟏)𝒏−𝟏 𝒏−𝟏 !
𝟏𝟏𝒏
Log(11) = 1 +
𝟏
𝟏𝟏
(x-10) -
(𝒙−𝟏𝟎)𝟐
𝟐𝟒𝟐
+
(𝒙−𝟏𝟎)𝟑
𝟑𝟗𝟗𝟑
-……+
(𝒙−𝟏𝟎)𝒏
𝒏
(−𝟏)𝒏−𝟏
𝟏𝟏𝒏

11. • This project has been beneficial for us , as it enabled us to gain a lot of knowledge about the use
of Taylor Series about a appoint , as well as its applications. It also helped us to develop a better
coordination among us as we shared different perspectives and ideas regarding the sub-topics
we organized in this project .

12. • https://en.wikipedia.org/wiki/Taylor_series
• https://math.stackexchange.com
• https://www.quora.com/What-is-the-expansion-of-log-1-x
• Engineering Mathematics I – B.K.Pal & K.Das