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These are study notes of engineering mathematics calculas which is amazing notes just see once
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Continuity of a Functions of One Variable A function y = ๐(๐ฅ) is said to be continuous at a point ๐ฅ 0 if
๐ฅโ๐ฅ 0 ๐(๐ฅ) exists
๐ฅโ๐ฅ 0
A function ๐ฆ = ๐(๐ฅ) is said to be continuous at a point ๐ฅ 0 , if for a given ๐ > 0 , there exist a real number ๐ฟ > 0 such that ๐ ๐ฅ โ ๐ ๐ฅ 0 < ๐ whenever |๐ฅ โ ๐ฅ 0 | < ๐ฟ Mathematically:
Example: Discuss the continuity of ๐ฆ = 21 /๐ฅ The given function is discontinuous at ๐ฅ = 0. There are two reasons:
1 ๐ฅ (^) = 0 lim ๐ฅโ 0 +
1 ๐ฅ (^) = +โ
Differentiability & Differentials A function ๐(๐ฅ) is said to be differentiable at the point ๐ฅ , if when ๐ฅ is given the increment ฮ๐ฅ (arbitrary increment), the increment ฮ๐ฆ can be expressed in the form ฮ๐ฆ = ๐ด ฮ๐ฅ + ๐ ฮ๐ฅ where ๐ด is independent of ฮ๐ฅ and ๐ โ 0 as ฮ๐ฅ โ 0_._ The first term on the right hand side (๐ด ฮ๐ฅ) is called differential (or Total differential) of ๐ฆ and is denoted by ๐๐ฆ. Thus ๐๐ฆ = ๐ด ฮ๐ฅ
Differentiability & Derivative The necessary and sufficient condition that the function ๐ฆ = ๐(๐ฅ) is differentiable at the point ๐ฅ is that it possesses a finite definite derivative at this point. Taking limit ฮ๐ฅ โ 0 , we get Suppose the function ๐ฆ = ๐(๐ฅ) is differentiable. This implies ฮ๐ฆ = ๐ด ฮ๐ฅ + ๐ ฮ๐ฅ. lim ฮ๐ฅโ 0
= ๐ด + lim ฮ๐ฅโ 0
โน if ๐(๐ฅ) is differentiable then ๐โฒ^ ๐ฅ exists and has definite value ๐ด
โฒ ๐ฅ = ๐ด ๐๐ข๐๐๐๐ซ๐๐ง๐ญ๐ข๐๐๐ข๐ฅ๐ข๐ญ๐ฒ โน Existence of Derivative
Geometrical Interpretation of Differentials ฮ๐ฅ or ๐๐ฅ dy ฮy
lim ฮ๐ฅโ 0
โฒ ๐ฅ = ๐ด ๐๐ฆ = ๐ด ๐๐ฅ Note: ๐๐ฆ and ๐๐ฅ measure changes along the tangent line While ฮ๐ฆ and ฮ๐ฅ measure changes for the function ๐(๐ฅ)
Geometrical Interpretation of Differentiability A function ๐ฆ = ๐(๐ฅ) is said to be differentiable at the point ๐(๐ฅ 0 , ๐ฆ 0 ) if it can be approximated in the neighborhood of this point by a linear function. Mathematically, ๐ ๐ฅ = ๐(๐ฅ 0 ) + (๐ฅ โ ๐ฅ 0 ) ๐ด + ๐ ๐ฅ โ ๐ฅ 0 linear function of ๐ฅ Equation of the tangent to the curve ๐ฆ = ๐(๐ฅ) at ๐ฅ 0 , ๐ ๐ฅ 0
Example 1 : Show that the function ๐(๐ฅ) = ๐ฅ^2 is differentiable. Let ๐ฆ = ๐ ๐ฅ = ๐ฅ 2 = 2 ๐ฅ ฮ๐ฅ + ฮ๐ฅ ฮ๐ฅ This implies the given function is differentiable and its derivative is 2 ๐ฅ.
Alternatively, lim ฮ๐ฅโ 0
= 2๐ฅ (^) OR lim ฮ๐ฅโ 0
Example 2 : Given the function ๐ฆ = ๐ฅ^2 , find ฮ๐ฆ and ๐๐ฆ at ๐ฅ = 2 and ฮ๐ฅ = 1 , ฮ๐ฅ = 0. 1 , ฮ๐ฅ = 0. 01. ฮ๐ฆ = ๐ ๐ฅ + ฮ๐ฅ โ ๐(๐ฅ) & ๐๐ฆ = ๐โฒ^ ๐ฅ ๐๐ฅ ๐ซ๐ ๐ซ๐ ๐ ๐ 1 5 4
KEY TAKEAWAY The function ๐ฆ = ๐(๐ฅ) is said to be differentiable at the point (๐ฅ, ๐ฆ) if, at this point where ๐ด is independent of ฮ๐ฅ and ๐ is a function of ฮ๐ฅ such that ๐ โ 0 as ฮ๐ฅ โ 0.
The linear function ๐ด ฮ๐ฅ is called the total differential of ๐ฆ at the point (๐ฅ, ๐ฆ) and is denoted by ๐๐ฆ. The value of ๐ด is the derivative of ๐ at ๐ฅ.
The value of the above limit is called the derivative of ๐ at ๐ฅ. We call a function ๐ฆ = ๐(๐ฅ) differentiable at the point ๐(๐ฅ, ๐ฆ) if lim ฮ๐ฅโ 0
Remark : Note that ๐๐ฆ ๐๐ฅ is not just a notation for ๐โฒ(๐ฅ) but it is a ratio of the two differentials. Therefore writing ๐๐ฅ and ๐๐ฆ alone makes sense. exists. KEY TAKEAWAY
