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ACMMM107

1.6 Estimating Derivatives (Not Required)

  • Any materials in this notes is not required in the scope of the course, but is included for your understanding.

Estimating Derivatives

  • After an extensive study of limit notations, continuity and differentiability, it is clear the derivatives is simply a limit notation, and there can be estimated as the value as x approaches a certain value.
  • This is useful when the function given is too difficult to obtain its derivative from first principles, or by directly differentiating (however, after “Derivatives of Functions” and “Differentiation Rules” sections in A3 – Calculus this problem should not exist anymore).

Example

If f(x)=3^{x}, find f^{\prime}(2.4), accurate to 3 decimal places

As a sneak peak, the answer is 3^{2.4} \ln 3=15.344(3 \mathrm{~d} \cdot \mathrm{p} \cdot)

We can estimate f^{\prime}(2.4) numerically by the following:

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1.5 Continuity and Differentiability (Not Required)

  • Any materials in this notes is not required in the scope of the course, but is included for your understanding.

Continuity at a Point

  • We only require an intuitive understanding of continuity. A function with rule f(x) is said to be continuous at x=a if the graph of y=f(x) can be drawn through the point with coordinates (a, f(a)) without a break. Otherwise, there is said to be a discontinuity at x=a.
  • Mathematically and formally, continuity of a point is assured by the following:

i) f(x) is defined at x=a

ii) \lim _{x \rightarrow a} f(x)=f(a)

Example

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1.4 Algebra of Limits (Not Required)

  • Any materials in this notes is not required in the scope of the course, but is included for your understanding.

Algebra of Limits

  • Below are some properties of limits, assuming both \lim _{x \rightarrow a}f(x) and \lim _{x \rightarrow a}g(x) exists:

i) Sum:

\lim _{x \rightarrow a}[f(x)+g(x)]=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a}g(x)

ii) Difference:

\lim _{x \rightarrow a}[f(x)-g(x)]=\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)

iii) Multiple:

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