Chapter 3: Regular Differentiation and its Applications

3.1 Understanding Velocity and Displacement

• When an object travels at a constant velocity:

  • Graph showing displacement as a function of time is linear (straight line).

  • Definition of velocity: It is the rate of change of displacement and can be calculated as the gradient of the linear graph.

• When velocity is not constant:

  • Displacement is not a linear function of time.

  • Slope cannot be calculated in the same manner as for the linear case.

  • Construct a tangent line to the graph at a specific coordinate to determine the velocity.

  • The gradient of the tangent line is referred to as the instantaneous velocity at that coordinate.

  • This concept is foundational to differentiation.

3.1.1 Tools Required: Pascal's Triangle for Expansion

• Pascal's Triangle is used for calculating coefficients in binomial expansions.

• To generate Pascal's Triangle:

  • Start with the base row [1].

  • Each subsequent row is created by adding the two values above, straddling the position to be filled.

• The entries in the rows correspond to coefficients of terms in the expansion of $(a + b)^n$ where $n$ is a positive whole number.

  • Row 0: 1 term (1)

  • Row 1: 2 terms ($a^1 + b^1$)

  • Row 2: 3 terms ($a^2 + 2ab + b^2$)

  • Row 3: 4 terms, etc.

  • In general: Row $n$ will have $n + 1$ terms.

• Example expansions using Pascal's Triangle:

  • Expansion of $(a + b)^2$: $1a^2 + 2ab + 1b^2 = a^2 + 2ab + b^2$

  • Expansion of $(a + b)^3$: $1a^3 + 3a^2b + 3ab^2 + 1b^3 = a^3 + 3a^2b + 3ab^2 + b^3$

3.1.2 The Binomial Expansion

• The binomial expansion is valid for any value of $n$, not just positive whole numbers.

• If $n$ is not a positive whole number, the expansion becomes infinite.

• Finite expansion occurs if $n$ is a positive whole number.

• For very small or bounded values of $x$, the binomial expansion becomes convergent as increasing powers approach zero.

Example: Binomial Expansion Calculation

1. Approximate a calculation using the Binomial expansion:

   • If $f(x) = (1 + x)^{n}$:

   • Substitute into the first few terms for approximation.

   • Example: Approximation when the calculator is broken and calculating $(1+x)^{-1/2}$ for $x$ very small.

   • Follow the steps:

     • Identify $n = -1/2$

     • Expand using the first four terms of Binomial formula:

     • $(1 + x)^{-1/2} ext{ up to 4th term}$

3.1.3 Limits

• Finding Limits:

  • For the function $f(x)$, first substitute $x$ into $f(x)$.

  • Example for limit calculation:

  • If $f(x) = (x^2 - 1)/(x - 1)$.

  • Step to evaluate the limit as $x → 1$:

    • Factor both numerator and denominator, eliminate common factors.

  • The limit approaches $2$ at $x = 1$.

Limit Evaluation Steps

1. Type 0/0 Limit:

   • Factorize and reduce.

   • Example: $ext{limit as } x <br>ightarrow 1$

2. Indeterminate Form Limit:

   • Identify the highest powered term.

   • For $f(x) = (x^2 - 2x + 5)/(2 - 3x^2)$, check dominant terms and simplify accordingly.

   • As $x → ext{inf}$, approach the limit.

3. Simplifying Division by Zero:

   • If limit leads to undefined, simplify.

   • Example: Calculate $ext{limit } rac{1}{s*(1/(2s)-3)}$ at $s=0$, yields $2$.