Comprehensive Notes on Advanced Mathematics Differentiation

Syllabus Outcomes and Core Competencies for Differentiation

• MAV-11-06: Proficiency in interpreting the meaning of the derivative and determining the derivative of functions to solve diverse problems.
• MAO-WM-01: Development of understanding and fluency through exploring and connecting mathematical concepts, applying mathematical techniques, and communicating reasoning clearly.

Concepts of Change: Estimating and Defining Rates

• Average Rate of Change: For a function $y = f(x)$ over the domain $[a, b]$, the average rate of change is defined as the change in $y$ divided by the change in $x$:     $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$
• Secant Gradient: The value $\frac{f(b) - f(a)}{b - a}$ is recognized geometrically as the gradient of the secant line passing through points $(a, f(a))$ and $(b, f(b))$ on the graph of $y = f(x)$.
• Speed as a Rate: Speed is defined as the rate of change of distance with respect to time.
• Instantaneous Speed: The speed of an object at a specific time $t$ is the limit of the average speed as the time interval approaches zero. It is approximated by identifying the average speed between position at time $t$ and a position some time later ($t + h$), where the approximation improves as $h$ decreases.
• Linear Modelling: In a linear function, the gradient represents a constant rate of change.
• Non-Linear Modelling: In non-linear functions, the rate of change is not constant and is represented by the gradient of the tangent to the curve at any specific point.

Numeric Estimation of Instantaneous Rates

• Cyclist Scenario: Consider a cyclist whose distance from the top of a hill is given by $d(t) = t^2$, where $d$ is in metres and $t$ is in seconds.
• Step 1 (Average Speed): Between $t = 1$ and $t = 3$:     $\text{Average Speed} = \frac{d(3) - d(1)}{3 - 1} = \frac{9 - 1}{2} = 4\,m/s$
• Step 2 (Shrinking the Interval): To find the speed at exactly $t = 2$, calculate the average speed for shrinking values of $h$:     $\text{Average Speed} = \frac{d(2 + h) - d(2)}{h}$
  • For $h = 1$ ($t = 2$ to $t = 3$): $(9 - 4) \div 1 = 5\,m/s$
  • For $h = 0.5$ ($t = 2$ to $t = 2.5$): $(6.25 - 4) \div 0.5 = 4.5\,m/s$
  • For $h = 0.1$ ($t = 2$ to $t = 2.1$): $(4.41 - 4) \div 0.1 = 4.1\,m/s$
  • For $h = 0.01$ ($t = 2$ to $t = 2.01$): $(4.0401 - 4) \div 0.01 = 4.01\,m/s$
• Conclusion: As $h \to 0$, the average speed approaches $4\,m/s$, which is the instantaneous speed at $t = 2$. This value corresponds to the gradient of the tangent at that point.

Formal Definitions and Notation of the Derivative

• The Derivative Function: Defined as $f'(x)$, the derivative represents the gradient of the tangent to the curve $y = f(x)$ at any point $(x, f(x))$, provided the tangent exists and is not vertical.
• Differentiation: The mathematical process of finding the derivative of a function.
• First Principles Definition: The derivative is the limiting value of the gradient of the secant as the interval $h$ approaches zero:     $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
• Notation Variants:
  • $\frac{dy}{dx}$ and $y'$ when $y$ is a function of $x$.
  • $\frac{d}{dx}[f(x)]$ and $f'(x)$ for a function $f(x)$.
  • $\frac{dx}{dt}$ or $\dot{x}$ for velocity (displacement $x$ as a function of time $t$).
• Differentiability Requirements: A function is differentiable at $x = a$ if and only if:
  1. It is smooth and continuous at $x = a$.
  2. The limit $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ exists and is finite.

Rules for Differentiation

• Constant Functions: If $f(x) = c$, then $f'(x) = 0$.
• Linear Functions: If $f(x) = mx + c$, then $f'(x) = m$.
• The Power Rule: For any real number $n$, if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
• Sum/Difference Rule: The derivative of a sum is the sum of the derivatives:     $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
• Constant Multiple Rule: The derivative of a multiple of a function is the multiple of its derivative:     $\frac{d}{dx}[k \cdot f(x)] = k \cdot f'(x)$
• The Chain Rule (Function of a Function): Used for composite functions. If $y$ is a function of $u$ and $u$ is a function of $x$:     $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$     Alternatively, if $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$.     Standard form for powers of a function: If $y = [f(x)]^n$, then $\frac{dy}{dx} = n \cdot f'(x) \cdot [f(x)]^{n-1}$.
• The Product Rule: Used for the product of two functions $u$ and $v$:     $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$     Also expressed as: $(uv)' = uv' + vu'$.
• The Quotient Rule: Used for the division of two functions $u$ and $v$:     $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$     Also expressed as: $(\frac{u}{v})' = \frac{vu' - uv'}{v^2}$.

Distance, Displacement, Speed, and Velocity

• Distance: The total path length travelled regardless of direction. This value is always non-negative ($\ge 0$) and never decreases over time.
• Displacement: The straight-line distance and direction from a fixed origin. It can be positive (right/forward), negative (left/backward), or zero (at the origin).
• Speed: The rate of change of distance. It is always positive or zero. Instantaneous speed is the magnitude (absolute value) of instantaneous velocity.
• Velocity: The rate of change of displacement. It includes direction; positive indicates motion in the positive direction, and negative indicates motion in the opposite direction.
• Average vs. Instantaneous Velocity:
  • $\text{Average Velocity} = \frac{\text{Change in Displacement}}{\text{Change in Time}} = \frac{x_2 - x_1}{t_2 - t_1}$
  • $\text{Instantaneous Velocity} = \frac{dx}{dt}$

Geometric Applications of the Derivative

• Angle of Inclination: The steepness of a tangent line can be expressed as an angle $\theta$ with the positive $x$-axis. The relationship is governed by:     $\tan(\theta) = m$     where $m$ is the gradient (the derivative at that point).
• Equations of Tangents: Once the gradient $m$ at point $(x_1, y_1)$ is found via the derivative, the equation of the tangent line is:     $y - y_1 = m(x - x_1)$
• Normals to a Curve: The normal is the line perpendicular to the tangent at the point of contact. Its gradient $m_{\text{normal}}$ is the negative reciprocal of the tangent gradient $m_{\text{tangent}}$, calculated as:     $m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}}$     The equation of the normal is: $y - y_1 = -\frac{1}{m}(x - x_1)$.

Increasing, Decreasing, and Stationary Functions

• Increasing Function: A curve is increasing at $x = a$ if $f'(a) > 0$. The tangent slopes upwards.
• Decreasing Function: A curve is decreasing at $x = a$ if $f'(a) < 0$. The tangent slopes downwards.
• Stationary Point: A curve is stationary at $x = a$ if $f'(a) = 0$. The tangent is horizontal.
• Justification of Behavior (NESA Standards): Conclusions about function shape must be explained in words based on the sign of $f'(x)$.
  • If $f'(x) > 0$, the curve is increasing.
  • If $f'(x) < 0$, the curve is decreasing.
  • If $f'(x) = 0$, the point is stationary.
• Turning Points: A stationary point is a maximum turning point if $f'(x)$ changes from positive to negative. It is a minimum turning point if $f'(x)$ changes from negative to positive.
• Stationary Point of Inflection: This occurs if $f'(a) = 0$ but the sign of $f'(x)$ is the same on both sides of $x = a$ (e.g., $f(x) = x^3$ at $x = 0$).

Physical Applications of Rates of Change

• Temperature (Case Study): If $T$ is temperature and $t$ is time, $\frac{dT}{dt}$ represents how fast the temperature is changing. A patient's temperature reaches a maximum when $\frac{dT}{dt} = 0$.
• Population Growth: A cockroach population $P = 7 + 6t - t^2$ (in millions) has a rate of change found by differentiation: $\frac{dP}{dt} = 6 - 2t$.
  • Population is increasing when $\frac{dP}{dt} > 0$.
  • Maximum population occurs when $\frac{dP}{dt} = 0$.
• Fluid Dynamics: Volume $V$ of fuel emptying from a tanker is given by $Q = 200(400 - t^2)$. The rate of emptying is $\frac{dQ}{dt} = -400t$.
• Projectile Motion: Height $h$ of a stone dropped from a building is $h = 80 - 5t^2$.
  • Building height is the initial height at $t = 0$.
  • Velocity is $\frac{dh}{dt} = -10t$. Speed at impact is calculated by finding the time when $h = 0$.