Gradient as Rate of Change

Introduction to Gradient as a Rate of Change

Introduction to Gradient

• Gradient: Reflects the steepness or slope of a curve, indicating how one variable changes concerning another.
• Rate of Change: Represents the speed at which a change occurs in a quantity.
  • Example: A car accelerates from 0 to 60 km/h in 10 seconds, yielding an average rate of 6 km/h per second.
• Average Rate of Change: The rate evaluated over a defined interval.
• Instantaneous Rate of Change: The change at a specific moment, closely associated with derivatives.

Key Definitions

• Gradient: An indicator of steepness or slope.
• Rate of Change: The velocity at which a quantity transforms over time.

Historical Context and Importance

• The development of calculus by Sir Isaac Newton and Gottfried Wilhelm Leibniz significantly advanced change calculations.
• Gradients and rates of change hold critical roles across multiple disciplines:
  • Physics: Understanding movement.
  • Economics: Analysing costs and benefits.
  • Biology: Forecasting growth patterns.

Difference Quotient

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$ evaluates the average rate of change over the interval $[x, x+h]$.
• It provides an estimation of the slope along the secant line connecting two points on a curve.

Practical Applications

• Areas of application include variations in speed, changes in plant growth, and tracking of resource usage.

Mathematical Interpretation

• Secant Line: A line that intersects two points on a function, useful for examining average trends.
• See Diagram: Points $(x, f(x)), (x+h, f(x+h))$, joined by a secant line.

Worked Examples

Linear Functions

• Example Calculation for $f(x) = 2x + 3$:
  • Step-by-Step: Utilise the Difference Quotient $\frac{f(a + h) - f(a)}{h}$.
  • Solution: $\frac{2a + 2h + 3 - 2a - 3}{h} = \frac{2h}{h} = 2$

Non-linear Functions

Quadratic Function

• Examination of $f(x) = x^2 + 4x + 4$:
  • Calculation: $\frac{f(a+h) - f(a)}{h} = \frac{(a+h)^2 + 4(a+h) + 4 - (a^2 + 4a + 4)}{h}$ $= \frac{a^2 + 2ah + h^2 + 4a + 4h + 4 - a^2 - 4a - 4}{h}$ $= \frac{2ah + h^2 + 4h}{h}$ $= 2a + h + 4$

Graphical Interpretation and Analysis

Techniques for Plotting Functions and Derivatives

Function and Derivative Plotting

• Investigate various functions, including linear and quadratic forms.
  • Quadratic: $f(x) = x^2$
  • Plot using key points for enhanced comprehension.

Comparing Graphical and Analytical Solutions

• Integrating both techniques deepens comprehension of mathematical results.
  • Scenario: Product cost reduction analysed both graphically and analytically.

Problem Set

• Beginner: Tackle basic linear functions.

  • Find the gradient of $f(x) = 3x - 2$ using the difference quotient.
  • Solution: $\frac{f(x+h) - f(x)}{h} = \frac{3(x+h)-2-(3x-2)}{h} = \frac{3x+3h-2-3x+2}{h} = \frac{3h}{h} = 3$

• Intermediate: Work on quadratic functions $f(x) = 2x^2 + 6x + 3$.

  • Calculate the rate of change using the difference quotient.
  • Solution: $\frac{f(x+h) - f(x)}{h} = \frac{2(x+h)^2 + 6(x+h) + 3 - (2x^2 + 6x + 3)}{h}$ $= \frac{2x^2 + 4xh + 2h^2 + 6x + 6h + 3 - 2x^2 - 6x - 3}{h}$ $= \frac{4xh + 2h^2 + 6h}{h}$ $= 4x + 2h + 6$

• Advanced: Find the gradient function for $f(x) = x^3 - 2x^2 + 4x - 1$.

  • Solution: $\frac{f(x+h) - f(x)}{h} = \frac{(x+h)^3 - 2(x+h)^2 + 4(x+h) - 1 - (x^3 - 2x^2 + 4x - 1)}{h}$

  Expanding the terms: $= \frac{x^3 + 3x^2h + 3xh^2 + h^3 - 2x^2 - 4xh - 2h^2 + 4x + 4h - 1 - x^3 + 2x^2 - 4x + 1}{h}$

  Simplifying: $= \frac{3x^2h + 3xh^2 + h^3 - 4xh - 2h^2 + 4h}{h}$ $= 3x^2 + 3xh + h^2 - 4x - 2h + 4$

  As $h$ approaches 0, the gradient function is: $f'(x) = 3x^2 - 4x + 4$