Related Rates Applications and Introduction to Integration

Conceptual Overview of Related Rates: * The process involves identifying the derivative given and the derivative required by reading the problem carefully. * Variables are often functions of time ( $t$ ), necessitating implicit differentiation.
Example: Rate of Change of the Area of a Circle: * Formula: The area of a circle is defined by the equation $A = \pi r^2$ . * Differentiation Step: When differentiating with respect to time ( $t$ ), we perform implicit differentiation: $\frac{dA}{dt} = 2 \pi r \cdot \frac{dr}{dt}$ . * The constant $\pi$ remains in the term. * The derivative of $r^2$ is $2r$ , which is then multiplied by the derivative of the inside function, $\frac{dr}{dt}$ , according to the chain rule/implicit differentiation. * Numerical Application: * Given: The rate of change of the area ( $\frac{dA}{dt}$ ) is $10$ . * Snapshot: We want to find the rate of change of the radius ( $\frac{dr}{dt}$ ) at the specific moment when the radius $r = 5$ . * Substitution: $10 = 2 \pi (5) \cdot \frac{dr}{dt}$ . * Final Answer: Solving for $\frac{dr}{dt}$ yields approximately $0.32$ .

Variable Definitions: * $n$ : Number of workers or laborers. * $r$ : Number of robots. * $q$ : Level of production (number of pairs of socks manufactured).
Problem Context: * The company currently produces $1,000$ pairs of socks ( $q = 1000$ ). * Production level $q$ is treated as a constant, reducing the formula from three variables to two ( $n$ and $r$ ). * The number of robots is increasing at a rate of one per month ( $\frac{dr}{dt} = 1$ ). * The question asks for the rate at which laborers are being retrenched ( $\frac{dn}{dt}$ ) as robots increase.
The Cobb-Douglas Function Properties: * The function typically features exponents that sum to one (e.g., $0.6$ and $0.4$ ). * Note on Differentiation: While the initial derivative of a Cobb-Douglas function appears messy, the property of exponents adding to one ensures the expression will simplify significantly during algebra. * Differentiation of the Labor Term: The derivative of the first term involving labor is given as $0.6 n^{-0.4} \cdot \frac{dn}{dt}$ .
Data for Calculation: * Current workers ( $n$ ): $20$ . * Rate of robot increase ( $\frac{dr}{dt}$ ): $1$ . * Calculated results might involve fractions (e.g., "two thirds of a worker"), which require practical interpretation in a business context.

Integration Timeline: * Integration is a two-week section of work. * Lectures on integration will continue for a few sessions after next Friday.
Exam Importance: * Integration is a major component of the final exam, as the final exam is the primary opportunity to test this specific section of calculus. * The lecturer emphasizes that despite student fatigue at the end of the semester (due to other subjects like "Ecos"), integration cannot be ignored.
Comparison of Topics: * Simple Interest: Included in the syllabus but often does not feature in the final exam. * Newton's Method: Appears in the final exam approximately 50% of the time, but is guaranteed to be in Class Test 2. * Integration: Guaranteed to feature heavily in the final exam with 100% certainty.

Definition: Integration is the reverse process of differentiation. It is often referred to as anti-differentiation.
The Concept of the Constant of Integration ( $C$ ): * Differentiation "kills" constants (the derivative of a constant is zero). * Functions like $y = x^2 - 1$ and $y = x^2 + 3$ are different formulas but yield the same derivative, $2x$ . * Geometric Interpretation: On a graph, shifting a function vertically (changing the constant) does not change the slope ( $dy/dx$ ) at any corresponding x-value. Because the slopes are identical, the reverse process (integration) cannot distinguish which specific vertical shift was intended without further information. * Notation: When finding the original function from a derivative like $\frac{dy}{dx} = e^x$ , we must write $y = e^x + C$ , where $C$ represents any possible constant.

The Power Rule for Integration: * To integrate a variable with a constant exponent ( $x^n$ ): 1. Add one to the exponent ( $n + 1$ ). 2. Divide by the new exponent ( $n + 1$ ). 3. Include the constant of integration ( $+ C$ ). * Formula: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ .
Constant Multiple Rule: * Constants multiplied by a function just "sit there" and are not removed during the integration process. * Example: Integrating $3x^7$ results in $3 \cdot \frac{x^8}{8} + C$ .
Integrating Constants: * If the derivative is a constant (e.g., $\frac{dy}{dx} = 9$ ), the original function is obtained by multiplying the constant by the variable. * Example: The integral of $9$ is $9x + C$ .
Composite Example: * If $\frac{dy}{dx} = 3x - 1$ , apply rules term-by-term: * Term 1 ( $3x$ ): $\frac{3x^2}{2}$ . * Term 2 ( $-1$ ): $-x$ (or $-1 \cdot \frac{x^1}{1}$ ). * Result: $y = \frac{3x^2}{2} - x + C$ .