Calculus: History, Definitions, and the Tangent Line Problem

Calculus was fundamentally investigated and developed by Isaac Newton ( $1642 - 1727$ ) and Gottfried Wilhelm Leibniz ( $1646 - 1716$ ), among other researchers.
The field originated from the pursuit of two central geometric and mathematical questions: - The Tangent Line Problem (Leibniz): What is the slope of a tangent line? A tangent line is defined as a line that "just touches" a curve at a specific point $P$ . - The Area Problem (Newton): What is the area of a region $R$ ? This typically refers to the area contained under a curve between two specific points on the x-axis, such as $a$ and $b$ .
Calculus techniques are essential for solving complex problems across a variety of scientific and professional fields, including: - Physics - Chemistry - Biology - Business

Differential Calculus: - Origins: This branch grew out of the investigation of the Tangent Line Problem (Problem #1). - Core Concept: It relies heavily on the concept of the derivative of a function, which is primarily concerned with the slope of a curve. - Key Applications: - Determining the velocity of a moving object. - Calculating the growth rate of a multiplying population of bacteria. - Finding the rate of change of a company's profit with respect to time.
Integral Calculus: - Origins: This branch was created as a solution to the Area Problem (Problem #2). - Core Concept: It relies on the concept of the antiderivative or the integral of $f(x)$ , which is primarily concerned with the calculation of area.

Rate of change is one of the most critical concepts in calculus. It describes how quickly and in what specific direction (positive or negative) one quantity changes in relation to another quantity.
Rates of Change in Linear Lines (Examples): - For the line $y = x + 1$ , the rate of change (r.o.c.) is $1$ . This means that $y$ increases by $1$ whenever $x$ increases by $1$ . - For the line $y = -2x + 4$ , the rate of change is $-2$ . The line passes through points such as $(0, 4)$ and $(1, 0)$ . This indicates that $y$ decreases by $2$ units for every $1$ unit increase in $x$ . - For the line $y = 3$ , the rate of change is $0$ . The line passes through points like $(0, 3)$ and $(1, 3)$ . In this case, $y$ neither increases nor decreases when $x$ increases by $1$ .
Definition of Slope for a Line: - The rate of change for any linear line is always constant. - Slope formulas include: $ext{Slope} = \frac{ ext{rise}}{ ext{run}} = \frac{ ext{\Delta} y}{ ext{\Delta} x}$ . - Broadly, slope can be defined for any point on any curve as "Steepness with direction." - Magnitude (Absolute Value): Indicates how steep the line is (large vs. small or high vs. low). - Direction: Indicates the orientation (up vs. down).

Linear Functions: - Represented by the general formula $y = mx + b$ . - Only two points are required to define a line. - Examples: $y = 2x + 253$ and $y = 2x - 114$ both have the same steepness/slope of $2$ . In linear functions, the steepness is constant for all values of $x$ .
Non-linear Functions: - In non-linear functions, the slope or rate of change is constantly changing as you move along the curve. - Different points on the curve will exhibit different characteristics: "quite steep positive," "less steep positive," "flat" (no steepness/slope of 0), or "negative slope."

A central question in calculus is: How can we find the slope or rate of change of a function if its slope is constantly changing?
We can estimate or approximate the slope at a single point (known as the instantaneous rate of change) using the average rate of change.
Average Rate of Change (Secant Slope): - The slope between two distinct points on a curve. - Visualized as a "secant line" passing through the curve. - Example: Given a function $f(x)$ with points $(1, 2)$ and $(4, -1)$ , the secant slope is calculated as: $m_{\text{secant}} = \frac{-1 - 2}{4 - 1} = \frac{-3}{3} = -1$ . - This provides an "alright" estimation of the slope at the point where $x = 1$ , as it at least moves in the same negative direction.
Instantaneous Rate of Change (The Derivative): - To find the exact slope of $f(x)$ at a specific point $x = a$ (denoted as $f'(a)$ ), the estimation must be improved. - The exact slope at a specific point is called the derivative of $f(x)$ at $x = a$ . - This derivative also represents the exact slope of the tangent line at the point $(a, f(a))$ .

General Formula for Secant Slope: - The secant line to a function $f(x)$ passing through points $(a, f(a))$ and $(x, f(x))$ has the slope: $m_{\text{sec}} = \frac{f(x) - f(a)}{x - a}$ .
Improving the Estimate: - The secant slope becomes a "better and better" approximation of the tangent slope the closer the second point comes to the first point $x = a$ . - Evaluating points progressively closer to $a$ (e.g., $x_1$ , then $x_2$ , then $x_3$ ) improves accuracy. - Example: A line using point $(x_3, f(x_3))$ to find the secant slope with $x = a$ provides the closest estimate to the tangent line slope (the derivative) if $x_3$ is the point nearest to $a$ .
The Concept of a Limit: - To bring the second point $(x, f(x))$ infinitely close to the point $x = a$ , a new mathematical concept is required called a LIMIT.