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1. What is Calculus? An Overview

Calculus is broadly divided into two interconnected branches:

• Differential Calculus: Deals with rates of change and slopes of curves. It helps us understand how quickly something is changing at a particular instant. Key concepts include derivatives.
  Think: How fast is a car going *right now*? What is the steepest point on a hill?
• Integral Calculus: Deals with accumulation of quantities and areas under curves. It helps us find the total amount of something when its rate of change is known. Key concepts include integrals.
  Think: How far has a car traveled given its speed over time? What is the total volume of an irregularly shaped object?

These two branches are linked by the Fundamental Theorem of Calculus, which shows that differentiation and integration are inverse operations.

Calculus was independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, revolutionizing mathematics and its applications to the physical world.

2. Limits: The Foundation of Calculus

The concept of a limit is the cornerstone upon which all of calculus is built. A limit describes the value that a function "approaches" as the input (variable) gets closer and closer to a certain value. It's about what happens *near* a point, not necessarily *at* the point itself.

Formal Notation:

For a limit to exist, the function must approach the same value from both the left side (values less than $a$) and the right side (values greater than $a$).

Evaluating Simple Limits (Direct Substitution):

For many continuous functions (functions without breaks or jumps), you can find the limit by simply substituting the value $x$ is approaching into the function.

Limits with Indeterminate Forms (Briefly):

Sometimes, direct substitution leads to indeterminate forms like $\frac{0}{0}$. In such cases, algebraic manipulation (like factoring or rationalizing) is often required before substitution.

Limits are essential for defining continuity, derivatives, and integrals.

3. Differential Calculus: Rates of Change and Slopes

Differential calculus is concerned with the concept of the derivative, which measures the instantaneous rate of change of a function. Geometrically, the derivative of a function at a point is the slope of the tangent line to the function's graph at that point.

The Derivative as a Slope:

For a linear function $y = mx + b$, the slope $m$ is constant. For a non-linear function, the slope changes at every point. The derivative gives us this instantaneous slope.

Definition of the Derivative (Limit Definition):

The derivative of a function $f(x)$, denoted as $f'(x)$ or $\frac{dy}{dx}$, is defined using a limit:

Basic Differentiation Rules:

While the limit definition is fundamental, we use rules to find derivatives more efficiently.

• Constant Rule: The derivative of a constant function is zero.
  If $f(x) = c$ (where $c$ is a constant), then $f'(x) = 0$.
  Example: If $f(x) = 7$, then $f'(x) = 0$.
• Power Rule: To differentiate $x$ raised to a power, bring the power down as a coefficient and subtract 1 from the power.
  If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
  Example 1: If $f(x) = x^3$, then $f'(x) = 3x^{3-1} = 3x^2$.
  Example 2: If $f(x) = x$, then $f'(x) = 1x^{1-1} = 1x^0 = 1$.
  Example 3: If $f(x) = \frac{1}{x} = x^{-1}$, then $f'(x) = -1x^{-1-1} = -x^{-2} = -\frac{1}{x^2}$.
• Constant Multiple Rule: If a function is multiplied by a constant, the constant remains as a multiplier of the derivative.
  If $f(x) = c \cdot g(x)$, then $f'(x) = c \cdot g'(x)$.
  Example: If $f(x) = 5x^2$, then $f'(x) = 5 \cdot (2x^{2-1}) = 5 \cdot 2x = 10x$.
• Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives.
  If $f(x) = g(x) \pm h(x)$, then $f'(x) = g'(x) \pm h'(x)$.
  Example: If $f(x) = 4x^3 - 2x + 1$, then $f'(x) = 4(3x^2) - 2(1) + 0 = 12x^2 - 2$.

Applications of Derivatives:

• Rates of Change: Velocity is the derivative of position with respect to time; acceleration is the derivative of velocity.
• Optimization: Finding maximum or minimum values of functions (e.g., maximizing profit, minimizing cost).
• Curve Sketching: Determining where a function is increasing or decreasing, and finding local maxima/minima.

4. Integral Calculus: Accumulation and Area

Integral calculus deals with the concept of the integral, which is essentially the reverse process of differentiation (finding the antiderivative) and also a method for calculating the accumulation of quantities, such as the area under a curve.

Antiderivatives:

If $F'(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$. Since the derivative of a constant is zero, the antiderivative of a function is not unique; it includes an arbitrary constant of integration, denoted by $C$.

Indefinite Integral (Antiderivative):

Basic Integration Rule (Power Rule for Integration):

To integrate $x$ raised to a power, increase the power by 1 and divide by the new power.

Definite Integral (Area Under a Curve):

A definite integral calculates the net signed area between a function's graph and the x-axis over a specific interval $[a, b]$.

Applications of Integrals:

• Area: Calculating the area of complex shapes or regions under curves.
• Volume: Finding the volume of solids.
• Total Change: If you know the rate at which a quantity is changing, integration can tell you the total change in that quantity over a period. (e.g., total distance traveled from velocity).
• Average Value: Calculating the average value of a function over an interval.

5. The Fundamental Theorem of Calculus: The Bridge Between Branches

The Fundamental Theorem of Calculus (FTC) is a monumental result that connects differential calculus and integral calculus. It establishes that differentiation and integration are inverse operations, much like addition and subtraction, or multiplication and division.

Part 1 of the FTC:

If $f$ is continuous on $[a, b]$, then the function $F(x)$ defined as:

This part states that the derivative of an integral is the original function.

Part 2 of the FTC (Evaluation Theorem):

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ on $[a, b]$, then:

This part provides a powerful method for evaluating definite integrals by simply finding an antiderivative and evaluating it at the upper and lower limits. This avoids the tedious process of using Riemann sums (summing infinitely many small rectangles to find area).

The FTC is the cornerstone of calculus, enabling the vast array of applications seen in science and engineering.

Practice Problems: Test Your Calculus Knowledge

Apply the concepts you've learned in this introductory lesson to solve the following practice problems.

1. Limits (Direct Substitution): Find $\lim_{x \to 4} (2x^2 - 5x + 3)$.
2. Limits (Factoring): Find $\lim_{x \to -1} \frac{x^2 - 1}{x + 1}$.
3. Derivative (Power Rule): Find the derivative of $f(x) = x^5$.
4. Derivative (Constant Multiple & Sum Rule): Find the derivative of $g(x) = 3x^4 - 7x^2 + 6x - 9$.
5. Antiderivative (Indefinite Integral): Find $\int (x^3 + 2x - 1) \, dx$.
6. Definite Integral (FTC Part 2): Evaluate $\int_{0}^{2} (3x^2) \, dx$.
7. Conceptual Question: What does the derivative of a position function represent?
8. Conceptual Question: What does the definite integral of a velocity function represent?
9. True or False: The limit of a function at a point always exists if the function is defined at that point.
10. Challenge (Derivative Definition): Use the limit definition of the derivative to find $f'(x)$ for $f(x) = x^2$.

(Solutions are not provided here, encouraging self-assessment, peer discussion, or seeking further assistance.)