## Topic Overview

**Main Concept/Theme:**

Calculus is the branch of mathematics that deals with continuous change. It is divided into two main branches: differential calculus and integral calculus. In grade 12, students primarily focus on differential calculus.

**Key Learning Objectives:**

– Understand the concept of a limit and continuity.

– Differentiate various functions using established rules.

– Apply differentiation to solve problems involving rates of change, tangents, and optimization.

– Understand basic integration and its applications.

## Key Terms and Definitions

**Limit:**The value that a function approaches as the input approaches a certain value.**Continuity:**A function is continuous if there are no breaks or holes in its graph.**Derivative:**The slope of the tangent to the graph of a function at a point; measures the rate of change.**Differentiation:**The process of finding a derivative.**Function:**A relationship where each input corresponds to exactly one output.**Integration:**The reverse process of differentiation; finding the area under a curve.

## Main Content Sections

### 1. Limits and Continuity

**Limits:**

– A limit is written as (\lim_f(x) = L), meaning as (x) approaches (c), (f(x)) approaches (L).

– Example: (\lim_(2x + 1) = 7)

**Continuity:**

– A function (f(x)) is continuous at (x = c) if (\lim_f(x) = f(c)).

– If there is a jump, hole, or vertical asymptote at (c), the function is not continuous at (c).

### 2. Differentiation Rules

**Basic Rules:**

– Constant Rule: ( \frac{d}{dx} [c] = 0 )

– Power Rule: ( \frac{d}{dx} [x^n] = nx^{n-1} )

– Sum Rule: ( \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) )

– Product Rule: ( \frac{d}{dx} [u \cdot v] = u’v + uv’ )

– Quotient Rule: ( \frac{d}{dx} \left[ \frac{u}{v} \right] = \frac{u’v – uv’}{v^2} )

– Chain Rule: ( \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) )

### 3. Applications of Differentiation

**Tangents and Normals:**

– The derivative ( f'(x) ) at ( x = a ) gives the slope of the tangent line to the curve at that point.

– Equation of the tangent: ( y = f(a) + f'(a)(x – a) )

**Rates of Change:**

– Velocity as the derivative of position with respect to time.

– Acceleration as the derivative of velocity with respect to time.

**Optimization:**

– Finding maxima and minima by setting the derivative equal to zero ( f'(x) = 0 ) and solving for ( x ).

– Determine whether it’s a max or min by the second derivative test ( f”(x) ).

### 4. Basic Integration

**Antiderivatives and Indefinite Integrals:**

– The antiderivative of ( f(x) ) is a function ( F(x) ) such that ( F'(x) = f(x) ).

– Example: ( \int x^n dx = \frac{x^{n+1}}{n+1} + C )

**Definite Integrals:**

– Represent the area under the curve from ( a ) to ( b ): ( \int_{a}^{b} f(x) dx ).

– Example: ( \int_{1}^{4} x dx = \left[ \frac{x^2}{2} \right]_{1}^{4} = 8.5 )

## Example Problems or Case Studies

### Example Problem 1: Differentiation

Find the derivative of ( f(x) = 3x^3 – 5x^2 + 2x – 4 ).

**Solution:**

[ f'(x) = 9x^2 – 10x + 2 ]

### Example Problem 2: Tangent Line

Find the equation of the tangent to the curve ( y = x^2 + 3x + 2 ) at ( x = 1 ).

**Solution:**

[ f'(x) = 2x + 3 ]

[ f'(1) = 5 ]

[ y = 6 ]

Equation of the tangent: ( y – 6 = 5(x – 1) )

[ y = 5x + 1 ]

## Summary or Review Section

**Limits and Continuity:**Determine the value a function approaches as the input nears a point; ensure no gaps or jumps in a function.**Differentiation:**Find the rate of change and apply rules such as power, product, quotient, and chain rules.**Applications:**Use derivatives to find tangents, rates of change, and optimize functions.**Integration:**Reverse process of differentiation to find antiderivatives and calculate the area under curves.

## Self-Assessment Questions

- Evaluate (\lim_(x^2 – 4)).
- Differentiate ( f(x) = 4x^5 – 3x^3 + x – 7 ).
- Find the derivative of ( g(x) = (2x^2 + 3)(x^3 – 4) ) using the product rule.
- Determine the tangent line to the curve ( y = x^3 – 2x + 1 ) at ( x = 2 ).
- Integrate ( \int (3x^2 – 4x + 5) dx ).

## Connections to Other Topics/Subjects

**Physics:**Calculus is used to describe motion—velocity and acceleration are derivatives of position with respect to time.**Economics:**Optimization in calculus is used to maximize profit or minimize cost.**Biology:**Calculus models population growth and rates of change in biological processes.**Engineering:**Engineering uses calculus for designing and analyzing systems and structures.

Remember to test your understanding regularly and ask for help when needed. Happy studying!