Mathematics - Applications and Interpretation | Calculus

🚀 Calculus: The Mathematics of Change (Applications and Interpretation)

Welcome to the exciting world of Calculus! Don't worry if this sounds intimidating; Calculus is simply the study of how things change. In Mathematics: Applications and Interpretation (AI), we focus less on abstract proofs and more on using these powerful tools to model and understand real-world phenomena—from predicting disease spread to optimizing business profit.

If you can confidently use your GDC (Graphing Display Calculator) to find roots, calculate areas, and analyze graphs, you're already halfway there!

Section 1: The Core Concept – Rates of Change

1.1 Average Rate of Change vs. Instantaneous Rate of Change

Imagine you're driving 100 km in 2 hours. Your average rate of change (average speed) is 50 km/h. But were you going 50 km/h the entire time? Probably not! Sometimes you sped up, sometimes you slowed down.

Calculus allows us to find the instantaneous rate of change—your exact speed at the moment you pass a specific kilometer marker.

• Average Rate of Change: Calculated over an interval. This is the familiar formula for the gradient of a secant line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\):
  $$ \text{Average Rate} = \frac{y_2 - y_1}{x_2 - x_1} $$
• Instantaneous Rate of Change: Calculated at a single point. This is the gradient of the tangent line at that point. We find this using differentiation.

Key Takeaway: Differentiation is the process of moving from average change (secant line) to instantaneous change (tangent line) by making the interval incredibly small (approaching zero).

Section 2: Differentiation (Finding the Slope)

Differentiation (or finding the derivative) gives us a formula, \(f'(x)\), which calculates the instantaneous gradient for any point on the original function, \(f(x)\).

2.1 Notation and Basic Rules

We use different ways to write the derivative:

• If the function is \(y = f(x)\), the derivative is \(f'(x)\) or \(\frac{dy}{dx}\).
• The operation of differentiating with respect to \(x\) is written as \(\frac{d}{dx}\).

While AI students often use the GDC for complex differentiation, knowing these basic rules helps you understand the underlying structure:

1. The Power Rule (The Most Important Rule):
If \(f(x) = ax^n\), then \(f'(x) = n a x^{n-1}\).
(Bring the power down and multiply, then reduce the power by 1.)

Example: If \(y = 4x^3\), then \(\frac{dy}{dx} = 4 \times 3 x^{3-1} = 12x^2\).

2. Derivatives of Standard Functions (AI Focus):

• Exponential: If \(f(x) = e^{kx}\), then \(f'(x) = k e^{kx}\).
• Natural Log: If \(f(x) = \ln(ax)\), then \(f'(x) = \frac{1}{x}\). (Yes, the \(a\) disappears!)
• Constant: If \(f(x) = c\) (a number), then \(f'(x) = 0\). (A horizontal line has zero slope.)

💡 Technology Note: For functions that require the Chain Rule, Product Rule, or Quotient Rule (especially required for HL students), you will rely heavily on your GDC’s differential calculus functions unless the specific question requires manual application.

2.2 Applications of the Derivative

The true power of Calculus in AI lies in its applications:

A. Finding Tangents and Normals

The derivative \(f'(a)\) gives the gradient of the tangent line at \(x=a\).

• Tangent Gradient: \(m_{tan} = f'(a)\)
• Normal Gradient: The normal line is perpendicular to the tangent. Its gradient is the negative reciprocal: \(m_{norm} = -\frac{1}{f'(a)}\).

B. Kinematics (Motion Modelling)

In physics and modelling, differentiation links position, velocity, and acceleration.

• Let \(s(t)\) be the Position (or displacement) at time \(t\).
• Velocity is the rate of change of position: \(v(t) = s'(t)\).
• Acceleration is the rate of change of velocity: \(a(t) = v'(t) = s''(t)\) (the second derivative).

C. Optimization (Finding Maxima and Minima)

This is critical for AI. Optimization is used to find the maximum profit, minimum cost, or maximum volume.

1. Find the derivative: Find \(f'(x)\).
2. Set \(f'(x) = 0\): Solving this equation gives you the \(x\)-values of the stationary points (where the slope is zero). *Use your GDC’s solver/root function here!*
3. Check Endpoints/Context: For real-world problems, always check the values at the boundaries of the domain (e.g., if \(0 \le x \le 5\)).
4. Verify Max/Min: Use the second derivative test or analyze the graph:
   • If \(f''(x) > 0\), it’s a Minimum (concave up, looks like a bowl ∪).
   • If \(f''(x) < 0\), it’s a Maximum (concave down, looks like a hill ∩).

Section 3: Integration (Finding the Accumulation)

Integration is the reverse process of differentiation. It serves two main purposes:

1. Finding the original function from its derivative (Indefinite Integral).
2. Finding the area under a curve (Definite Integral).

3.1 Indefinite Integrals (Antiderivatives)

When we integrate \(f(x)\), we find its antiderivative, \(F(x)\).

Notation: \(\int f(x) \, dx = F(x) + C\)

The Constant of Integration (\(+ C\)):
When you differentiate, any constant disappears (\(\frac{d}{dx}(x^2+5) = 2x\); \(\frac{d}{dx}(x^2-10) = 2x\)). Since we don't know what that original constant was, we must always add \(+ C\) when performing indefinite integration. This is often solved using initial conditions (e.g., "when \(x=1, y=5\)") to find the specific value of \(C\).

Reverse Power Rule:
If \(f(x) = ax^n\), then \(\int f(x) \, dx = \frac{a x^{n+1}}{n+1} + C\) (as long as \(n \neq -1\)).
(Increase the power by 1, then divide by the new power.)

Example: \(\int 6x^2 \, dx = \frac{6x^3}{3} + C = 2x^3 + C\).

Reverse Standard Functions:

• \(\int e^{kx} \, dx = \frac{1}{k} e^{kx} + C\)
• \(\int \frac{1}{x} \, dx = \ln|x| + C\) (This is the special case where \(n=-1\))

3.2 Definite Integrals (Area and Accumulation)

The Definite Integral is used when we calculate the integral between two defined boundaries, \(a\) and \(b\).

Notation: \(\int_a^b f(x) \, dx\)

The Fundamental Theorem of Calculus:
$$ \int_a^b f(x) \, dx = [F(x)]_a^b = F(b) - F(a) $$ (Where \(F(x)\) is the antiderivative of \(f(x)\)).

Did you know? When calculating a definite integral, the \(+ C\) constant cancels itself out (\((F(b)+C) - (F(a)+C) = F(b) - F(a)\)). This is why we only need the constant \(C\) for indefinite integrals!

3.3 Applications of Definite Integration

A. Area Under a Curve

The definite integral calculates the net signed area between the function \(f(x)\) and the \(x\)-axis from \(x=a\) to \(x=b\).

• If the curve is above the \(x\)-axis, the area is positive.
• If the curve is below the \(x\)-axis, the integral result is negative.
• To find the total area (ignoring negative results), you must split the integral at any \(x\)-intercepts and take the absolute value of the sections below the axis.

B. Area Between Two Curves

If \(f(x)\) is the top function and \(g(x)\) is the bottom function over the interval \([a, b]\): $$ \text{Area} = \int_a^b (f(x) - g(x)) \, dx $$

C. Kinematics (The Reverse Process)

Integration allows us to move "up" the kinematics chain:

• \(\int a(t) \, dt = v(t) + C\) (Velocity from Acceleration).
• \(\int v(t) \, dt = s(t) + C\) (Position/Displacement from Velocity).
• Displacement: \(\int_{t_1}^{t_2} v(t) \, dt\) (Net change in position).
• Total Distance Traveled: \(\int_{t_1}^{t_2} |v(t)| \, dt\) (You must use the absolute value of velocity to count movement backwards as positive distance).

D. Volumes of Revolution (HL Specific Topic)

(For HL students and advanced applications only) If a curve \(y=f(x)\) is rotated around the \(x\)-axis, the volume \(V\) generated is given by: $$ V = \pi \int_a^b (f(x))^2 \, dx $$