🔬 Calculus Foundation
📊 Average Rate of Change
• Measures change over an interval
• Represents secant line slope
• Finite difference approach
• Foundation for derivatives
⚡ Instantaneous Rate of Change
• Measures change at a point
• Represents tangent line slope
• Limit-based approach
• Definition of derivative
🌉 The Bridge to Derivatives
From Average to Instantaneous
🎯 The Fundamental Connection
The derivative is the limit of average rates of change as the interval approaches zero!
📈 Geometric Interpretation
Secant Lines vs Tangent Lines
Secant Line (Average Rate):
- Connects two distinct points on the curve
- Slope = average rate of change over interval
- Represents overall trend between points
Tangent Line (Instantaneous Rate):
- Touches the curve at exactly one point
- Slope = derivative at that point
- Represents instantaneous behavior
🧮 Worked Examples
Example 1: Polynomial Function
Function: f(x) = x³ - 2x² + x
Task: Compare average rates over shrinking intervals around x = 2
f(3) = 27 - 18 + 3 = 12
f(2) = 8 - 8 + 2 = 2
Average rate = (12 - 2) / 1 = 10
f(2.5) = 15.625 - 12.5 + 2.5 = 5.625
f(2) = 2
Average rate = (5.625 - 2) / 0.5 = 7.25
f(2.1) = 9.261 - 8.82 + 2.1 = 2.541
f(2) = 2
Average rate = (2.541 - 2) / 0.1 = 5.41
f'(x) = 3x² - 4x + 1
f'(2) = 3(4) - 4(2) + 1 = 12 - 8 + 1 = 5
Example 2: Exponential Function
Function: f(x) = e^x
Task: Show that average rate approaches derivative at x = 0
(e^h - e^0) / h = (e^h - 1) / h
lim[h→0] (e^h - 1) / h = 1
f'(x) = e^x, so f'(0) = e^0 = 1 ✓
🎯 Advanced Applications
🏃♂️ Physics: Motion
Position: s(t)
Average Velocity: Δs/Δt
Instantaneous Velocity: ds/dt
Average Acceleration: Δv/Δt
Instantaneous Acceleration: d²s/dt²
💰 Economics: Marginal Analysis
Cost Function: C(x)
Average Cost Rate: ΔC/Δx
Marginal Cost: dC/dx
Revenue Optimization: dR/dx = 0
🧬 Biology: Growth Rates
Population: P(t)
Average Growth: ΔP/Δt
Instantaneous Growth: dP/dt
Growth Models: P' = kP
🌡️ Chemistry: Reaction Rates
Concentration: [A](t)
Average Rate: Δ[A]/Δt
Instantaneous Rate: d[A]/dt
Rate Laws: Rate = k[A]^n
📊 Mean Value Theorem
🎯 The Mean Value Theorem
Statement: If f is continuous on [a,b] and differentiable on (a,b), then there exists some c in (a,b) such that:
Interpretation: The instantaneous rate at some point equals the average rate over the interval!
MVT Example
Function: f(x) = x² on [1, 4]
f'(x) = 2x, so 2c = 5
c = 2.5
🔍 Numerical Methods
Approximating Derivatives
When analytical derivatives are difficult, use average rates with small intervals:
⚠️ Common Calculus Pitfalls
🚫 Mistake 1: Confusing Average and Instantaneous
Problem: Using f'(x) when the problem asks for average rate over an interval
Solution: Read carefully - "over interval [a,b]" means average rate
🚫 Mistake 2: Incorrect Limit Notation
Wrong: lim[x→0] instead of lim[h→0]
Right: The variable in the limit should match the one approaching zero
🚫 Mistake 3: Forgetting Domain Restrictions
Problem: Applying MVT without checking continuity/differentiability
Solution: Always verify function properties before applying theorems
🎓 Advanced Practice Problems
Problem 1: For f(x) = sin(x), show that the average rate over [0, π/2] is 2/π and find where the instantaneous rate equals this value.
Problem 2: A particle's position is s(t) = t³ - 6t² + 9t. Find all times when the instantaneous velocity equals the average velocity over [0, 4].
Problem 3: Use the definition of derivative to find f'(x) for f(x) = 1/x by taking the limit of average rates.
Problem 4: For the function f(x) = x^(2/3), explain why the Mean Value Theorem doesn't apply on [-1, 1], but average rate of change can still be calculated.