🔬 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?] (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?] instead of lim[h?]
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.