Chapter 14: Applications of Differentiation

Question 1

Define rate of change and use derivative and second-derivative functions to solve problems

Answer 1

Rate of change:
- dy/dx: rate at which y changes with respect to x
- d²y/dx²: rate at which dy/dx changes

Eg. dy/dx may be positive (indicating that the rate of change always increases), but d²y/dx² may be negative (indciating that the rate of change slows over time)

Question 2

Solve optimisation problems

Answer 2

1) Construct a formula with 1 unknown variable as the subject to be optimised
2) Find the first derivative (f’(x)) and the zeroes of the first derivative (f’(x) = 0)
3) Perform the sign diagram test (f’(x)) to determine if x=a is a local maxima (+ to -) or a local minima (- to +)
OR
3) Use the second derivative to determine if x=a is a local maxima (f”(x) ≤0) or a local minima (f”(x)≥0)
4) Identify optimal solution