Calculus and Rates of Change: Slopes, Accumulation, and Interpretation
Core concepts
At foundation level, the essential ideas are average and instantaneous rate of change, the derivative as a limit of slopes of secants, differentiation of polynomials, and the definite integral as accumulated change. You should connect a graph's slope to the sign and size of the derivative, and connect area under a rate graph to a total quantity.
Differentiation rules you need
- Power rule:
d/dx (x^n) = n x^(n-1). - Constant multiples and sums: differentiate term by term.
- The derivative of a constant is zero.
Traps to watch
- Confusing average rate of change with the derivative. The average rate over
[a, b]is(f(b) - f(a))/(b - a); the derivative is the limit as the interval shrinks. - Forgetting the units of a rate: metres per second, litres per minute. A numerical answer without units is incomplete.
- Assuming a zero derivative means a maximum. It may be a minimum or a stationary inflection; check the sign change.
- Sign errors in the power rule when the exponent is negative or fractional.
Worked example
A tank's volume is V(t) = 3t^2 + 2t litres after t minutes. Find the rate at t = 4.
Differentiate: V'(t) = 6t + 2. At t = 4, V'(4) = 26 litres per minute. The average rate from 0 to 4 minutes is (56 - 0)/4 = 14 litres per minute - clearly different, which is exactly the distinction examiners test.
Rule of thumb
Name the quantity, name the units, then differentiate or integrate. If the question says "rate", differentiate; if it says "total" or "accumulated", integrate.