To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time The actual change in y is Δ y = f ( a + d x ) - f ( a )

In terms of the quantities, state the information given and the rate to be found Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates

related rates — are rates of change associated with two or more related quantities that are changing over time. actual change in y — Δ y = f ( a + d x ) - f ( a ). concavity test — suppose f is twice differentiable over an interval I ; if f ″ > 0 over I , then f is concave up over I ; if f ″ < 0 over I , then f is concave down over I. differential — the differential d x is an independent variable that can be assigned any nonzero real number; the differential d y is defined to be d y = f ′ ( x ) d x.

Be able to find an equation relating the quantities; use differentiation, applying the chain rule as necessary, to find an equation that relates the rates; find an equation relating the variables.