5.9 Differentiation rules and related ratesIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 5.9 Differentiation rules and related rates

1. The depth of water, $h$ metres, in a harbour $t$ hours after midnight is modelled by $h(t)=6+2\sin(0.5t)$ for $0\le t\le 12$, where the angle is in radians.
   Find the first time after midnight at which the depth is greatest, and state the greatest depth.2 marks
2. The concentration of a drug in a patient's blood, $C$ mg l$^{-1}$, is modelled by $C(t)=\dfrac{20t}{t^2+4}$ for $t\ge0$, where $t$ is the time in hours after the drug is given.
   Find the rate of change of the concentration at $t=1$ and state whether the concentration is increasing or decreasing.2 marks
3. The concentration of a chemical in a reaction vessel, $g$ mol dm$^{-3}$, is modelled by $g(t)=\sqrt{t}\,e^{-0.5t}$ for $t>0$, where $t$ is the time in minutes after mixing.
   Find an expression for $\dfrac{dg}{dt}$.3 marks