5.1 Limits and the derivativeIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 5.1 Limits and the derivative

1. The function $f$ is defined by $f(x)=\frac{\sqrt{x+9}-3}{x}$ for $x\neq0$. A table of values gives $f(-0.1)=0.16713$, $f(-0.01)=0.16671$, $f(0.01)=0.16662$ and $f(0.1)=0.16621$ (5 s.f.). Use your GDC where needed.
   Use your GDC to evaluate $f(0.0001)$ and $f(-0.0001)$, and hence write down the limit as $x\to0$ to $3$ significant figures.2 marks
2. A ball is thrown upwards. Its height above the ground is $h(t)=20t-5t^2$ metres, $t$ seconds after it is thrown. Use your GDC or calculator.
   Find the average rate of change of $h$ between $t=1$ and $t=1.001$ and hence estimate $h'(1)$, stating its units.2 marks
3. The volume of water in a tank is $V$ litres, $t$ minutes after a tap is opened. The model gives $V(5)=60$, $V(5.1)=61.18$ and $\frac{dV}{dt}=12$ when $t=5$.
   Interpret $\frac{dV}{dt}=12$ when $t=5$, including units, and use it to estimate the volume of water when $t=5.5$.3 marks