5.1 Limits and the derivativeIB Maths: Applications and Interpretation HL: Subtopic test

1. 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.

   (a)

   Use the table to estimate $\lim_{x\to0}f(x)$.

   [1 mark]

   • A$0$
   • B$3$
   • CThe limit does not exist
   • D$0.167$

   (b)

   Which statement about $f$ is correct?

   [1 mark]

   • AThe limit is $f(0)$
   • B$f(0)=0.167$
   • CThe limit exists although $f(0)$ is not defined
   • DThe values approach different numbers from the left and right

   (c)

   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]

   Total for question 1: 4 marks

2. 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.

   (a)

   Find the average rate of change of $h$ between $t=1$ and $t=1.1$.

   [1 mark]

   • A$0.95$ m s$^{-1}$
   • B$9.5$ m s$^{-1}$
   • C$15.95$ m s$^{-1}$
   • D$0.864$ m s$^{-1}$

   (b)

   The time interval starting at $t=1$ is made smaller and smaller. The average rate of change over the interval approaches which quantity?

   [1 mark]

   • AThe gradient of the tangent at $t=1$, written $h'(1)$
   • BThe height $h(1)$
   • CThe average height over the interval
   • DThe area under the curve

   (c)

   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]

   Total for question 2: 4 marks

3. 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 $t=5$.

   (a)

   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]

   (b)

   (i) A student writes '$\frac{dV}{dt}=12$ means there are $12$ litres in the tank when $t=5$'. Explain the error.
   (ii) Find the average rate of change of $V$ between $t=5$ and $t=5.1$, and explain why it is not exactly $12$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The distance, $s$ metres, travelled by a car $t$ seconds after it starts to accelerate is modelled by $s(t)=0.2t^3$ for $0\leq t\leq8$. Use your GDC.

   (a)

   (i) Find the average velocity of the car between $t=6$ and $t=8$.
   (ii) Find the average velocity between $t=6$ and $t=6.1$, and between $t=6$ and $t=6.01$.
   (iii) Hence estimate the velocity of the car at $t=6$, and explain how your values from (ii) lead to this estimate.

   [6 marks]

   (b)

   It is given that $\frac{ds}{dt}=21.6$ when $t=6$ and $\frac{ds}{dt}=38.4$ when $t=8$.
   (i) State what $\frac{ds}{dt}$ represents, including its units.
   (ii) Comment on what the two given values show about the motion of the car.
   (iii) Use $\frac{ds}{dt}$ at $t=6$ to estimate $s(6.5)$. State whether this is an underestimate or an overestimate, with a reason.

   [6 marks]

   Total for question 4: 12 marks

End of questions