5.10 Second derivative and concavityIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The profit, $P$ thousand euros, of a company in year $t$ is modelled by $P(t)=t^3-9t^2+15t+20$ for $0\le t\le8$.

   (a)

   Which value of $t$ gives a local minimum of $P$?

   [1 mark]

   • A$t=5$
   • B$t=1$
   • C$t=3$
   • D$t=8$

   (b)

   Which statement describes the concavity of the graph of $P$?

   [1 mark]

   • AConcave-up for $0\le t<3$ and concave-down for $3<t\le8$
   • BConcave-up for all $t$ in the domain
   • CConcave-down for all $t$ in the domain
   • DConcave-down for $0\le t<3$ and concave-up for $3<t\le8$

   (c)

   Find the time at which the profit is falling most rapidly, and the rate of change of the profit at that time.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A factory's average cost per unit, $C$ euros, when it makes $x$ hundred units is modelled by $C(x)=x+\dfrac{16}{x}$ for $x>0$.

   (a)

   Find $C''(x)$.

   [1 mark]

   • A$\dfrac{16}{x^3}$
   • B$\dfrac{32}{x^3}$
   • C$-\dfrac{32}{x^3}$
   • D$1+\dfrac{32}{x^3}$

   (b)

   Which statement about the graph of $C$ is correct?

   [1 mark]

   • AIt has a point of inflexion at $x=4$.
   • BIt is concave-down for $x>4$.
   • CIt is concave-up for all $x>0$ and has no point of inflexion.
   • DIt is concave-up only for $x<4$.

   (c)

   Find the number of hundreds of units that gives the least average cost, using the second derivative test to justify your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The concentration of a drug in a patient's blood, $c$ mg l$^{-1}$, is modelled by $c(t)=4te^{-t/2}$ for $t\ge0$, where $t$ is the time in hours after the drug is given.

   (a)

   Find $c'(t)$ and hence show that $c''(t)=(t-4)e^{-t/2}$.

   [3 marks]

   (b)

   (i) Use the second derivative test to show that the concentration is greatest at $t=2$.
   (ii) Show that the graph of $c$ has a point of inflexion at $t=4$ and interpret this in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves in a straight line. Its displacement from a fixed point $O$ is $s$ metres at time $t$ seconds, where $s(t)=t^3-9t^2+24t$ for $t\ge0$.

   (a)

   (i) Find expressions for the velocity $v=\dfrac{ds}{dt}$ and the acceleration $a=\dfrac{d^2s}{dt^2}$.
   (ii) Find the times at which the particle is at rest.
   (iii) Find the time at which the acceleration is zero, and describe the concavity of the graph of $s$ before and after this time.

   [6 marks]

   (b)

   (i) Use the second derivative test to find whether $t=2$ and $t=4$ give local maximum or minimum displacements, and find the displacement at each.
   (ii) Find the total distance travelled in the first $5$ seconds.

   [6 marks]

   Total for question 4: 12 marks

End of questions