CalculusIB Maths: Applications and Interpretation HL: Topic test

1. 1
   A function is defined by $f(x)=x^4-8x^2+3$.

   (a)

   Which expression is $f'(x)$?

   [1 mark]

   • A$4x^3-8x$
   • B$4x^3-16x+3$
   • C$4x^3-16x$
   • D$x^3-16x$

   (b)

   For which values of $x$ is $f$ increasing?

   [1 mark]

   • A$-2<x<0$ and $x>2$
   • B$x<-2$ and $0<x<2$
   • C$x>2$ only
   • D$-2<x<2$

   (c)

   Use the second derivative to classify the stationary point at $x=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $y=f(x)$, for $x>-\frac12$, passes through the point $(0,3)$ and has gradient function $f'(x)=\frac{6}{2x+1}$.

   (a)

   What is the gradient of the normal to the curve at the point where $x=1$?

   [1 mark]

   • A$2$
   • B$-2$
   • C$\frac12$
   • D$-\frac12$

   (b)

   Which expression is $f(x)$?

   [1 mark]

   • A$6\ln(2x+1)+3$
   • B$3\ln(2x+1)+3$
   • C$3\ln(2x+1)$
   • D$-\frac{12}{(2x+1)^2}+15$

   (c)

   Find $f(4)$, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An oil slick spreads in a circle of radius $r$ metres at time $t$ minutes. Its area $A$ increases at a constant rate of $12$ m$^{2}$ per minute.

   (a)

   Show that $\frac{dr}{dt}=\frac{6}{\pi r}$.

   [3 marks]

   (b)

   Given that $r=0$ when $t=0$, solve $\frac{dr}{dt}=\frac{6}{\pi r}$ by separation of variables, and hence find the radius of the slick when $t=10$. Give your answer in metres to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An open-topped box has a square base of side $x$ metres and a volume of $32$ m$^{3}$. Its surface area is $A$ m$^{2}$.

   (a)

   (i) Show that $A=x^2+\frac{128}{x}$.
   (ii) Find the value of $x$ for which $A$ is a minimum.
   (iii) Use the second derivative to justify that this value of $x$ gives a minimum.

   [6 marks]

   (b)

   The side $x$ is now increasing at $0.02$ m per minute.
   (i) Find the rate of change of $A$ when $x=2$.
   The material costs $15$ AED per m$^{2}$.
   (ii) Find the minimum cost of the material for the box.
   (iii) Explain why $A$ has no maximum value for $x>0$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A buoy bobs vertically. Its displacement from its rest level is $x=3\sin2t$ metres at time $t$ seconds, for $t\ge0$, where the angle is in radians.

   (a)

   What is the velocity of the buoy when $t=\frac{\pi}{3}$?

   [1 mark]

   • A$-3$ m s$^{-1}$
   • B$3$ m s$^{-1}$
   • C$-10.4$ m s$^{-1}$
   • D$5.20$ m s$^{-1}$

   (b)

   What is the acceleration of the buoy when $t=\frac{\pi}{4}$?

   [1 mark]

   • A$-6$ m s$^{-2}$
   • B$0$ m s$^{-2}$
   • C$12$ m s$^{-2}$
   • D$-12$ m s$^{-2}$

   (c)

   Use your GDC to find the total distance travelled by the buoy in the first $\frac{\pi}{2}$ seconds.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A slope field is drawn for the differential equation $\frac{dy}{dx}=y-2x$. A solution curve passes through the point $(0,1)$.

   (a)

   At which point is the slope segment horizontal?

   [1 mark]

   • A$(2,1)$
   • B$(1,2)$
   • C$(1,-2)$
   • D$(0,2)$

   (b)

   Along which line do all the slope segments have gradient $1$?

   [1 mark]

   • A$y=2x-1$
   • B$y=2x$
   • C$y=2x+1$
   • D$y=\frac12x+1$

   (c)

   Use Euler's method with step length $h=0.5$ to estimate the value of $y$ when $x=1$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The region $R$ is bounded by the curve $y=e^{x/2}$, the $x$-axis, the $y$-axis and the line $x=2$. Lengths are in centimetres. A GDC may be used.

   (a)

   Use the trapezoidal rule with four intervals of equal width to estimate the area of $R$. Give your answer to 3 significant figures.

   [3 marks]

   (b)

   The region $R$ is rotated through $360^\circ$ about the $x$-axis. Find the exact volume of the solid formed, and give it to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two linked tanks hold salt solution. At time $t$ minutes, the deviations of the salt concentration from equilibrium, in g l$^{-1}$, in tank 1 and tank 2 are $x$ and $y$. They satisfy $\frac{dx}{dt}=-2x+y$ and $\frac{dy}{dt}=x-2y$.

   (a)

   (i) Find the eigenvalues of the matrix $\begin{pmatrix}-2&1\\1&-2\end{pmatrix}$.
   (ii) Find an eigenvector for each eigenvalue.
   (iii) Describe the long-term behaviour of the concentrations, referring to the phase portrait.

   [6 marks]

   (b)

   Initially $x=4$ and $y=0$.
   (i) Use Euler's method with step length $0.5$ to estimate $x$ and $y$ when $t=1$.
   (ii) Use your eigenvectors from part (a) to find the exact solution for $x$, and hence find $x(1)$ to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions