5.15 Slope fieldsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A slope field is drawn for the differential equation $\frac{dy}{dx}=x-y$.

   (a)

   Find the gradient of the line segment at the point $(3,1)$.

   [1 mark]

   • A$2$
   • B$4$
   • C$3$
   • D$-2$

   (b)

   All line segments are horizontal along which line?

   [1 mark]

   • A$y=-x$
   • B$x=0$
   • C$y=0$
   • D$y=x$

   (c)

   A solution curve passes through the point $(0,1)$. Find the equation of the tangent to the solution curve at this point.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A slope field is drawn for the differential equation $\frac{dy}{dx}=\frac{x}{y}$, for $y\neq0$.

   (a)

   Find the gradient of the line segment at the point $(-3,6)$.

   [1 mark]

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

   (b)

   What is the gradient of the line segments at all points on the line $y=2x$ (with $x\neq0$)?

   [1 mark]

   • A$2$
   • B$0$
   • C$\frac12$
   • D$-\frac12$

   (c)

   A solution curve passes through the point $(1,2)$. Use the gradient of the slope field at this point to estimate the value of $y$ on this curve when $x=1.2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The population $P$, in hundreds, of fish in a lake at time $t$ years is modelled by $\frac{dP}{dt}=0.2P(5-P)$ for $P\ge0$. A slope field is drawn for this differential equation.

   (a)

   Find the values of $P$ at which the line segments are horizontal, and interpret your answer in context.

   [3 marks]

   (b)

   (i) Find the gradient of the line segments at $P=1$ and at $P=6$.
   (ii) Hence describe how the population changes over time for a solution curve starting at $P=1$, and for one starting at $P=6$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A slope field is drawn for the differential equation $\frac{dy}{dx}=x+y$. A GDC may be used.

   (a)

   (i) Find the gradient of the line segment at the point $(1,2)$ and at the point $(-2,2)$.
   (ii) Show that every line segment at a point on the line $y=2-x$ has gradient $2$.
   (iii) Find the equation of the line along which every line segment has gradient $-1$.

   [6 marks]

   (b)

   (i) Show that $y=e^x-x-1$ satisfies the differential equation and passes through the origin.
   (ii) Find the value of $y$ when $x=2$ on this solution curve.
   The line $L$ has equation $y=-x-1$, and every line segment on $L$ has gradient $-1$.
   (iii) Explain why the solution curve $y=e^x-x-1$ stays above $L$ for all values of $x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions