IB Mathematics Applications & Interpretation Topic…

An object is released from rest at the top of a long vertical tube filled with a thick viscous fluid at time $t = 0$ seconds. Take $x(0)=0$ and $v(0)=0$ , where is the object’s displacement from the top of the tube measured in metres, and is its velocity in .

(a) By substituting $v = \frac{\text{d}x}{\text{d}t}$ into the equation, find an expression for the velocity of the object at time $t$ . Give your answer in the form $v = f ($ .

(b) From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.

(e) the behavior of the solution curve passing through $\left(\pi,-\frac32\right)$ as $x$ increases, and whether it has any asymptotes ().

(e) Find the values of $x$ and $y$ when $t = 2$ .

(f) the limiting value of the ratio $\frac{y}{x}$ as $t \to \infty$ and hence the equation of the line passing through the origin to which the trajectory becomes parallel.

(g) the quadrant in which the trajectory lies as $t \to \infty$ and its motion relative to the origin.

(d) Use Euler’s method, with a step length of $0.2\,\text{s}$ , to find the displacement and velocity of the object when $t = 0.6$ .

(e) By repeated application of Euler’s method, an approximation for the terminal velocity, to five significant figures.

(f) Use the differential equation to find the exact terminal velocity for the object predicted by the second model.

(g) Use your answers to parts (d), (e) and (f) to on the accuracy of the Euler approximation to this model.

Consider the differential equation $\frac{dy}{dx}=y-3\sin x$ and the solution curve passing through $\left(\pi,-\frac32\right)$ .

(b) Find the equation of the curve where $\frac{dy}{dx}=0$ .

(d) Find the equation of the tangent line to the solution curve passing through $\left(\pi,-\frac32\right)$ .

Consider the coupled system of differential equations

\dot{x} = 5x + 4y

\dot{y} = -x + y

A general solution to the coupled system is given by

\begin{pmatrix} x \\ y \end{pmatrix} = A\begin{pmatrix} 2 \\ -1 \end{pmatrix}e^{3t} + B\begin{pmatrix} 2t+1 \\ -t \end{pmatrix}e^{3t}

(a) Show that the matrix

\begin{pmatrix} 5 & 4 \\ -1 & 1 \end{pmatrix}

has a repeated eigenvalue (that is, only one distinct eigenvalue). Find this eigenvalue and an associated eigenvector.

(b) Using your eigenvalue and eigenvector, verify that

\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}e^{3t}

is a solution to the above system.

\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2t+1 \\ -t \end{pmatrix}e^{3t}

is also a solution.

(d) If initially at $t = 0$ , $x = 7$ , $y = -4$ , find the particular solution.

(a) the value of $\frac{dy}{dx}$ at the point $\left(\pi,-\frac32\right)$ .

IB Mathematics Applications & Interpretation AHL 5.15—slope Fields Questionbank

IB Mathematics Applications & Interpretation AHL 5.15—slope Fields Questionbank

IB Mathematics Applications & Interpretation AHL 5.15—slope Fields Questionbank

IB Mathematics Applications & Interpretation AHL 5.15—slope Fields Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.15—slope Fields Questionbank