IB Mathematics Applications & Interpretation Topic…

The motion of a particle is given by:

\ddot{x}+5 \dot{x}^{2}=2 \sin (t),

(a) Convert to a system of first-order differential equations.

(b) With $x(0)=1, \dot{x}(0)=-1$ , use Euler's method with step size 0.2 to $x$ and $\overset{}{}$ at .

(d) one limitation of Euler's method in this context.

Consider the function $f(x)=\frac{x^{3}}{x^{2}+4}$ for $x \in \mathbb{R}$ .

(a) Find $f^{\prime \prime}(x)$ .

(c) Find the coordinates of any points of inflection and verify by showing that $f^{\prime\prime}(x)$ changes sign at each point.

A particle moves along a straight line. Its displacement $x$ (in metres) at time $t$ (in seconds), $t\geq 0$ , satisfies

\ddot{x}+4 \dot{x}^{2}=e^{-2 t}.

(a) Express this as a system of first-order differential equations.

A spring system is governed by:

\ddot{x}+3 \dot{x}+2 x=\cos (2 t),

with $x$ in metres, $t \geq 0$ .

(a) Write the differential equation as a system of first-order differential equations.

A system is modeled by:

\ddot{x}+2 \dot{x}+3 x=t^{3},

with $x$ in metres, $t$ in seconds, $t \geq 0$ .

(a) Write as a system of first-order differential equations.

(b) the interval(s) of $x$ for which the graph of $f$ is concave down.

(d) the graph of $f^{\prime \prime}(x)$ , indicating points where $f^{\prime \prime}(x)=0$ .

(b) With $x(0)=0, \dot{x}(0)=1$ , use Euler's method with step size $0.2$ to $x$ and $\dot{x}$ at $t=0.4$ .

(d) why Euler's method might overestimate or underestimate the velocity at $t=0.4$ .

(b) Given $x(0)=0$ and $\dot{x}(0)=0$ , use Euler's method with step size $0.1$ to $x$ and $\dot{x}$ at $t=0.5$ .

(b) With $x(0)=0, \dot{x}(0)=0$ , use Euler's method with step size $0.1\,\text{s}$ to $x$ and $\dot{x}$ at $t=0.3\,\text{s}$ .

IB Mathematics Applications & Interpretation Calculus Questionbank