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Question 1

HLPaper 2

A function $f$ is defined for $-2 \leq x \leq 2$ , with the behavior of $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ given below:

$x$	$-2<x<-1$	$x=-1$	$-1<x<1$	$1<x<2$
$f^{\prime}(x)$	Positive	0	Negative	Negative
$f^{\prime \prime}(x)$	Negative	0	Positive	Negative

1.

Identify the x -coordinates of the local extrema of $f$ .

[3]

2.

Determine the intervals where $f$ is concave up.

[2]

3.

Given $f(-2)=3$ , sketch the graph of $f$ , indicating the local extrema and points of inflection.

[4]

Question 2

HLPaper 2

A particle moves in a straight line with displacement given by $s(t)=t^{4}-8 t^{3}+20 t^{2}-16 t+5$ , where $t \geq 0$ is time in seconds and $s$ is in meters.

1.

Find expressions for the velocity $v(t)$ and acceleration $a(t)$ of the particle.

[3]

2.

Determine the times when the particle is stationary.

[3]

3.

Use the second derivative test to classify the nature of each stationary point.

[3]

4.

Given that the particle's acceleration is zero at $t=p$ , find $p$ and determine whether this is a point of inflection.

[4]

5.

Sketch the graph of $a(t)$ , indicating the point where $a(t)=0$ .

[3]

Question 3

HLPaper 1

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

1.

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

[5]

2.

Determine the points where the graph of $f$ is concave down.

[3]

3.

Find the coordinates of any points of inflection and verify using the second derivative test.

[4]

4.

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

[4]

Question 4

HLPaper 2

A particle moves along a straight line with displacement given by $s(t)=2 t^{3}-9 t^{2}+12 t+1$ , where $t \geq 0$ is time in seconds and $s$ is in meters.

1.

Find the velocity $v(t)$ and acceleration $a(t)$ of the particle.

[3]

2.

Determine the times when the particle is stationary.

[2]

3.

Use the second derivative test to classify the nature of the stationary points of $s(t)$ .

[3]

Question 5

HLPaper 2

A particle moves along a curve defined by $y=x^{3} e^{-2 x}$ for $x \geq 0$ , where $x$ is time in seconds and $y$ is displacement in meters.

1.

Find the velocity $v(x)=\frac{d y}{d x}$ .

[3]

2.

Determine the exact coordinates of the stationary points of $y$ .

[4]

3.

Sketch the graph of the acceleration $a(x)=\frac{d^{2} y}{d x^{2}}$ , indicating where $a(x)=0$ .

[2]

Question 6

HLPaper 1

The wind chill index $W$ is a measure of the temperature, in $°C$ , felt when taking into account the effect of the wind.

When Frieda arrives at the top of a hill, the relationship between the wind chill index and the speed of the wind $v$ in kilometres per hour $(km h^{-1})$ is given by the equation

W = 19.34 - 7.405v^{0.16}

1.

Find an expression for $\frac{dW}{dv}$ .

[2]

2.

When Frieda arrives at the top of a hill, the speed of the wind is $10$ kilometres per hour and increasing at a rate of $5$ $\text{km h}^{-1} \text{ minute}^{-1}$ .

Find the rate of change of $W$ at this time.

[5]

Question 7

HLPaper 2

A satellite moves along a straight path in space, and its displacement at time ( $t$ ) is given by the function $s(t) = t^3 - 6t^2 + 9t$ meters.

1.

Calculate the velocity and acceleration functions of the satellite.

[3]

2.

Determine the time(s) when the satellite is at rest.

[4]

3.

Use the second derivative test to explain whether the satellite is accelerating or decelerating at t = 2.

[4]

4.

Find the total distance travelled by the satellite from t = 0 to t = 3.

[5]

5.

Sketch the displacement-time graph for the satellite, indicating the key features of its motion.

[4]

Question 8

HLPaper 2

At an archery tournament, a particular competition sees a ball launched into the air while anarcher attempts to hit it with an arrow.

The path of the ball is modelled by the equation

$(\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 5 \\ 0 \end{matrix}) + t (\begin{matrix} u_{x} \\ u_{y} - 5 t \end{matrix})$

where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacementfrom the ground, both measured in metres, and $t$ is the time, in seconds, since the ballwas launched.

$u_{x}$ is the horizontal component of the initial velocity
$u_{y}$ is the vertical component of the initial velocity.

In this question both the ball and the arrow are modelled as single points. The ball is launchedwith an initial velocity such that $u_{x} = 8$ and $u_{y} = 10$ .

An archer releases an arrow from the point $(0, 2)$ . The arrow is modelled as travelling in astraight line, in the same plane as the ball, with speed $60 {m s}^{- 1}$ and an angle of elevation of $10 °$ .

1.

Find the initial speed of the ball.

[2]

2.

Find the angle of elevation of the ball as it is launched.

[2]

3.

Find the maximum height reached by the ball.

[3]

4.

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

[3]

5.

For the path of the ball, find an expression for $y$ in terms of $x$ .

[3]

6.

Determine the two positions where the path of the arrow intersects the path of the ball.

[4]

7.

Determine the time when the arrow should be released to hit the ball before the ballreaches its maximum height.

[4]

Question 9

HLPaper 2

A point Q moves in a straight line with velocity $v$ ms⁻¹ given by $v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}$ at timet seconds, wheret≥ 0.

1.

Find the value of the acceleration of Q at time t₁.

2.

Determine the first time t₁ at which Q has zero velocity.

3.

Find an expression for the acceleration of Q at time t.

Question 10

HLPaper 1

A sustainable farmer is designing a rectangular pen for livestock, ensuring that the total fencing used does not exceed 40 metres. The farmer wants to maximize the area available for the animals.

1.

Express the area $A$ of the pen as a function of its length $x$ .

[4]

2.

Find the value of $x$ that maximizes the area.

[5]

3.

Use the second derivative to confirm that the area is maximized at the value of $x$ found in part (b).

[4]

4.

What is the maximum area of the pen?

[3]

AHL 5.10—Second derivatives, testing for max and min Questionbank