1. Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Question 1

Graphing calculator required

A survey drone moves in a horizontal plane. For $0\leq t\leq6$ , where $t$ is measured in hours, its velocity components are

$v_x(t)=2\cos(0.6t)+0.4$

and

$v_y(t)=1.2\sin(0.8t)-0.3$ .

Velocity is measured in $\mathrm{km/hour}$ . At time $t=0$ , the drone is at the position $(1,-2)$ kilometres. The graph shows the two velocity components. Round calculator results to three decimal places.

Part (a)(i)

[1]

Write an expression for the drone’s position vector at time $t=5$ .

Part (a)(ii)

[2]

Calculate the drone’s position at time $t=5$ .

Part (b)

[1]

Determine the first time in $0<t<6$ at which the drone’s horizontal direction changes from eastward to westward.

Part (c)

[2]

Calculate the total distance travelled by the drone from $t=0$ to $t=5$ .

Part (d)

[2]

Justify whether the drone’s speed is increasing or decreasing at $t=4$ .

Part (e)

[1]

Interpret the horizontal motion of the drone immediately after the time found in part (b).

Question 2

Graphing calculator not permitted

The graph shows a continuous function $g$ . The curved portion from $x=4$ to $x=6$ is a semicircle of radius $1$ lying below the $x$ -axis. Define

$\displaystyle h(x)=3+\int_0^x g(t),dt$ .

What is $h(6)$ ?

A. $\displaystyle 3-\frac{\pi}{2}$

B. $\displaystyle 5-\frac{\pi}{2}$

C. $\displaystyle 5+\frac{\pi}{2}$

D. $\displaystyle 7-\frac{\pi}{2}$

[1]

Select Answer

Question 3

Graphing calculator required

The polar curve is defined by

$r=2+\sin(2\theta)$

for $0\leq\theta\leq2\pi$ . The region $R$ is bounded by the curve and the rays $\theta=0$ and $\theta=\frac{\pi}{2}$ . Point $P$ corresponds to $\theta=\frac{\pi}{4}$ . Round calculator results to three decimal places.

Part (a)(i)

[1]

Calculate the rectangular coordinates of $P$ .

Part (a)(ii)

[2]

Calculate $\frac{dy}{dx}$ at $P$ .

Part (b)

[1]

Represent the line tangent to the polar curve at $P$ with an equation in $x$ and $y$ .

Part (c)(i)

[1]

Write an expression for the area of region $R$ .

Part (c)(ii)

[1]

Evaluate the area of region $R$ exactly.

Part (d)(i)

[1]

Write an expression for the length of the curve from $\theta=0$ to $\theta=\frac{\pi}{2}$ .

Part (d)(ii)

[1]

Approximate the length of this part of the curve.

Part (e)

[1]

Explain the direction in which the point on the curve is moving at $P$ as $\theta$ increases.

Question 4

GRAPHING CALCULATOR NOT PERMITTED

A curve is defined parametrically by $x=t^2+1$ and $y=t^3-3t$ for $t>0$ . What is the value of $\frac{dy}{dx}$ at $t=2$ ?

A. $\frac{3}{2}$

B. $\frac{9}{4}$

C. $3$

D. $\frac{9}{2}$

[1]

Select Answer

Question 5

Graphing calculator not permitted

A curve is defined parametrically by

$x(t)=t^3-3t$

and

$y(t)=t^2$

for $-2\leq t\leq2$ . Point $P$ corresponds to $t=\sqrt2$ .

Part (a)(i)

[1]

Verify that the curve has a horizontal tangent at $t=0$ .

Part (a)(ii)

[2]

Verify that the curve has vertical tangents at $t=-1$ and $t=1$ .

Part (b)(i)

[1]

Calculate $\frac{dy}{dx}$ at $P$ .

Part (b)(ii)

[2]

Calculate $\frac{d^2y}{dx^2}$ at $P$ .

Part (b)(iii)

[1]

Determine whether the curve is concave up or concave down at $P$ .

Part (c)

[2]

Justify that on $1<t<2$ , $y$ can be treated as a differentiable function of $x$ and that this function is concave down.

Question 6

Graphing calculator not permitted

A ground robot moves along a path described by

$x(t)=t^2$

and

$y(t)=\frac23t^3$

for $0\leq t\leq2$ , where position is measured in metres and time is measured in seconds. Point $P$ corresponds to $t=1$ .

Part (a)

[2]

Calculate the velocity vector and acceleration vector at $t=1$ .

Part (b)

[2]

Represent the line tangent to the robot’s path at $P$ .

Part (c)

[2]

Calculate the total distance travelled from $t=0$ to $t=2$ .

Part (d)

[2]

Determine the robot’s displacement vector and the magnitude of its displacement from $t=0$ to $t=2$ .

Part (e)

[1]

Justify that the robot’s speed is increasing for $0<t\leq2$ .

Question 7

Graphing calculator not permitted

The graph shows the polar curves

$r=2\cos\theta$

and

$r=1$ .

Region $R$ lies inside $r=2\cos\theta$ and outside $r=1$ .

Part (a)

[2]

Determine the two angles at which the curves intersect and that bound region $R$ .

Part (b)

[2]

Write an expression for the area of region $R$ .

Part (c)

[2]

Evaluate the area of region $R$ .

Part (d)

[2]

Calculate the slope of the line tangent to $r=2\cos\theta$ at the upper intersection point.

Part (e)

[1]

Represent the tangent line at the upper intersection point with an equation in $x$ and $y$ .

Question 8

Graphing calculator not permitted

A particle has position $\mathbf r(t)=\langle x(t),y(t)\rangle$ for $0\leq t\leq4$ . The graph shows the velocity components $x'(t)$ and $y'(t)$ . At $t=0$ , the particle is at $(0,1)$ . Time is measured in seconds, position in metres, and velocity in $\mathrm{m/s}$ .

Part (a)

[2]

Calculate the particle’s position at $t=2$ .

Part (b)

[2]

Represent $x(t)$ and $y(t)$ algebraically for $0\leq t\leq1$ .

Part (c)

[2]

Represent the portion of the particle’s path for $0\leq t\leq1$ with a Cartesian equation relating $x$ and $y$ .

Part (d)

[2]

Calculate the particle’s average velocity vector on $0\leq t\leq4$ .

Part (e)

[1]

Interpret the particle’s motion at $t=3.5$ .

Question 9

Graphing calculator not permitted

A curve is defined parametrically by differentiable functions $x(t)$ and $y(t)$ for $-2\leq t\leq4$ . The graph shows $x'(t)$ and $y'(t)$ . Both component graphs are linear on each displayed segment.

Part (a)(i)

[1]

Determine all values of $t$ at which the curve has a horizontal tangent.

Part (a)(ii)

[1]

Verify that the horizontal-tangent condition is satisfied at those times.

Part (b)(i)

[1]

Determine the value of $t$ at which the curve has a vertical tangent.

Part (b)(ii)

[1]

Verify that the vertical-tangent condition is satisfied at that time.

Part (c)

[3]

Justify whether the speed of the particle is increasing or decreasing at $t=0$ .

Part (d)(i)

[1]

Calculate $\frac{d^2y}{dx^2}$ at $t=0$ .

Part (d)(ii)

[1]

Determine the concavity of the curve at $t=0$ .

Question 10

Graphing calculator not permitted

The polar curve is

$r=1+\cos\theta$ .

The shaded region $R$ is the part of the curve lying above the $x$ -axis. What is the area of $R$ ?

A. $\displaystyle \frac{\pi}{2}$

B. $\displaystyle \frac{3\pi}{4}$

C. $\displaystyle \frac{3\pi}{2}$

D. $\displaystyle 2\pi$

[1]

Select Answer

Practice Questions

1. Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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