Unit 04 - Essential problems

Parametric curves

01

Convert parametric curve to function graph

Write the following curves as the graphs of functions $y=f(x)$. (Find $f(x)$ for each case.)

(a) $x=t+3$, $y=4t$ and $0<t<1$

(b) $x=\cos t$, $y={\sin }^{2}t$ and $0<t<2\pi$

Sketch each curve.

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Solution

Solutions - 0265-01

(a) From the first equation, $t=x-3$.

Plug that in for $t$ in $y(t)$:

$$\begin{array}{c}y=4t\gg \gg y=4(x-3)\\ \\ \gg \gg y=4x-12\end{array}$$

The sketched curve should be the portion of the line with $x\in (\mathrm{3,4})$.

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(b)

For this one, best not to solve for $t$. Instead, notice the trig identity:

$$x^2+y={(\cos t)}^2+{\sin }^{2}t=1$$

Therefore the points on the curve satisfy the equation $x^2+y=1$. Solve this for the function:

$$y=1-x^2$$

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03

Convert function graph to parametric curve

Find parametric curves $c(t)=(x(t),y(t))$ whose images are the following graphs:

(a) $y=3x-4$ and $c(0)=(\mathrm{2,2})$

(b) $y=3x-4$ and $c(3)=(\mathrm{2,2})$

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Solution

Solutions - 0265-03

(a)

First choose a function $x(t)$, then set $y(t)=3x(t)-4$ to ensure the equation is satisfied.

When choosing $x(t)$, we want $x$ to cover the whole domain of $y=3x-4$ which is $x\in (-\infty ,\infty )$. We also need $x(0)=2$ to satisfy the initial condition.

Start by trying $x(t)=t$:

$$y=3t-4$$

But then $x(0)=-4$. Since $c(0)=(\mathrm{2,2})$ we should have $x(0)=2$. We can arrange for this by setting $x(t)=t+a$ and solving for $a$:

$$x(0)=2\gg \gg 0+a=2\gg \gg a=2$$

Therefore we define $x(t)=t+2$. Then:

$$y(t)=3(t+2)-4\gg \gg 3t+2$$

So we use:

$$\begin{array}{cc}x(t)& =t+2\\ y(t)& =3t+2\end{array}$$

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(b)

Same method but different condition:

$$x(3)=2\gg \gg 3+a=2\gg \gg a=-1$$

Therefore we define $x(t)=t-1$. Then:

$$y(t)=3(t-1)-4\gg \gg 3t-7$$

So we use:

$$\begin{array}{cc}x(t)& =t-1\\ y(t)& =3t-7\end{array}$$Link to original

03

Parametric concavity

Find the interval(s) of $t$ on which the parametric curve $c(t)=(t^2,t^3-4t)$ is concave up.

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Solution

Solutions - 0260-03

First derivative:

$$\begin{array}{c}\frac{dy}{dx}=\frac{y^{\prime }(t)}{x^{\prime }(t)}\\ \\ \gg \gg \frac{3t^2-4}{2t}\gg \gg \frac{3}{2}t-2t^{-1}\end{array}$$

Second derivative:

$$\begin{array}{c}\frac{d^{2}y}{d{x}^2}=\frac{d}{dt}\left(\frac{dy}{dx}\right)\cdot \frac{1}{x^{\prime }(t)}\\ \\ \gg \gg \frac{d}{dt}(\frac{3}{2}t-2t^{-1})\cdot \frac{1}{2t}\\ \\ \gg \gg \frac{3}{4t}+\frac{1}{t^3}\gg \gg \frac{3t^2+4}{4t^3}\end{array}$$

This is positive if-and-only-if $t>0$. (Numerator always positive, denominator same sign as $t$.) So:

$$\text{Concave up:}t\in (0,+\infty )$$Link to original

06

Cycloid - Arclength and surface area of revolution

Consider the cycloid given parametrically by $c(t)=(t-\sin t,1-\cos t)$.

(a) Find the length of one arch of the cycloid.

(b) Suppose one arch of the cycloid is revolved around the $x$-axis. Find the area of this surface of revolution.

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Solution

Solutions---0260-06

Polar curves

01

Convert points: Cartesian to Polar

Convert the Cartesian (rectangular) coordinates for these points into polar coordinates:

(a) $(\mathrm{1,0})$ $$ (b) $(3,\sqrt{3})$ $$ (c) $(-\mathrm{2,2})$ $$ (d) $(-1,\sqrt{3})$

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Solution

Solutions---0270-01

01

Polar curve - Vertical or horizontal tangent lines

Find all points on the given curve where the tangent line is horizontal or vertical.

$$r=\cos \theta \theta \in [\mathrm{0,2}\pi )$$

Hint: First determine parametric Cartesian coordinate functions using $\theta$ as the parameter.

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Solution

Solutions---0280-01

04

Convert equations: Cartesian to Polar

Convert the Cartesian equation to a polar equation. Be sure to simplify.

(a) $x^2+y^2=25$ $$ (b) $x=5$ $$ (c) $y=x^2$

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Solution

Solutions---0270-04

04

Polar coordinates - lunar areas

(a) Find the area of the green region.

(b) Find the area of the yellow region.

(You can find these in either order.)

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Solution

Solutions---0280-04

06

Area of an inner loop

A limaçon is given as the graph of the polar curve $r=1+2\sin \theta$.

Find the area of the inner loop of this limaçon.

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Solution

Solutions---0280-06

Complex numbers

01

Complex forms - exponential to Cartesian

Write each number in the form $a+bi$.

(a) $2e^{i\frac{\pi }{4}}$ $$ (b) $e^{\ln 4+i\frac{\pi }{2}}$

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Solution

Solutions---0300-01

04

Complex products and quotients using polar

For each pair of complex numbers $z$ and $w$, compute:

$$zw,\frac{z}{w},\frac{1}{z}$$

(a) $z=1+\sqrt{3}i,w=\sqrt{3}+i$

(b) $z=2\sqrt{3}-2i,w=6i$

(Use polar forms with $\theta \in [\mathrm{0,2}\pi )$.)

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Solution

Solutions---0300-04

05

Complex powers using polar

Using De Moivre’s Theorem, write each number in the form $a+bi$.

(a) ${(1+i)}^{16}$ $$ (b) ${(\sqrt{3}-i)}^5$

(First convert to polar/exponential, then compute the power, then convert back.)

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Solution

Solutions---0300-05

02

Complex roots using polar

Find each of the indicated roots.

(a) The four $4^{\text{th}}$ roots of $1$.

(b) The three cube ($3^{\text{rd}}$) roots of $\sqrt{2}+\sqrt{2}i$.

Try to write your answer in $a+bi$ form if that is not hard, otherwise leave it in polar form.

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Solution

Solutions---0310-02