Calculus for Team-Based Inquiry Learning: 2022 Edition

(a)

Draw a right triangle with the red dashed line segment as its hypotenuse, one leg parallel to the $x$-axis, and the other parallel to the $y$-axis.

How long are these legs?

1. $3$ and $7\text{.}$

2. $4$ and $8\text{.}$

3. $3$ and $8\text{.}$

4. $4$ and $7\text{.}$

(b)

The Pythagorean theorem states that for a right triangle with leg lengths $a,b$ and hypotenuse length $c\text{,}$ we have...

1. $a=b=c\text{.}$

2. $a+b=c\text{.}$

3. $a^2=b^2=c^2\text{.}$

4. $a^2+b^2=c^2\text{.}$

(c)

Using the leg lengths and Pythagorean theorem, how long must the red dashed hypotenuse be?

1. $\sqrt{20}\approx 4.47\text{.}$

2. $\sqrt{56}\approx 7.48\text{.}$

3. $\sqrt{67}\approx 8.19\text{.}$

4. $\sqrt{100}=10\text{.}$

(d)

Compared with the blue parametric curve connecting the same two points, is the red dashed line segement length an overestimate or underestimate?

1. Overestimate: the blue curve is shorter than the red line.

2. Underestimate: the blue curve is longer than the red line.

3. Exact: the blue curve is exactly as long as the red line.

(a)

Let $\mathrm{\Delta }{L}_1,\mathrm{\Delta }{L}_2,\mathrm{\Delta }{L}_3$ describe the lengths of each of the three segements. Which expression describes the total length of these segments?

1. ${\displaystyle \mathrm{\Delta }{L}_1\times \mathrm{\Delta }{L}_2\times \mathrm{\Delta }{L}_3}$

2. ${\displaystyle \mathrm{\Delta }{L}_1+2\mathrm{\Delta }{L}_2+3\mathrm{\Delta }{L}_3}$

3. ${\displaystyle \sum _{i=1}^3\mathrm{\Delta }{L}_i}$

(b)

We can let each $\mathrm{\Delta }{L}_i=\sqrt{(\mathrm{\Delta }{x}_i{)}^2+(\mathrm{\Delta }{y}_i{)}^2}\text{.}$ But we will find it useful to involve the parameter $t$ as well, or more accurately, the change $\mathrm{\Delta }{t}_i$ of $t$ between each point of the subdivision.

Which of these is algebraically the same as the above formula for $\mathrm{\Delta }{L}_i\text{?}$

1. ${\displaystyle \sqrt{{\left(\frac{\mathrm{\Delta }{x}_i}{\mathrm{\Delta }{t}_i}\right)}^2+{\left(\frac{\mathrm{\Delta }{y}_i}{\mathrm{\Delta }{t}_i}\right)}^2}}$

2. ${\displaystyle \sqrt{[{\left(\frac{\mathrm{\Delta }{x}_i}{\mathrm{\Delta }{t}_i}\right)}^2+{\left(\frac{\mathrm{\Delta }{y}_i}{\mathrm{\Delta }{t}_i}\right)}^2]\mathrm{\Delta }{t}_i}}$

3. ${\displaystyle \sqrt{{\left(\frac{\mathrm{\Delta }{x}_i}{\mathrm{\Delta }{t}_i}\right)}^2+{\left(\frac{\mathrm{\Delta }{y}_i}{\mathrm{\Delta }{t}_i}\right)}^2}\mathrm{\Delta }{t}_i}$

(c)

Finally, we'll want to increase $N$ from $3$ so that it limits to $\mathrm{\infty }\text{.}$ What can we conclude when that happens?

1. Each segment is infintely small.

2. ${\displaystyle \mathrm{\Delta }{x}_i\to 0}$

3. ${\displaystyle \mathrm{\Delta }{t}_i\to 0}$

4. ${\displaystyle \frac{\mathrm{\Delta }{x}_i}{\mathrm{\Delta }{t}_i}\to \frac{dx}{dt}}$

5. All of the above.

(a)

Find $dx/dt$ and $dy/dt\text{,}$ and substitute them into the formula above along with $a=1$ and $b=3\text{.}$

(b)

Show that the value of this formula is $10\text{.}$

(c)

Show that the length of the line segment connecting $(2,-2)$ to $(8,-10)$ is $10$ by applying the distance formula directly instead.

(a)

The portion of $x=\sin 3t,y=\cos 3t$ where $0\le t\le \pi /6\text{.}$

(b)

The portion of $x=e^t,y=\ln t$ where $1\le t\le e\text{.}$

(c)

The portion of $x=t+1,y=t^2$ between the points $(3,4)$ and $(5,16)\text{.}$

(a)

Let $x=t\text{.}$ How can $\frac{dx}{dt}$ be simplified?

1. ${\displaystyle dx}$

2. ${\displaystyle dt}$

3. ${\displaystyle 1}$

4. ${\displaystyle 0}$

(b)

Given $x=t\text{,}$ how should $\frac{dy}{dt}$ and $dt$ be rewritten?

1. $\frac{dy}{dt}=\frac{dy}{dx}$ and $dt=dx\text{.}$

2. $\frac{dy}{dt}=\frac{dx}{dt}$ and $dt=dx\text{.}$

3. $\frac{dy}{dt}=\frac{dy}{dx}$ and $dt=1\text{.}$

4. $\frac{dy}{dt}=\frac{dy}{dt}$ and $dt=1\text{.}$

(c)

Write a modified, simplified formula for ${\int }_{t=a}^{t=b}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt$ with $t$ replaced with $x\text{.}$