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 \(\Delta L_1,\Delta L_2,\Delta L_3\) describe the lengths of each of the three segements. Which expression describes the total length of these segments?

1. \(\displaystyle \Delta L_1\times \Delta L_2\times \Delta L_3\)

2. \(\displaystyle \Delta L_1+ 2\Delta L_2+ 3\Delta L_3\)

3. \(\displaystyle \sum_{i=1}^{3} \Delta L_i\)

(b)

We can let each \(\Delta L_i=\sqrt{(\Delta x_i)^2+(\Delta y_i)^2}\text{.}\) But we will find it useful to involve the parameter \(t\) as well, or more accurately, the change \(\Delta t_i\) of \(t\) between each point of the subdivision.

Which of these is algebraically the same as the above formula for \(\Delta L_i\text{?}\)

1. \(\displaystyle \sqrt{\left(\frac{\Delta x_i}{\Delta t_i}\right)^2+\left(\frac{\Delta y_i}{\Delta t_i}\right)^2}\)

2. \(\displaystyle \sqrt{\left[\left(\frac{\Delta x_i}{\Delta t_i}\right)^2+\left(\frac{\Delta y_i}{\Delta t_i}\right)^2\right]\Delta t_i}\)

3. \(\displaystyle \sqrt{\left(\frac{\Delta x_i}{\Delta t_i}\right)^2+\left(\frac{\Delta y_i}{\Delta t_i}\right)^2}\Delta t_i\)

(c)

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

1. Each segment is infintely small.

2. \(\displaystyle \Delta x_i\to 0\)

3. \(\displaystyle \Delta t_i\to 0\)

4. \(\displaystyle \frac{\Delta x_i}{\Delta t_i}\to\frac{dx}{dt}\)

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 \(0\leq t\leq \pi/6\text{.}\)

(b)

The portion of \(x=e^t, y=\ln t\) where \(1\leq t\leq e\text{.}\)

(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 \(x\text{.}\)