Math 214 - Multivariable Calculus - Prep

Task 4.1

Consider the curve $\vec r(t) = (2t+3, 4(2t-1)^2)$.

Construct a graph of $\vec r$ for $0\leq t\leq 2$.
If this curve represents the path of a rover (meters for distance, minutes for time), find the velocity of the rover at any time $t$, and then specifically at $t=1$. What is the rover's speed at $t=1$?
Give a vector equation of the tangent line to $\vec r$ at $t=1$. Include this on your graph.
State the rover's acceleration vector.
Explain how to obtain the slope of the tangent line, and then write an equation of the tangent line using point-slope form. [Hint: How can you turn the direction vector, which involves $(dx/dt)$ and $(dy/dt)$, into the number given by the slope $(dy/dx)$?]

Task 4.2

We are ready to tackle the problem of finding the length of a path. This length we call arc length. If a rover moves at a constant speed, then the distance traveled is simply $$\text{distance} = \text{speed}\times\text{time}.$$ This requires that the speed be constant. What if the speed is not constant? Over a really small time interval $dt$, the speed is almost constant, so we can still use the idea above.

Suppose a rover (or other object) moves along the path given by $\vec r(t)=(x(t),y(t))$ for $a\leq t\leq b$. We know that the velocity is $\dfrac{d\vec r}{dt}$, and so the speed is just the magnitude of this vector.

Show that we can write the rover's speed at any time $t$ as $$\ds\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}.$$
If the rover moves at speed $\ds\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$ for a little time length $dt$, what's the little distance $ds$ that the rover traveled?
Explain (Riemann sums may help) why the length of the path given by $\vec r(t)$ for $a\leq t\leq b$ is $$s=\int ds=\int_a^b \left|\frac{d\vec r}{dt}\right| dt=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt.$$
The path $\vec r(t) = (3\cos t, 3\sin t)$ for $0\leq t\leq 2\pi$ is a circle of radius 3. Verify that the formula above does in fact yield the circumference of this circle.
If the curve is in space (so $\vec r(t)=(x(t),y(t),z(t))$ is the path), then how does the arc length formula above change?
Are there any requirements we must know about the parametrization $\vec r$ so that the formula above is valid?

Task 4.3

Gravity is often the first example we encounter of a vector field. Other important vector fields arise when we study magnetism, electricity, fluid flow, and more. To analyze how a river flows, we can construct a plot of the river and at each point in the river we draw a vector that represents the velocity at that point. This creates a collection of many vectors drawn all at once, where the base of each velocity vector is placed at the point where the velocity occurs. For gravity, a similar picture can be drawn, though all the vectors will point down with the same magnitude. This task has us construct a plot of a vector field.

Consider the function $\vec F(x,y) = \left<x-2y,x+y\right>$. This is a function where the input is a point $(x,y)$ in the plane, and the output is the vector $\left<x-2y,x+y\right>$. For example, if we input the point $(1,0)$, then the output is $\left<1-2(0),1+0\right> = \left<1,1\right>$. To construct a vector field plot, we draw the vector $\left<1,1\right>$ with its base located at the input $(1,0)$. In the picture below, based at $(1,0)$ we draw a vector that points right 1 and up 1.

Complete the table below and add the other 7 vectors to the graph.
$\begin{array}{c|c} (x,y)&\left<x-2y,x+y\right>\\\hline (1,0)&\left<1,1\right>\\ (1,1)&\\ (1,-1)&\\ (0,1)&\\ (0,-1)&\\ (-1,0)&\\ (-1,1)&\\ (-1,-1)& \end{array}$
Repeat the above for the vector field $\vec F(x,y)=(-2y,3x)$, constructing a vector field plot consisting of 8 vectors.

Task 4.4

The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below.

section 3.2: checkpoint 3.7, exercises 75-92
Arc Length Practice: section 3.3: checkpoint 3.9, exercises 102-112

Big View
Prep Day 4 Task 4.1 Consider the curve $\vec r(t) = (2t+3, 4(2t-1)^2)$. Construct a graph of $\vec r$ for $0\leq t\leq 2$. If this curve represents the path of a rover (meters for distance, minutes for time), find the velocity of the rover at any time $t$, and then specifically at $t=1$. What is the rover's speed at $t=1$? Give a vector equation of the tangent line to $\vec r$ at $t=1$. Include this on your graph. State the rover's acceleration vector. Explain how to obtain the slope of the tangent line, and then write an equation of the tangent line using point-slope form. [Hint: How can you turn the direction vector, which involves $(dx/dt)$ and $(dy/dt)$, into the number given by the slope $(dy/dx)$?] Task 4.2 We are ready to tackle the problem of finding the length of a path. This length we call arc length. If a rover moves at a constant speed, then the distance traveled is simply $$\text{distance} = \text{speed}\times\text{time}.$$ This requires that the speed be constant. What if the speed is not constant? Over a really small time interval $dt$, the speed is almost constant, so we can still use the idea above. Suppose a rover (or other object) moves along the path given by $\vec r(t)=(x(t),y(t))$ for $a\leq t\leq b$. We know that the velocity is $\dfrac{d\vec r}{dt}$, and so the speed is just the magnitude of this vector. Show that we can write the rover's speed at any time $t$ as $$\ds\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}.$$ If the rover moves at speed $\ds\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$ for a little time length $dt$, what's the little distance $ds$ that the rover traveled? Explain (Riemann sums may help) why the length of the path given by $\vec r(t)$ for $a\leq t\leq b$ is $$s=\int ds=\int_a^b \left\|\frac{d\vec r}{dt}\right\| dt=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt.$$ The path $\vec r(t) = (3\cos t, 3\sin t)$ for $0\leq t\leq 2\pi$ is a circle of radius 3. Verify that the formula above does in fact yield the circumference of this circle. If the curve is in space (so $\vec r(t)=(x(t),y(t),z(t))$ is the path), then how does the arc length formula above change? Are there any requirements we must know about the parametrization $\vec r$ so that the formula above is valid? Task 4.3 Gravity is often the first example we encounter of a vector field. Other important vector fields arise when we study magnetism, electricity, fluid flow, and more. To analyze how a river flows, we can construct a plot of the river and at each point in the river we draw a vector that represents the velocity at that point. This creates a collection of many vectors drawn all at once, where the base of each velocity vector is placed at the point where the velocity occurs. For gravity, a similar picture can be drawn, though all the vectors will point down with the same magnitude. This task has us construct a plot of a vector field. Consider the function $\vec F(x,y) = \left<x-2y,x+y\right>$. This is a function where the input is a point $(x,y)$ in the plane, and the output is the vector $\left<x-2y,x+y\right>$. For example, if we input the point $(1,0)$, then the output is $\left<1-2(0),1+0\right> = \left<1,1\right>$. To construct a vector field plot, we draw the vector $\left<1,1\right>$ with its base located at the input $(1,0)$. In the picture below, based at $(1,0)$ we draw a vector that points right 1 and up 1. Complete the table below and add the other 7 vectors to the graph. \(\begin{array}{c\|c} (x,y)&\left<x-2y,x+y\right>\\\hline (1,0)&\left<1,1\right>\\ (1,1)&\\ (1,-1)&\\ (0,1)&\\ (0,-1)&\\ (-1,0)&\\ (-1,1)&\\ (-1,-1)& \end{array}\) Repeat the above for the vector field $\vec F(x,y)=(-2y,3x)$, constructing a vector field plot consisting of 8 vectors. Task 4.4 The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below. section 3.2: checkpoint 3.7, exercises 75-92 Arc Length Practice: section 3.3: checkpoint 3.9, exercises 102-112 Edit Page History Source Attach File Backlinks List Group Page last modified on September 16, 2022, at 04:13 PM
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Day 4

Task 4.1

Task 4.2

Task 4.3

Task 4.4