with(plots): I enter the vector-valued function that gives the cow's position, as a function of time. Rather than just graphing it right away, I name the graph and don't display it ... that way I can graph other things on the same set of axes later. Notice that I added the plot option "numpoints=200" -- without it, I found the graph to be jagged. r:= t -> [cos(2*t),sin(3*t)^3, cos(3*t)-sin(3*t)+19];

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In order to graph a 3D vector-valued function, I use the spacecurve command. (I also have to load the package plots, above.) Notice that I added the plot option "numpoints=200" -- without it, I found the graph to be jagged. spacecurve([cos(2*t),sin(3*t)^3, cos(3*t)-sin(3*t)+19], t=0..5, color=blue, numpoints=200, axes=normal);

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To find where the cow move's horizontally, I need to find where the z-coordinate isn't changing, or where the derivative of the z-component is zero: diff(r(t),t);

NyUsJC1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiMhIiMsJComLUYlNiMsJEYsIiIkRi0tSSRjb3NHRiZGMiIiIiIiKiwmRjEhIiRGNUY6

solve(-3*sin(3*t)-3*cos(3*t)=0,t);

LCRJI1BpRyUqcHJvdGVjdGVkRyMhIiIiIzc=

evalf(%);

JCEreVEqemgjISM1

Unfortunately, the result Maple gave me isn't in the time range I'm interested in. From here, I can either use my trig knowledge, or I can cop out and use fsolve. I'll use trig: the deriv of the z-component will be zero if sin(3t)=- cos(3t), or if 3t=3Pi/4, 7Pi/4, 11Pi/4, 15Pi/4, or 19Pi/4, so if t=3Pi/12, 7Pi/12, 11Pi/12, 15Pi/12, or 19Pi/12. I got 5 zeros, let me just look at the graph to check: plot(-3*sin(3*t)-3*cos(3*t),t=0..5);

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At any given time, the direction the cow is moving is given by the velocity vector, or in other words, by r' at that time. If I just tired to plug a specifid time into the derivative of r, Maple will get confused (it will try plugging the number in before taking the derivative. Here's one way to do it: v:= T -> subs(t=T,diff(r(t),t));

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v(3*Pi/12); evalf(%);

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NyUkISIjIiIhJCErOTApPj0kISIqJEYlRiU=

v(7*Pi/12);evalf(%);

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NyUkIiIiIiIhJCIrOTApPj0kISIqJEYlRiU=

v(11*Pi/12); evalf(%);

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NyUkIiIiIiIhJCErOTApPj0kISIqJEYlRiU=

v(15*Pi/12); evalf(%);

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NyUkISIjIiIhJCIrOTApPj0kISIqJEYlRiU=

v(19*Pi/12); evalf(%);

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NyUkIiIiIiIhJCErOTApPj0kISIqJEYlRiU=

Next, I want to find where the cow is moving straight up and down. In order for the cow to be moving like this, the only component that can be changing is the z-component. So I want to find those times when both the x and y components are not changing. For this, I need the derivative again. v(t);

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I want to find when both the x and y component of v(t) are zero. I can tell Maple to work with just the first component, or just the second, as follows: v(t)[1];v(t)[2];

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So now I can use that to have Maple solve for when both are zero at the same time: solve([v(t)[1]=0,v(t)[2]=0],t);

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From this, I can see that it's zero at every multiple of Pi/2 between 0 and 5, so at t=0, t=Pi/2, t=Pi, t=3*Pi/2. Now here's the really cool part: finding maximum speed Speed = ||velocity|| = the length of the derivative of the position function. Recall we've already defined the velocity function: v(t);

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and then I have Maple find the length. If I load the vector calculus package, I could just used the "norm" command, but I'll again have Maple pick out the first, second, and third components of the velocity vector: speed:=sqrt( v(t)[1]^2 + v(t)[2]^2 + v(t)[3]^2);

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I can get a sense of where the max is by looking at the graph: plot(speed,t=0..5);

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Once I look at that graph, I see there are loads of local maxima, but one around 2.5 that's higher than the rest. I need to find out exactly where it is. So I need to take the derivative and set it equal to zero. First I have Maple do the differentiation, and then I give that derivative a name. diff(speed,t);

LCQqJiwoKiQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YrIiIjRi8iIiUqJi1GJzYjLCRGLiIiJEYwLUkkY29zR0YoRjNGLyIjIikqJCwmRjIhIiRGNkY7Ri8iIiIjISIiRi8sKiomRiZGPC1GN0YsRjwiIzsqJkYyRjVGNkY1IiRzKiomRjIiIiZGNkY8ISQnWyomRjpGPCwmRjYhIipGMiIiKkY8Ri9GPCNGPEYv

dspeed:=%;

LCQqJiwoKiQtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YrIiIjRi8iIiUqJi1GJzYjLCRGLiIiJEYwLUkkY29zR0YoRjNGLyIjIikqJCwmRjIhIiRGNkY7Ri8iIiIjISIiRi8sKiomRiZGPC1GN0YsRjwiIzsqJkYyRjVGNkY1IiRzKiomRjIiIiZGNkY8ISQnWyomRjpGPCwmRjYhIipGMiIiKkY8Ri9GPCNGPEYv

Since there are so many maxima and minima close together, I want to get a better sense of where the global max is, so I look at the graph of the derivative, so I can see where it crosses 0. plot(dspeed, t=0..5);

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The x-intercept I'm interested in occurs between t=2.3 and 2.5 (I can tell by clicking on the graph and looking at the little box of coordinates that appears on the upper left of the Maple window). But I want to know exactly, so I ask Maple to find the solution that lies between 2.3 and 2.5 for me. For that, I need fsolve, rather than just solve. fsolve(dspeed=0, t, 2.3..2.5);

JCIreEliK0MhIio=

then since what I really want to know is what the maximum speed is, I ask Maple to take that time and substitute it into the speed function for me: subs(t=%,speed);

KiQpLCgqJiIiJSIiIiktSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyQiK2FoNSxbISIqIiIjRidGJyooIiMiKUYnKS1GKjYjJCIrSiNmOz8oRjJGJkYnKS1JJGNvc0dGK0Y4RjNGJ0YnKiQpLCYqJiIiJEYnRjdGJyEiIiomRkJGJ0Y8RidGQ0YzRidGJyNGJ0YzRic=

That's a mess y combination of approximation and exactness, so I ask Maple to approximate it for me: evalf(%);

JCIrQnpuI3omISIq