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"source": [
"# Vector Valued Functions\n",
"\n",
"## Section 12.1\n",
"\n",
"From [our text](https://www.contemporarycalculus.com/dh/Calculus_all/CC12_1.pdf), we have:\n",
"\n",
"Definition: A **vector-valued function** is a rule that assigns a vector to each input number. Typically a vector-valued function has the form\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\boldsymbol{r(t)} = x(t)i + y(t)j + z(t)k = \\boldsymbol{\\langle x(t), y(t), z(t)\\rangle}\n",
"\\end{align}\n",
"$$\n",
"\n",
"where x, y and z are scalar-valued functions. \n",
"\n",
"The domain consists of those $t$ in the domain of x, y, and z.\n",
"\n",
"So the range is a collection of vectors. \n",
"\n",
"Below is an example of a vector valued function plotted in Sage:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "a10ef030-9d0d-482a-bdb2-d365cab32b52",
"metadata": {},
"outputs": [
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"data": {
"text/html": [
"\n",
"\n"
],
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"Graphics3d Object"
]
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"output_type": "display_data"
}
],
"source": [
"t = var('t')\n",
"# Define the tuple or vector\n",
"r = (cos(t), sin(t), t)\n",
"\n",
"# Use parametric_plot3d for curves\n",
"# Syntax: parametric_plot3d( (x, y, z), (variable, start, end) )\n",
"parametric_plot3d(r, (t, 0, 6*pi), thickness=2).show(aspect_ratio=(4,4,1))"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "482eb148-728e-4e3e-86ec-e994e9ff413d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
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"output_type": "display_data"
}
],
"source": [
"t = var('t')\n",
"f1 = 4 - 2*t\n",
"f2 = 2 - t\n",
"r2 = vector((t, f1, f2))\n",
"parametric_plot3d(r2, (t, 0, 2), thickness=2).show()"
]
},
{
"cell_type": "markdown",
"id": "eb1a4d57-670a-4a07-ad41-07839519020a",
"metadata": {},
"source": [
"## Limits\n",
"\n",
"The limit of a vector valued function (so it appears) is also a vector:\n",
"\n",
"\n",
"\n",
"We can't find such limits directly in sage, but we can write a function to do it easily:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "49ee135c-3ba7-4b9e-80e6-a2d8a19ff2e9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\\left(4 \\, t^{2} - t - 2,\\,\\frac{t^{2} + 2 \\, t - 63}{t - 7},\\,2 \\, \\cos\\left(t\\right)\\right)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Do assignment 12.1 with our new function \n",
"\n",
"t = var('t')\n",
"V = vector([4*t^2 - t - 2, (t^2 + 2*t -63)/(t-7), 2*cos(t)])\n",
"latex(V)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "2fbd9f5b-1d97-4fe7-a7eb-7448029474fe",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 2, 3]\n"
]
}
],
"source": [
"def vvf_limit(fn, v, a):\n",
" \"\"\"Returns a list with the limit of a vector valued function\n",
"\n",
" Parameters: \n",
" fn: The vector, list, or tuple representing the vector valued function\n",
" v: The variable used in the function\n",
" a: The \"endpoint\" or target of the limit, e.g. oo for infinity, 0, etc.\n",
"\n",
" Returns:\n",
" A list of values with the limit of each component of the function\n",
" \"\"\"\n",
" l = list(fn)\n",
" return [limit(x, v, a) for x in l]\n",
" \n",
" t = var('t')\n",
"r5 = [cos(t), sqrt(4 + t), 3*e**(2*t)]\n",
"print(vvf_limit(r5, t, 0))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "883d3c90-d493-4218-880e-c0e76b774f30",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"12.1 question 5\n",
"[4*t^2 - t - 2, (t^2 + 2*t - 63)/(t - 7), 2*cos(t)]\n",
"[187, 16, 2*cos(7)]\n",
"\n",
"12.1 question 5\n",
"[2*t^2 + 3*t - 1, (t^2 + 2*t - 24)/(t - 4), 4*cos(t)]\n"
]
},
{
"data": {
"text/plain": [
"1"
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"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%display plain\n",
"t = var('t')\n",
"x(t) = 4*t^2 - t -2\n",
"y(t) = (t^2 + 2*t -63)/(t-7)\n",
"z(t) = 2 * cos(t)\n",
"V = [x(t), y(t), z(t)]\n",
"print(\"12.1 question 5\")\n",
"print(V)\n",
"print(vvf_limit(V, t, 7))\n",
"print(\"\\n12.1 question 5\")\n",
"V2 = [2*t^2 + 3*t -1, (t^2 + 2*t -24)/(t-4), 4*cos(t)]\n",
"print(V2)\n",
"vvf_limit(V2, t, 4)\n",
"\n",
"lim(sin(t)/t, t, 0)"
]
},
{
"cell_type": "markdown",
"id": "1a5058c7-6992-4328-b414-434952467163",
"metadata": {},
"source": [
"## Continutity\n",
"A vector valued function $r(t)$ is continuous at $t_0$ if $\\lim\\limits_{t \\to t_0} r(t) = r(t_0)$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "4206e20f-04f7-473c-a89a-a72d384b115b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# An ellipse and an eliptical helix, comment out one or the other\n",
"x = var('x')\n",
"# parametric_plot([sin(x), 2*cos(x)], (x, 0, 2*pi))\n",
"parametric_plot3d([sin(x), 2*cos(x), x], (x, 0, 8*pi))"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "c8ab6f74-5d8e-4c06-b82b-90109bfef582",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(cos(9), sqrt(13), 3*e^18) (-0.911130261884677, 3.60555127546399, 1.96979907411992e8)\n"
]
}
],
"source": [
"# Sustitution example:\n",
"\n",
"r5(t) = [cos(t), sqrt(4 + t), 3*e**(2*t)]\n",
"\n",
"print(r5(9), N(r5(9)))"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "9f43bc88-8413-4ce5-aa17-719e8393c82b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
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}
],
"source": [
"# 1. Import the Cylinder shape\n",
"from sage.plot.plot3d.shapes import Cylinder\n",
" \n",
"# 2. Create the cylinder object: Cylinder(radius, height)\n",
"C = Cylinder(1, 2, color='blue', opacity=0.4)\n",
" \n",
"# 3. Display the graph\n",
"C.show(aspect_ratio=(1,1,1)) \n",
"# aspect_ratio=(1,1,1) ensures the axes are scaled equally, so it looks circular"
]
},
{
"cell_type": "markdown",
"id": "3d584521-da4c-4b05-a122-b8bd814d32da",
"metadata": {},
"source": [
"# Section 12.2 - The Derivative of vector-valued functions\n",
"\n",
"The derivative of a vector valued function is simply the derivative of each component, so as in the case of limits, we can implement a function to find one easily in Sage."
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "43e51b48-1a33-4e3c-8c40-359b4c5a2bfc",
"metadata": {},
"outputs": [],
"source": [
"def vvf_diff(fn, v, diff_n=1):\n",
" \"\"\"Returns a list representing the limit of a vector valued function\n",
"\n",
" Parameters: \n",
" fn : The vector, list, or tuple representing the vector valued function\n",
" v : The variable used in the function\n",
" diff_n : 1 for first derivative, 2 for second, etc.\n",
"\n",
" Returns:\n",
" A list of values with the limit of each component of the function\n",
" \"\"\"\n",
" l = list(fn)\n",
" return [diff(x, v, diff_n) for x in l]"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "b525fb7e-0100-41da-9cff-814c861ddee9",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[3*x^2 + 3, 1/x, 5]\n"
]
}
],
"source": [
"# Test out the vvf_diff function\n",
"\n",
"var('x')\n",
"V = vector([x^3 + 3*x, ln(x), 5*x + 3])\n",
"V_prime = vvf_diff(V, x)\n",
"assert V_prime == [3*x^2 + 3, 1/x, 5]\n",
"print(V_prime)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "cf21c0c9-5588-4004-ab93-5fa18c81169c",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Problem 12.2, q 3\n",
"(-3*t^5 - 2, 4*e^(3*t), -5*sin(-3*t))\n",
"[-15*t^4, 12*e^(3*t), 15*cos(-3*t)]\n",
"\n",
"Problem 12.2, q 4\n",
"(4/(t - 6), 2*t/(3*t^2 + 1), t^2/(7*t^3 + 2))\n",
"[-4/(t - 6)^2, -12*t^2/(3*t^2 + 1)^2 + 2/(3*t^2 + 1), -21*t^4/(7*t^3 + 2)^2 + 2*t/(7*t^3 + 2)]\n",
"\n",
"Problem 12.2, q 5 (second derivative)\n",
"(6*t + 3*sin(t), 5*t + 6*cos(t))\n",
"[-3*sin(t), -6*cos(t)]\n",
"[-3*sin(t), -6*cos(t)]\n"
]
}
],
"source": [
"# Solve a problem (12.2, q 3):\n",
"t = var('t')\n",
"\n",
"print(\"Problem 12.2, q 3\")\n",
"V = vector([-3*t^5 - 2, 4*e^(3*t), -5*sin(-3*t)])\n",
"print(V)\n",
"print(vvf_diff(V, t))\n",
"\n",
"print(\"\\nProblem 12.2, q 4\")\n",
"V = vector([(-4)/(-t+6), (2*t)/(3*t^2 + 1), (-t^2)/(-7*t^3 - 2)])\n",
"print(V)\n",
"print(vvf_diff(V, t))\n",
"\n",
"print(\"\\nProblem 12.2, q 5 (second derivative)\")\n",
"V = vector([6*t + 3*sin(t), 5*t + 6*cos(t)])\n",
"print (V)\n",
"print(vvf_diff(V, t, 2))\n",
"print(vvf_diff(vvf_diff(V, t), t))"
]
},
{
"cell_type": "markdown",
"id": "a8725a13-b901-4743-801d-4f0863cd4c03",
"metadata": {},
"source": [
"## Rules and Properties Related to Differentials of Vector Valued Functions\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\begin{array}{|c|c|c|}\n",
"\\hline\n",
"\\textbf{Operation} & \\textbf{Rule} & \\textbf{Result Type} \\\\\n",
"\\hline\n",
"\\text{Sum} & \\frac{d}{dt} \\left[ \\mathbf{r}_1(t) + \\mathbf{r}_2(t) \\right] = \\mathbf{r}_1'(t) + \\mathbf{r}_2'(t) & \\text{Vector} \\\\\n",
"\\hline\n",
"\\text{Scalar Multiple} & \\frac{d}{dt} \\left[ c \\mathbf{r}(t) \\right] = c \\mathbf{r}'(t) & \\text{Vector} \\\\\n",
"\\hline\n",
"\\text{Product (Dot)} & \\frac{d}{dt} \\left[ \\mathbf{r}_1(t) \\cdot \\mathbf{r}_2(t) \\right] = \\mathbf{r}_1'(t) \\cdot \\mathbf{r}_2(t) + \\mathbf{r}_1(t) \\cdot \\mathbf{r}_2'(t) & \\text{Scalar} \\\\\n",
"\\hline\n",
"\\text{Product (Cross)} & \\frac{d}{dt} \\left[ \\mathbf{r}_1(t) \\times \\mathbf{r}_2(t) \\right] = \\mathbf{r}_1'(t) \\times \\mathbf{r}_2(t) + \\mathbf{r}_1(t) \\times \\mathbf{r}_2'(t) & \\text{Vector} \\\\\n",
"\\hline\n",
"\\text{Chain Rule} & \\frac{d}{dt} \\left[ \\mathbf{r}(f(t)) \\right] = f'(t) \\mathbf{r}'(f(t)) & \\text{Vector} \\\\\n",
"\\hline\n",
"\\end{array}\n",
"\\end{align}\n",
"$$\n",
"\n",
"## Velocity/Acceleration of Vector Valued Functions\n",
"\n",
"In either $\\mathbb{R}^2$ or $\\mathbb{R}^3$ if a vector-valued function, $r(t)$ represents the curve of position (e.g. at point $t$), then $v(t) = r^{\\prime}(t)$ is the velocity, and $r^{\\prime\\prime}(t)$ is the acceleration. The speed is given by the magnitude of the velocity vector, $\\|v(t)\\|$.\n",
"\n",
"To do this in code, see the following example based on functions we've already defined."
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "88255b0a-d2a3-4e2b-9294-56540c4e3ade",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(-pi*sin(pi*t), pi*cos(pi*t), t)\n",
"[-pi^2*cos(pi*t), -pi^2*sin(pi*t), 1]\n",
"sqrt(pi^2*cos(pi*t)^2 + pi^2*sin(pi*t)^2 + t^2)\n"
]
}
],
"source": [
"# Find the velocity vector and speed given a position vector\n",
"# Our results agree with the example shown in the Video assignment 12.2 Q4, so they are correct,\n",
"# but the simplification on the speed value leaves room for improvement.\n",
"t = var('t')\n",
"pos_vector = vector([cos(pi * t), sin(pi*t), t^2/2])\n",
"velocity_vector = vector(vvf_diff(pos_vector, t)) # Need to wrap list in a vector again for \"norm\"\n",
"acceleration_vector = vvf_diff(pos_vector, t, 2)\n",
"speed = velocity_vector.norm().simplify()\n",
"print(velocity_vector)\n",
"print(acceleration_vector)\n",
"print(speed)"
]
},
{
"cell_type": "markdown",
"id": "fc3ac654-12a9-4102-8e04-429e9731be93",
"metadata": {},
"source": [
"## The Unit Tangent Vector.\n",
"\n",
"A unit tangent vector for a VVF, $r(t)$ is given by the derivative of the vector over the magnitude of that derivative. (So it's similar to the formula for a unit vector.)\n",
"\n",
"That is, assuming $r^{\\prime}(t) \\ne 0$,\n",
"\n",
"$$\n",
"\\begin{align}\n",
"T(t) = \\frac{r^{\\prime}(t)}{\\|r^{\\prime}(t)\\|}\n",
"\\end{align}\n",
"$$\n",
"\n",
"### Calculating and Plotting a Unit Tangent Vector in SageMath\n",
"\n",
"Let's calculate one for $r(t) = \\langle t^3, 2t^2\\rangle$, with a tangency point at t = 1. We get the correct values, but this vector would need to be graphed with the origin at the tangency point, (1,2). W"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "34ab1619-86f8-4bd5-9b42-e9cc06551d61",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Tangency point: (1, 2)\n",
"Tangent vector: (3/5, 4/5)\n"
]
},
{
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",
"text/plain": [
"Graphics object consisting of 2 graphics primitives"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Set up the curve\n",
"t = var('t')\n",
"r = vector([t**3, 2*t^2])\n",
"\n",
"# Get the derivative, and plug in the point at which we want to draw the unit tangent vector\n",
"r_prime = vector(vvf_diff(r, t))\n",
"tangency_point = r(t=1)\n",
"print(\"Tangency point: \", tangency_point)\n",
"\n",
"# Calculate the tangent vector using the formula above\n",
"tangent_vector = r_prime/r_prime.norm()\n",
"print(\"Tangent vector:\", tangent_vector(t=1))\n",
"\n",
"# Get a plot for the curve\n",
"plt = parametric_plot(r, (t, .75, 1.4), linestyle=\"dashed\")\n",
"\n",
"# Plot the unit tangent vector. By setting start equal to our tangency point, we place\n",
"# the vector tail where it belongs\n",
"plt2 = plot(tangent_vector(t=1), start=tangency_point, color=\"red\")\n",
"plt += plt2 \n",
"plt.show(aspect_ratio=1)"
]
},
{
"cell_type": "markdown",
"id": "a9934319-88fa-479c-9946-023111fd76c3",
"metadata": {},
"source": [
"## Anti-Derivatives: Integration of Vector Valued Functions\n",
"\n",
"Given our discussion of limits and derivatives, we might have guessed that the integral of a vector valued function is calculated by taking the integral of each of the components. Thus, for example, given an indefinite integral for $r(t) = \\langle f(t), g(t), h(t)\\rangle$, then the definite integral is as follows:\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"\\int \\mathbf{r}(t) \\, dt = \\left( \\int f(t) \\, dt \\right) \\mathbf{i} + \\left( \\int g(t) \\, dt \\right) \\mathbf{j} + \\left( \\int h(t) \\, dt \\right) \\mathbf{k} + \\mathbf{C}\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"For the definite integral, we have:\n",
"$$\n",
"\\begin{aligned}\n",
"\\int_{a}^{b} \\mathbf{r}(t) \\, dt = \\left( \\int_{a}^{b} f(t) \\, dt \\right) \\mathbf{i} + \\left( \\int_{a}^{b} g(t) \\, dt \\right) \\mathbf{j} + \\left( \\int_{a}^{b} h(t) \\, dt \\right) \\mathbf{k}\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"As we did before for limits and derivatives, we can easily implement this in sage"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "82dc509b-1716-4bc3-b8f9-0c4c14684534",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[sin(t), 2/3*(t + 4)^(3/2), 3/2*e^(2*t)]\n",
"[1, 2/3*(1/2*pi + 4)^(3/2) - 16/3, 3/2*e^pi - 3/2]\n"
]
}
],
"source": [
"def vvf_integrate(fn, v, limits=None):\n",
" \"\"\"Returns a list with the integral of each component of a vector-valued function\n",
"\n",
" Parameters: \n",
" fn: The vector, list, or tuple representing the vector valued function\n",
" v: The variable used in the function\n",
" limits: Either none or a two-element iterable for a definite integral.\n",
"\n",
" Returns:\n",
" A list of values with the limit of each component of the function\n",
"\n",
" Notes: \n",
" Initial condition problems are not yet handled by this function. \n",
" \"\"\"\n",
" l = list(fn)\n",
" \n",
" if limits is None:\n",
" return [integrate(x, v) for x in l]\n",
" else:\n",
" return [integrate(x, v, limits[0], limits[1]) for x in l]\n",
" \n",
" t = var('t')\n",
"r5 = [cos(t), sqrt(4 + t), 3*e**(2*t)]\n",
"\n",
"# Test it out\n",
"print(vvf_integrate(r5, t))\n",
"print(vvf_integrate(r5, t, (0, pi/2)))"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "5f7f9da6-c739-4f04-8197-2c383e8016af",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"12.2 Q6 [-4/3*t^3 + t, e^(-t), 5/4*cos(4*t)]\n",
"\n",
"Add initial conditions manually\n",
"to each term for the next problem:\n",
"\t12.2 Q7 [2*t^2 + 3*t, sin(t), 2*e^(3*t)]\n"
]
}
],
"source": [
"# Solve some problems with it. Let's start with 12.2, Q6\n",
"\n",
"t = var('t')\n",
"r = vector([-4*t^2 + 1, -1*e^(-1 * t), 5 * sin(-4*t)])\n",
"print(\"12.2 Q6\", vvf_integrate(r, t))\n",
"\n",
"print(\"\\nAdd initial conditions manually\\nto each term for the next problem:\")\n",
"r = vector([3 + 4*t, cos(t), 6*e^(3*t)])\n",
"print(\"\\t12.2 Q7\", vvf_integrate(r, t))"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "c2f12065-edfb-481d-a1c3-53c0bc6870e6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/t"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"diff(ln(2*t))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "90b7dbef-b184-40de-a82c-415fe3f79676",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2.30052398302186\n"
]
}
],
"source": [
"# Solving 12.2 Q9. We're asked to find the angle of \n",
"# intersection between two vectors. Originally I though we could just \n",
"# use get_angle_between_vectors with respect to the two vectors \"as is\", \n",
"# but it turns out we needed the angle between the tangent vectors\n",
"# of each funciton, so get the derivative first.\n",
"\n",
"# Copied from another notebook -- at some point this needs to be a library!\n",
"def get_angle_between_vectors(v: vector, w: vector):\n",
" \"\"\"returns the angle between two vectors in radians.\"\"\"\n",
" cos_theta = (v*w)/(v.norm() * w.norm())\n",
" return acos(cos_theta)\n",
"\n",
"r1 = vector([2*t, t^3, -1*t^4])\n",
"r2 = vector([sin(-2 * t), sin(2*t), t - pi])\n",
"\n",
"angle_as_fn = get_angle_between_vectors(diff(r1, t), diff(r2, t))\n",
"print(N(angle_as_fn(t=0)))"
]
},
{
"cell_type": "markdown",
"id": "71386c8e-d204-4cb4-9e72-8851aa6493ff",
"metadata": {},
"source": [
"## Solving another unit tangent vector problem\n",
"\n",
"Question 12.2 # 10 asks us to find the unit tangent vector at the point $t = 0$ for the function $r(t) = \\langle 4t^5 -2, -2 e^{3t}, -2 sin(5t) \\rangle$\n",
"\n",
"Let's do this using the technique we developed earlier:"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "a14c1850-3768-49c2-952a-ab268bb8dc1b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(4*t^5 - 2, -2*e^(3*t), -2*sin(5*t))\n",
"Tangency point: (-2, -2, 0)\n",
"Unit tangent vector: (0, -3/34*sqrt(34), -5/34*sqrt(34))\n"
]
}
],
"source": [
"# Set up the curve\n",
"t = var('t')\n",
"r = vector([4*t^5 - 2, -2*e^(3*t), -2 * sin(5*t)])\n",
"print(r)\n",
"\n",
"\n",
"# Get the derivative, and plug in the point at which we want to draw the unit tangent vector\n",
"r_prime = vector(vvf_diff(r, t))\n",
"tangency_point = r(t=0)\n",
"print(\"Tangency point: \", tangency_point)\n",
"\n",
"# Calculate the tangent vector using the formula above\n",
"tangent_vector = r_prime/r_prime.norm()\n",
"print(\"Unit tangent vector:\", tangent_vector(t=0))"
]
}
],
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