# Curve sketching chart

The Organic Chemistry Tutor 120,653 views 41:29 Curve sketching with calculus: logarithm. In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. However Sal doesn’t use the sign diagram/chart approach that I like and so I am including that as well in the handout to help you. Draw a rough sketch of the graph. 1. You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. 006/016, Calculus I New York University April 1, 2010 . You may need to hit Enter to wrap your typing to the next line. The graph of the derivative function. Curve Sketching Learning Outcomes Make tables and draw the graphs of various equations to include: Linear Functions Quadratic Functions Cubic Functions – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. SOLUTIONS TO GRAPHINGOF FUNCTIONS USING THE FIRST AND SECOND DERIVATIVES Now determine a sign chart for the second derivative, f'' . The graph shows that the values of the  Newton's diagram (also known as Newton's the shape of an algebraic curve close to and far away from the origin. Connecting a function, its first The normal distribution curve is also referred to as the bell curve, and it's used to calculate data trends using a curve-and-scatter chart against an X and Y axis. Put the critical numbers in a sign chart to see where the first derivative is positive or negative (plug in the first derivative to get signs). If you are having any trouble with these problems, it is recommended that you review the curve sketching tutorial at the link below. Homework: Pg. Graph is symmetry about  Solve Curve sketching problems with our Curve sketching calculator and problem solver. Sign In. Due to most graphing calculators’ poor resolution, it can also be difficult to get detailed information about the shape of a graph. SECTION 3. 12 Jun 2019 Learn how to sketch curves using differentiation and axis intercepts. Curve Sketching packet. Due to most graphing calculators' poor resolution, it can also be difficult to get detailed information about the shape of a graph. Symmetry about the Axis: Graph is symmetry about y-axis, if we replace x by -x and the given equation of the curve does not change. too big for my taste; I don't feel like drawing my graph that wide, so I'll quit at x = 16. Do any values of 𝑥𝑥 correspond to holes or vertical asymptotes? Find 𝑥𝑥-intercepts of 𝑓𝑓(𝑥𝑥): (b) Determine 𝑥𝑥 for which 𝑓𝑓(𝑥𝑥) = 0. Graphing y = cos x. First, one should know how to solve partial problems, for instance how to find the domain, determine asymptotes,  Intervals x-axis: y-axis: Reticule lines x-axis: y-axis: Dashes length x-axis: y-axis: Decimal places: Gap at origin: Graph thickness: Circle at origin: Log. You get only two attempts at each question. NOTES 05. This will be useful when finding vertical asymptotes and determining critical numbers. However, this page came about because I have often been asked specifically how to create a Normal distribution curve in Excel. Look below to see them all. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Please try again later. g. Under your name type a few sentences about what you learned. The sign chart of f ' is −∞ −3 −1 2 ∞ is: − + − + A sign chart is a real number line that shows where the derivative is positive (+) and negative (−). a) Domain: Find the domain of the function. Choose from different chart types, like: line and bar charts, pie charts, scatter graphs, XY graph and pie charts. This is the currently selected item. by M. Curve sketching . The diagram shows the same parametric curve we have just studied where we have included some arrows to illustrate the orientation. f. Curve Sketching A good graphing calculator can show you the shape of a graph, but it doesn’t always give you all the useful information about a function, such as its critical points and asymptotes. day 11 ­­ intro to curve sketching n2017. 4. 6. Intercepts: 0)0(. When curve  )3,2( is a point of inflection. We can obtain a good picture of the graph using certain crucial information provided by derivatives of the function and certain 1 Graphing the Derivative of a Function Warm-up: Part 1 - What comes to mind when you think of the word 'derivative'? Part 2 - Graph . In the list below, you’ll see some steps grouped if they are based on similar methods. Next lesson. The tutorial includes numbers, graphs, and examples of how the PPF is created. of 𝑓𝑓: (a) Determine 𝑥𝑥 for which 𝑓𝑓(𝑥𝑥) is undefined. We'll cover two types of curves. Curve Sketching: General Rules. It is an application of the theory of curves to find their main features. Selection File type icon File name Description Critical Points Chart. a. ECON 103 - exam 2 study guide by daniellevillalobos includes 11 questions covering vocabulary, terms and more. 3. Here are instruction for establishing sign charts (number line) for the first and second derivatives. To graph a rational function, find the asymptotes and intercepts, plot a few points on each side of each vertical asymptote and then sketch the graph. Plot a The function is discontinuous at x = 1, because ln 1 = 0. Get smarter on Socratic. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Complete the square and find composite functions for Higher Maths. f ′(x). GeoGebra Applet Press Enter to start activity  Demonstrates, step-by-step and with illustrations, how to graph logarithms. Curve Sketching unit plan (blue) Graphs for notes today (stapled together but you need to cut out) Orange chart (also for notes today) WHAT YOU NEED TO BE DOING… Get our your Unit 3 review HW & unit plan so you can show me that you did at least 40 problems. scale x: No Our all Data Visualization GUI Charts Graphs Diagrams Tables free resources for Sketch App by Bohemian. Curve sketching overview in chart form Quiz review with solutions Quiz Trig 3 review Quiz Trig 3 review answer key Test rvw w/answrs (no Newton or prop err) Extra Test review Optimization practice Kahoot questions Detailed Example of Curve Sketching x Example Sketch the graph of f(x) = . 11. It is very easy to do, and once you master it, the possibilities are endless! Chart and Diagram Slides for PowerPoint - Beautifully designed chart and diagram s for PowerPoint with visually stunning graphics and animation effects. One of the easiest curves to create using curve stitching is a parabola. Use your graph to answer the following question. Background for black and white chart 8. Let’s put it all together; here are some general curve sketching rules: Find critical numbers (numbers that make the first derivative 0 or undefined). Graphing Parabolas With Microsoft Excel Mr. Create online graphs and charts. Graphing Quadratic Equations. 5 Choosing the Scales for a Graph or Chart. jpg Curve sketching with calculus: logarithm. Create professional flowcharts, process maps, UML models, org charts, and ER diagrams using our templates or import feature. Graph of the Function. Let's look at a couple of techniques for making our curve-drawing life a little easier. 3c2a7ee4a106e660a0d79eaa74f13cef. pdf from AP CALC at New Milford High School. pdf. 4 1st and 2nd Derivative Test · 3. What is a Z Table: Standard Normal Probability. = f x xf. While such a curve is an excellent approximation when there are many producers (or consumers), each of the curves is actually made up of many small discrete steps. Be sure you get feedback on Functions and Their Graphs Jackie Nicholas Janet Hunter Jacqui Hargreaves c 1997 University of Sydney. NOTES: There are now many tools for sketching functions (Mathcad, Scientific Notebook, graphics calculators, etc). See the many curve sketching she loves math. Give a complete graph of f(x) = 1 3 x3 1 2 x2 2x+ 1: Be sure to show on a sign chart where the function is increasing/decreasing, concave up/concave down, and identifying (as ordered pairs) all relative extrema and in ection points. The parabola is the envelope of the straight lines. Curve Sketching: Classwork The one problem that comes up repeatedly on AP exams is sketching a possible graph of f (x) AP Calculus AB/BC - M. After completing the chart, graph the ordered pairs in the chart. Which of the EXAMPLE: Sketch the graph of a twice-differentiable function y=f(x) with the following  To plot quality graphs that can be used for academic and research publication purposes, which software application will you recommend? Graph Drawing. CURVE SKETCHING I. About: Beyond simple math and grouping (like "(x+2)(x-4)"), there are some functions you can use as well. Curve Sketching - First & Second Derivatives - Graphing Rational Functions & Asymptotes - Calculus - Duration: 41:29. Our new CrystalGraphics Chart and Diagram Slides for PowerPoint is a collection of over 1000 impressively designed data-driven chart and editable diagram s guaranteed to impress any audience. Clausen Algebra 2 California State Standard for Algebra 2 #10. Thus, a curve is a generalization of a line, in that it may be curved. The ten steps of curve sketching each require a specific tool. To sketch the curve we will need to know the nature of this asymptote. They are mostly standard functions written as you might expect. . Here is my smoothed line chart of the curve chosen, as depicted above, with the addition of some little "bumps" along the x-axis which will be discussed later: Lucidchart is your solution for visual communication and cross-platform collaboration. The z-table helps by telling us what percentage is under the curve at any particular point. A parabolic curve is a two-dimensional drawing that seems to make a curve. Derivatives can help graph many functions. geometry, curve sketching (or curve tracing) includes techniques that can be used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot. It is important in this section to learn the basic shapes of each curve that you meet. Learning to recognize the formulas of these equations will help in sketching  The graph of the derivative off. 51, 3. GraphPlot and GraphPlot3D are suitable for straight line drawing of general graphs. 0. Graph of function of two variables. They are interesting curves because they have discontinuities. I want to do a really challenging curve sketching example. Make a sign chart for this function. Bourne. MAFA chart Plotter is a server based function plotting program which allows you to plot your function graphs online without any installation. The following example shows us how to sketch the  Graph - Graph is a program for drawing graphs of mathematical functions in a coordinate system. There are many ways to create the graph, using line charts, bar charts, area charts, and scatter plots. ∈ . Here’s how you can test the circles and semi-circle functions Create online graphs and charts. the second derivative test can also be used in curve sketching to find relative minima and relative maxima and is the following Explore math with desmos. notebook October 12, 2017 Aim: Introduction to Curve Sketching Do Here graphs of numerous mathematical functions can be drawn, including their derivatives and integrals. 5, with 6 variations of a and c given. Justifications and Curve Sketching in AP Calc We returned to their chart over and over throughout the lesson and it's something they can keep to study from in the We look at a number of examples of circle and semi-circle functions, sketch their graphs, work out their domains and ranges, determine the centre and radius of a circle given its function, etc. is shown in the figure above. As with all of Sal’s films (Sal is the founder of Khan Academy), these are solid and thorough. In this section we will discuss what the first derivative of a function can tell us about the graph of a function. (Note: this function is only deﬁned ln x for x > 0) 1. set up a sign chart with cps Curve Sketching. f(x) = p 3 x2 ln(x + 1) ( 1;0) [ 0; p 3 i Where might you expect f(x) to have a vertical asymptote? What does the function look like nearby? In the latter instance, if the intercept is of, say, multiplicity 2, you know that the graph "bounces" at the axis, and then heads off the same way it came. If you still use Excel 2003 or before, you should read How to Create Normal Curves with Shaded Areas in Classic Excel. The straight lines do not actually create the curve, they merely approximate it. Relative (local) maximums and minimums and first derivative test 3. After you create the graph, you can add a trendline, and then right-click the extra points and set them to have no marker visible. com info@thinkwell. How to draw curves, pfft! Not so fast! Curves are in a great deal of things you'll want to draw. Get feedback on your graphs. Intervals where function is increasing or decreasing (with use of sign chart and use f' to verify) a. This mathematical paper proves that the curve formed by the method below is a parabola. This means that the graph of f(x) is concave down for x > 3 and concave up for x < 3. Wow! What an exciting topic this sounds like. The computer can areas of the graph to look at: the computer might default to showing us some. = = ⇒. Quizlet flashcards, activities and games help you improve your grades. Is the function increasing or decreasing at x=−5? Increasing. Unit 1: Limits & Continuity The best videos and questions to learn about Examples of Curve Sketching. But at the same time, don't forget that you could already say a lot about this graph with just those pre-calculus skills. Sketch a graph of f(x) using all the information obtained above. Once a parabolic section has been created, you can CURVE SKETCHING I. These are general guidelines for all curves, so each step may not always apply to all functions. Start studying Curve Sketching/Derivatives. Proper graph sketching works on two levels. 5 Man vs machine. The scales have been  For example, the graph below is said to be symmetric about the y-axis (the line x = 0) (2) a spatial perspective so that you could draw a sketch of a graph that  Functions for graph drawing. ,0. Solving a curve sketching task. To start with simply, choose a curve where everything is set equal to y, as above. In the examples considered so far, the scales on both axes have been provided for you. Curve Sketching Table - Kara Klinke - TeachersPayTeachers. So you can draw a dot at the intercept, and then sketch in the portion of the graph which is right around that intercept, showing the curve bouncing off the axis. e. Try the quiz at the bottom of the page! go to quiz. Chapter 3. Further we use this algorithm for the investigation of functions. 20. Use Wolfram|Alpha to generate plots of functions, equations and inequalities in one, two and three dimensions. The graphs of tan x, cot x, sec x and csc x are not as common as the sine and cosine curves that we met earlier in this chapter. Analyzing a function with its derivative. Curve Sketching - transforming f(x) to f(a-x) 0. However, they do occur in engineering and science problems. 024, -3. 6 Summary of Curve Sketching · 3. thinkwell. Whoops! There was a problem previewing NOTES 05. Essentially, the first and second derivatives are used to determine which of the following pieces will be used where to form the graph: Figure %: The Four Easy Pieces Then, intercepts and asymptotes Guide to Curve Sketching. f(x). 5T Curve Sketching. These cannot be graded by WebAssign. Instead, WebAssign will ask limited submission questions about your graphs. Therefore, this point is not an extremum. 1: Domain, Intercepts, and Asymptotes Curve Sketching Example: Sketch 1 Review: nd the domain of the following function. 1 Some General strategies for graphing polynomials The following steps may be helpful in sketching a general polynomial. As x How to Draw a Parabolic Curve (a Curve with Straight Lines). 5: Curve Sketching Workalong Example 1 to 4 are based on Khan Academy videos. LayeredGraphPlot attempts to draw the  In this section we will explore the graphs of the six trigonometric functions, beginning with the graph of the cosine function. Then find and graph it. Math 170 Curve Sketching I Notes All homework problems will require that you create both a sign chart and a graph. Determine . And a lot of people struggle with them even when they don't realise it. STEP 1 Create A Table Of Points Save your Excel file as LASTNAME FIRSTNAME Parabolas, and save this in your “S:” network directory. Poor‎ > ‎Notes‎ > ‎ Curve Sketching. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Mark these on your graph. To establish a sign chart (number lines) for f' , first set f' equal to zero and then solve for x. 16 Mar 2017 f(x)=f(−x). This article explains how to create the figures in New Excel. You get the shape of the curve. At the same time, we know that we also have to be concave down in this range. Also, iden-tify the y-intercept. Instead, find the intervals for increasing and decreasing, find intervals for concave up and concave down, find x-values for relative max and mins, find points of inflection, and sketch. Curve Sketching: Ex 3. In the examples above, the chart contained smooth curves. = = x x xfy . In the last section, we learned how to graph a point with polar coordinates (r, θ). Type your name in a cell under the chart. \) Domain Find the domain of the function and determine the points of discontinuity (if any). . CALCULUS Curve Sketching (I) Sign Chart for the Second Derivative f00(x) x 1:000 f(x) _ 3:000 ^ f00(x) 0 + In ection Points There is an in ection point at P View 20 - Curve Sketching. This is an input location where either f0(x Welcome; Class Calendars & Syllabi; AP Calculus. Beginning with This feature is not available right now. $$2. Symmetry:. It is simple to use and highly customizable with many parameters at the same time. Intuitively, a curve may be thought as the trace left by a moving point. But just to show where it might matter, I'll animate the same thing again, another function that draws the same curve. 0)(. We say that the function is increasing on the open interval (a, p). And if you just want, you know, an analytical way of describing curves, you find some parametric function that does it. The area under the whole of a normal distribution curve is 1, or 100 percent. 0121. 2: CURVE SKETCHING POLYNOMIALS Example 3. Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function looks like. View Notes - 1Intro to curve sketching. 232 #13, 17, 21, 26, 43; Ignore directions in book. Curve Sketching using Differentiation. )Here is an example: Graphing. Get step-by-step solutions to your Curve sketching problems, with easy . (1) Find the vertical and horizontal asymptotes sure to justify your answers using limits. Even if you have no particular reason to chart a normal curve, you might find the techniques interesting. It is sometimes helpful to use your pencil as a tangent line. 1) Given the graph of f(x) below, complete the chart, estimating the derivative (slope of the tangent line) at the given values of x. Find f′ and form its sign chart. Contents 1Functions 1 illustrate this by sketching the graph. Section 3. Sketch derived, inverse or other related functions using graph translations. Domain: Rx. AP Calculus FAQ's, Exam Results, & College Credit Equivalents; AP Calculus AB. Retrying. Why do I need to learn to sketch curves if I can always just graph them on a 29 Jul 2008 Curve Sketching Using Calculu In this video I discuss the following topics to help produce the graph of a function: domain, x-y intercepts, Curve sketching is not my favorite subject in Calculus, since it's so abstract, but For the differentiable graph, do you see how if the graph goes up and comes Curve sketching is another practical application of differential calculus. com The orientation of a parameterized curve is the direction determined by increasing values of the parameter. 1 Feb 2017 In this article, you'll see a list of the 10 key characteristics that describe a graph. Although these problems are a little more challenging, they can still be solved using the same basic concepts covered in the tutorial and examples. Sketching a complicated function. The general procedure for curve sketching is based on the material learned in the last few sections.$$ Sketching a curve from knowledge of the signs of the first and second derivatives is a useful way to find the approximate shape of a function's graph. Why bother? Graphing utilities are very accessible,  Curve Sketching Warmup on Brilliant, the largest community of math and science problem solvers. 200X Calculus I: Curve Sketching Worksheet November 7, 2012 Techniques for carefully sketching functions When sketching a graph of a function f(x), you want to clearly indicate all the important features of the function, including: its domain, the x- and y-intercepts (maybe), intervals on which the Lesson 21: Curve Sketching (handout) Find when f is posi ve, nega ve, zero, not deﬁned. A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0. An inflection point (or point of inflection) is the point at which the concavity of the graph changes sign. A. We also investigate the concavity/convexity of the given curve. What others are saying When the design team couldn't find a coffee table large enough for the living room, they created a custom-sized table by grouping two industrial-looking tables made from rebar and used glass for the top. Locate critical points algebraically and with calculator graphically 2. Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function looks like. The curve is used as a visual Calculus One – Graphing the derivative of a function. 4 Curve Sketching V63. Plotting and graphing are methods of visualizing the behavior of mathematical functions. Be prepared – these problems take a significant amount of time and care! Start by looking at the domain, range, intercepts, and asymptotes. Mathematics Learning Centre, University of Sydney 1 1 Curve sketching using calculus 1. This chart shows that as x gets increasingly Curve Sketching Module: Horizontal Asymptotes and Infinite Limits www. So I’m going to have to find that in the process of my planning for the graph. Graphs may be added with different color and line styles. The first derivative of a function is the slope of the tangent line for any point on the function! Hopefully you can see that by augmenting your pre-calculus curve sketching skills with calculus, you can learn a little more about the graph of a function. Summary of Curve Sketching. The first derivative will allow us to identify the relative (or local) minimum and maximum values of a function and where a function will be increasing and decreasing. The extra simplification here is that this graph has an asymptote. Analysis of graphs and curve sketching 1. Sketch the graph for. THE FIRST DERIVATIVE TEST: Let's take an arbitrary function like the one whose graph is given below: As x goes from a to p, the graph rises as x moves to the right towards the interval P, (a, p) and the value of f increases. Show Answer. 5. graph{(y-(x^2)/(x^2+9))(y-1)=0 [-7. 7 L'Hopital's Rule · Curve Sketching Chart  asymptotes. Author: matheagle. While you may not be tested on your artistic ability to sketch a  There is no difference in this case. aaaa x . Be careful when creating a sign chart for some radicals, don’t forget find any critical points for the function. 3 & 3. 6 – Curve Sketching 1 . gives us interesting information about the original function. (If. But some of the steps are closely related. In this case, the second  Adding the first and second derivatives to our toolkit of curve sketching will be very . domain. com - id: 49b3e4-M2RiO CALCULUS Curve Sketching (V) Concavity Intervals The second derivative of the function f(x) is given by: f00(x) = d dx x2 + 0x+ 4 x2 8 (x 0)3 The second derivative of the function f(x) is not zero at any x. Check from the equation of the function whether the graph has any type of symmetry. d. When passing through the point $$t = {\large\frac{1}{3}\normalsize}$$ the derivative also changes sign from plus to minus, but the curve $$y\left( x \right)$$ is not a single-valued function in this area. Every set of data has a different set of values. In this section, we learn methods of drawing graphs by hand. So, we can start off with sketching an increasing curve that has is also concave down until we reach $$x = - 1$$. Therefore, the derivative is positive from B to C. Don’t let your lines get too long or it won’t fit on one page. Example 1: The graph of f(x) is shown below. Add HW points so we can figure out your Unit 3 HW grade Click to share this graph on your favourite social network: Add Math 132 Curve Sketching Stewart x3. The curve is symmetric about the y-axis We can build a chart. Learn the vocabulary term critical point. in this worksheet students are presented with a graph and asked to evaluate several limits based on that graph right and left hand limits are included This is a cheesy workaround, but add additional data points that are on the asymptote to your chart's source data. It makes easy to publish networks on Web pages, and allows developers to integrate network  Graph of function of two variables. Curve Sketching If you have been keeping up with my posts, then you will truly appreciate this one as it combines MANY skills we’ve learned to allow us to sketch these different functions based on their derivatives, which tell us our critical values, which are the points on the graph which have a horizontal tangent line. And you don't really care about the rate. Make a . Graphing in polar coordinates is still a bit clunky in GeoGebra 4, so this sketch does a bit of the clunky work for the student by parameterizing the polar equation. Finally, look at slope and concavity information. Finally, sketch a rough graph of just the asymptotes. You lose the input information. Determine the domain of the function. Sometimes arrows are drawn on the curve to denote the orientation. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. 0: Students graph quadratic functions and determine the maxima, minima, and zeros of the function. Curve Sketching. png. Graphs of tan, cot, sec and csc. Mark these x-values underneath the sign chart, and write a zero above each of these x-values on the sign chart. Intervals where function is concave up or concave down How do you sketch the curve #y=x^2/(x^2+9)# by finding local maximum, minimum, inflection points, asymptotes, and intercepts? Calculus Graphing with the Second Derivative Examples of Curve Sketching 1 Answer There is more to distribution fitting than just overlaying a distribution on top of the histogram. I’m going to go through each of the steps you see here in detail. To sketch a graph  we combine them here to produce an accurate graph of the function without plotting lots of extraneous points. The following steps are taken in the process of curve sketching: $$1. 6: Sketching Graphs 3. 2. A Quick Guide to Curve Sketching: Note: You can use this chart to help you solve LOTS of problems, even if you are not asked to ﬁnd the curve. What can you say about the ﬁrst and second derivative. Choose nice colors for the background, gridlines, curve, and axes. When curve sketching making a sign chart of the derivatives is an easy way to spot possible inflection points and to find relative maxima and minima , which are both key in sketching the path of Curve Sketching. com, a free online graphing calculator So, if we start with \(x < - 1$$ we know that we have an increasing function. I’m asked to graph the function f(x) equals 144 over x² plus 12. COM . com. sign chart Chapter 20 - 2 Derivatives in Curve Sketching. )( 2 +. calculus questions with answers 2. Connecting a function, its first derivative, and its second derivative. DAVID LIAO. 6 Homework Solutions · 8. Find the critical points and mark these on your graph. Curve sketching is a  Sigma is a JavaScript library dedicated to graph drawing. Section 4. The following steps are helpful when sketching curves. The computer can do this much better simply by plotting many points, so why The first curve of the section is the Hyperbola, y=c(x^2 - a^2)^. An interactive applet that allows you to see the effects of changing the coeeficients in a cubic function using sliders. pdf from MATH MCS22XA-01 at Midwood High School. 3. 51]}  22 Nov 2017 Chart — the most powerful data visualization plugin for Sketch you will find the file with common parameters like curve type, color palette or  Sketching a curve from knowledge of the signs of the first and second derivatives is a useful way to find the approximate shape of a function's graph. This post goes over the process of how to draw a PPF (production possibilities frontier) or PPC (curve) given a table or opportunity costs. 02, 7. Enter the function of the radius in terms of theta as r(t), and then manually progress t or press play to see the graph. curve sketching chart

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