Sketch the curve with the given vector equation. We can think of a curve in the plane as a path of a moving point. Space Curves and Vector-Valued Functions In Section 10. Tangents of Parametric Curves When a curve is described by an equation of the form y= f(x), we know that the slope of the. 9 Exercise: The alpha curve is given implicitly by y2 = x3 + x. r(t)=t^2i + t^4j + t^6k Please show work. (a) Sketch the plane curve r(t) = (2sint)i+(3cost)j. the formula for the spectral curve; and (d) prove the existence of the quantum curve, or the stationary Schr odinger equation. (a) Make a large sketch of the curve described by the vector function , , and draw the vectors r (1), r (1. Vector Fields. Start studying Math 80 - Calculus III - Chapter 13. Solution to Problem Set #4 1. r(t) = 2 \cos t i + 2 \sin t j + k. (a) Sketch the plane curve with the given vector equation. Then sketch a graph locating points at 0 ≤ t ≤ 3 and indicate the orientation of the curve. Question: Sketch the curve with the given polar equation by first sketching the graph of r as a function of {eq} \theta {/eq} in Cartesian coordinates. Solution: When t is less than 0 the x, y, and z coordinates are all decreasing functions of t, so the vector is moving down the curve toward the origin. We have already proved this in the case n=1,and the general case is not any harder. We have dealt extensively with vector equations for curves, ${\bf r}(t)=\langle x(t),y(t),z(t)\rangle$. Select the rectangular sketch you just made for the profile and the equation given curve sketch for the path. Therefore the surface is a union of all such circles, that is, a circular cylinder. r(t) = (t, sin t, 2 cos t) 16. They are mostly standard functions written as you might expect. Consider the space curve given by the graph of the vector function r(t) = h1 + cost;sint;ti; 0 t 2ˇ: Sketch the curve and indicate the direction of increasing t in your graph. Clearly identify the direction of motion. Consider the surface given by 9x 2 +y -4z 2 = 36. The parametric equations are: x t cost y t sint,0≤t. x = 5t , y = t + 2. 9 Exercise: The alpha curve is given implicitly by y2 = x3 + x. We need a second such vector, then their cross product will give us a vector perpendicular to the plane. This is called a parameter and is usually given the letter t or θ. Download Flash Player. t sin 5t, t2, t cos 5t. The polar curve C has equation. A vector-valued function, or vector function, is simply a function whose domain is a set of real numbers and whose range is a set of vectors. Sketch the curve with the given vector equation. Space Curves and Vector-Valued Functions In Section 10. (c) Sketch the position vector and the tangent vector for the given value of t rt1 cos ti 2 sin tj t 6 rte" is broken down into a number of easy to follow steps, and 37 words. Sketch the curve with the given vector equation. Show that the curve of intersection. (9pts) A curve is given by the equation r + 1 r = 4cosµ in polar coordinates. 2, a plane curvewas defined as the set of ordered pairs. Click below to download the free player from the Macromedia site. Find r'(t). 1 Calculus of Vector-Functions Find a vector equation for the curve of intersection of two surfaces. solve for t in one equation than the other. Explain your reasoning. We are given one vector paralell to the plane, namely the direction vector of the line; that is, the vector h2;¡1;3i. Find an equation of the osculating plane of the curve 2 3 xt yt zt at t 1. Write down the coordinates of the minimum point of the curve with equation. z = y2 2x: A trough with parabolic walls, and. How do i go about solving this problem and eventually leading to a sketch of the graph?. (Notice that in some of the pictures all of the vectors have been uniformly scaled so that the picture is more clear. x t y t 2 and 3 5. EXAMPLE 6 2Find a vector function that represents the curve of the intersection of the cylinder x. As q varies between 0 and 2 p, x and y vary. A tangent is a line that just touches a curve at one point, without cutting across it. This will be useful when finding vertical asymptotes and determining critical numbers. Oblique Cartesian coordinate system ( u, v ) defined by: u = x + y, v = x Acˆ’ y. Compute the value of Z C ~F d~r. GeoGebra can create implicit curves defined by equations, if the equations are written using polynomials in \(x\) and \(y\). We can draw a picture of these slopes by drawing a small line (or arrow )indicating the direction of the curve at each point we have considered. The position vector r(t) of a particle is given by r(t)=sinti+tj+costk. 1 Space Curves and Vector-Valued Functions 12. About: Beyond simple math and grouping (like "(x+2)(x-4)"), there are some functions you can use as well. Plane curves. • Find a set of parametric equations to represent a curve. (a)Find and sketch the equation of the curve obtained by intersecting the surface with the xy-plane. •Let P(x,y,z) be any point in space and r,r 0 is the position vector of point P and P 0 respectively. All points with r = 2 are at. Sketch the curve with the given vector equation. Sketch the position vector r(t) and the tangent vector r'(t) for the value t = 1. The curve is given parametrically by x = e2t;y = et. projection is (iii), rather than the two other graphs. Sketch the curve with the given vector equation. ” Change the path alignment type to “Direction Vector. At what point? First answer this question by making a careful sketch on graph paper, and then nd a way to solve it using a system of equations. asked by Melissa on January 13, 2016; calculus. 2x23y2=1 3x2+y2=15 Hint: Put u=x2 and v=y2. The Attempt at a Solution So far, I have <1, -1, 2> and points (0, 2, 0). The vectors and are unit vectors along the positive the x and y axes respectively. Show that the curve of intersection. (a) Make a large sketch of the curve described by the vector function , , and draw the vectors r (1), r (1. Indicate with an arrow the direction in which t increases. Take a look at the blue and red graph and their equations. Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. Consider the space curve given by the graph of the vector function r(t) = h1 + cost;sint;ti; 0 t 2ˇ: Sketch the curve and indicate the direction of increasing t in your graph. From definition, write the parametric equations of vector function r ( t ) = 〈 t , 2 − t , 2 t 〉. 49)x = -4z2, no limit on y A)Cylinder B)Hyperboloid of two sheets C)Sphere D)Parabolic cylinder 49) Find the curvature κ and radius of curvature R for the curve at the given point. The derivation of the equation for the tangent plane just involves showing that the tangent plane is normal to the vector {\bf n} = (f_x (x_0 , y_0 ),. Parametric vectorial equations of lines and planes. The following steps are helpful when sketching curves. x = 5t , y = t + 2. Indicate with an arrow the direction in which the curve is traced as t increases. Use this fact to help sketch Rhe. If the differential equation is represented as a vector field or slope field, then the corresponding integral curves are tangent to the field at each point. Remark: It is quite labor-intensive, but it is possible to sketch the phase. At this point, you may notice a similarity between vector-valued functions and parameterized curves. In order to get the sketch will assume that the vector is on the line and will start at the point in the line. – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. Autonomous equations. (a) (7pts) Find an equation of the curve in Cartesian coordinates. methods of separable equations and Linear First order equations. Indicate with an arrow the direction in which t increases. From definition, write the parametric equations of vector function r ( t ) = 〈 t , 2 − t , 2 t 〉. •Understand vector valued functions and be able to sketch the vectors and curves associated with a vector valued function. Use this fact to help sketch the curve. 700) Sketch the curve with the given vector equation. Find a vector equation and parametric equations for the line segment that joins P to Q. We indicate the location of the projectile at several times in Figure 9. if t ranges from 0 to infinity, then the domain of f is x ≥ 1. Hence equations (1) and (2) together also represent a circle centred at the origin with radius a and are known as the parametric equations of the circle. x t y t 2 and 2 4. Without eliminating the parameter, be able to nd dy dx and d2y dx2 at a given point on a parametric curve. The parametric equations are: x t cost y t sint,0≤t. I want to be able to type that equation with the variable t in it. 1 Vectors in the Plane PreCalculus 6 - 2 Example 3: Sketch vector v in standard position with magnitude 41 and direction 220. Use your calculator to find enough number of points using the conventional parametric form and sketch the curve. Now, we're going to tackle solving those equations to find the roots of a curve - that is, the places where the curve crosses a given straight line. Sketch the curve with the given vector equation. We can think of a curve in the plane as a path of a moving point. com - id: 7b9eb1-ZjE3Y. Then graph, showing that the 2 equations graph the same curve. x t y t 2 and 3 5. The direction of motion of a parametric curve Evaluation of parametric equations for given values of the parameter Sketching parametric curve Eliminating the parameter from parametric equations Parametric and rectangular forms of equations conversions: P arametric equations definition. (a) Find (b) Sketch the plane curve together with position vector and the tangent vector for the given value of. The difference between a vector and a vector field is that the former is one single vector while the latter is a distribution of vectors in space and time. a) Domain: Find the domain of the function. Consider a vector function such as r = r 0 + t v, then the vector function possesses space curve is a line passing through the point r 0 and parallel to vector v. However, a two-dimensional coordinate system is insufficient for modeling many physical phenomena. As we will see below, the gradient vector points in the direction of greatest rate of increase of f(x,y) In three dimensions the level curves are level surfaces. x e y t t= + =2t 1 3 1 2 2, ,− − ≤ ≤. That trail is the parametric curve, the point is the. The solution manual basically says: Since $(x+2)^2 = t^2 = y-1 => y = (x+2)^2 - 1$, the curve is a parabola. Also notice that there are eight vector elds but only six pictures. Co-planar and collinear points. When working by hand, one useful approach is to consider the "projections'' of the curve onto the three standard coordinate. 1 of the text discusses equilibrium points and analysis of the phase plane. •Let P(x,y,z) be any point in space and r,r 0 is the position vector of point P and P 0 respectively. A knot vector, , must be specified. Indicate with an arrow the direction in which increases rt1 t, 3t, t 1. I am completely lost on finding the length though. Sketch the graph of the plane curve given by the vector - valued function, and, at the point on the curve determined by r (t 0), sketch the vectors T and N. " - Bruce Horn, creator of the Macintosh Finder. Sketch the curve. (a) Sketch some level curves of this surface. Learn exactly what happened in this chapter, scene, or section of Calculus AB: Applications of the Derivative and what it means. Indicate with an arrow the direction in which t increases. Eliminating t from the equations, we get x = y2, which is the. The vector function then tells you where in space a particular object is at any time. x t y t 2 and 2 4. Make a table of values and sketch the curve, indicating the direction of your graph. It's going to be perpendicular to the tangent of the level curve and pointing in the direction where f(x,y) is increasing. Sketch the vector function f(t) = < t 2, t 3 >. The Attempt at a Solution So far, I have <1, -1, 2> and points (0, 2, 0). Given a function sketch, the derivative, or integral curves Use the language of calculus to discuss motion Explain what the effect of a discontinuity in a function has on the derivative and the integral curves. This equation can be used to modeled the growth of a population in an environment with a nite carrying capacity P max. Notice that the curve in part h is just the sum of the curves in f and g. (6)What is the area between the curve and the x-axis for the curve described by the parameteric equations. vector T(t) at the point. Represent a parameterized curve using a vector-valued function. This is the equation of a circle of radius r, with center at the origin (0, 0). (b) Find r'(t). Define the limit of a vector-valued function. 6Let r(t) = eti+e tj. At this point, you may notice a similarity between vector-valued functions and parameterized curves. This vector is normal to the level curve f = 7. A similar technique can be used to represent surfaces in a way that is more general than the equations for surfaces we have used so far. 6] Curves and Surfaces Goals • How do we draw surfaces? – Approximate with polygons – Draw polygons • How do we specify a surface? – Explicit, implicit, parametric • How do we approximate a surface? – Interpolation (use only points) – Hermite (use points and tangents). ) (b) Sketch level curves of the graph of a function of two variables (c) Match algebraic formulae, sketches of graphs, and sketches of level curves 8. Consider the surface given by 9x 2 +y -4z 2 = 36. Best Answer: notice that the j-coordinate is the cube of the i-coordinate. Control points. Part (a) asked students to find the area of the region common to the interiors of both graphs. Vectors and Geometry in Two and Three Dimensions §I. The derivative (or gradient function) describes the gradient of a curve at any point on the curve. Illustrate the concept by examining the superbola , ,. Use your calculator to find enough number of points using the conventional parametric form and sketch the curve. b) find r'(t) c) Sketch the position vector r(t) and the tangent vector r'(t) for the tiven value of t. EXAMPLE 6 2Find a vector function that represents the curve of the intersection of the cylinder x. Be able to nd the domain of vector-valued functions. This Curve Sketching Worksheet is suitable for 12th Grade. Sketch the curve with the given vector equation and indicate with the arrow the direction of the curve: a) r(t)=<1+t, 2+t, 3+2t>. [3800335]- Consider the given vector equation. Control points. It's going to be perpendicular to the tangent of the level curve and pointing in the direction where f(x,y) is increasing. For each curve give vector and parametric equations, and sketch the curve. Let C1 and C2 be defined by the polar equations r = cos 8 + 1 and r = cos 8 -1, respectively. Of course, doing this at just one point does not give much information about the solutions. 2 Example (Section 9. Given a unit length vector ^, consider an axis oriented in the direction of ^. Write down their equations and sketch their graphs. Part (a) asked students to find the area of the region common to the interiors of both graphs. The finite region. To obtain a tangent vector, you can either use the fact that this vector is parallel to lines of slope ( 5=4), or you could directly try to nd a tangent vector T by algebraically determining a. Thus z= 1 k (l mx ny) and so x= acost y= asint z= 1 k (l macost nasint): 4. Repeating the above operations for all other points on the two silhouette contours, we obtain all deformed curves that describe a new 3D deformed shape. If g(x, y) = x2 + y2 − 6x, find the gradient vector ∇g(2, 4) and use it to find the tangent line to the level curve g(x, y) = 8 at the point (2, 4). r(t)=t^2i + t^4j + t^6k Please show work. (a) Sketch graphs of functions of two variables or level sets of functions of three variables. [10 points] Sketch the level curves of z = exy for z = 0, 1, and 2. Construct parametric equations for curves in the plane and in space. Find more Mathematics widgets in Wolfram|Alpha. Then eliminate the parameter. 2, a plane curvewas defined as the set of ordered pairs. How do i go about solving this problem and eventually leading to a sketch of the graph?. (c) Sketch the position vector r(t) and the tangent vector r/(t,) for the value t (use the same graph as for (a) 5. the ball was farther up the incline when it turned around. A vector-valued function, or vector function, is simply a function whose domain is a set of real numbers and whose range is a set of vectors. Thus an equation of the line is given by the vector equation 2 HOMEWORK 2 SOLUTIONS, MATH 175 - FALL 2010 sketch this. Example 10. (Simple cases only. Indicate with an arrow the direction in which t increases. Window Settings. Parametric Vector Functions. Find more Mathematics widgets in Wolfram|Alpha. [10 points] Find a vector function that represents the curve formed by the intersection of the cylinder given by x2 +y2 = 16 and the plane give by x+z = 5. You can visualize a vector field by plotting vectors on a regular grid, by plotting a selection of streamlines, or by using a gradient color scheme to illustrate vector and streamline densities. (6 Points) Find a vector parallel to the line of intersection for the two planes x+ 2y+ 3z= 0 and x 3y+ 2z= 0: Solution: A vector which gives the direction of the line of intersection of these planes is. y 1 2 3 x 1 2 3 (b) Find an equation for the plane tangent to this surface at. Study Guide Block 2: Vector Calculus Unit 4: Polar Coordinates I 2. Find two curves lying on the sphere S: x2 + y2 + z2 = 1. To get the second. Indeed, given a vector-valued function we can define and If a restriction exists on the values of t (for example, t is restricted to the interval for some constants then this restriction is enforced on the parameter. Space Curves and Vector-Valued Functions In Section 10. (a) Sketch some level curves of this surface. Sketch the curve for d d44t Solution ± ± 5 10 2 4 6 8 10 12 14 16 x y Finding the cartesian equation of a curve To find the cartesian equation of a curve from its parametric equations you need to eliminate the parameter. I need a response within next two days! Could you give me the command to use if there is one or how to do it? Thanks!. [10 points] Find an equation for the line in R3 going through the points (1,−2,3) and (7,2,1). Let Cbe the intersection of x2 +y2 = 1 and y+z= 1. equation that contains x and the given quantities. Question Details SCalcET8 13. Elimination of the parameter gives us , which is the equation of a parabola with vertex at ( 1, 0). So on the graph that was provided it will point in negative y and positive x. Let s find out what happens when those values change. ~F = x ıˆ+y ˆâ C is the semicircle of radius 1 from ( 1,0) to (1,0) with y 0 Answer: 0 2. (c) Sketch the position vector r(t) and the tangent vector r0(t) for t = ˇ=3. Get an answer for 'How can I find the coordinates of the points at the curve y=x2-x-12 where it cuts the x axis and y axis?' and find homework help for other Math questions at eNotes. Write down the coordinates of the minimum point of the curve with equation. With C1 and C2 as in part (a), sketch these two curves. 2, a plane curvewas defined as the set of ordered pairs. (θ is normally used when the parameter is an angle, and is measured from the positive x-axis. Homework Equations 3. As for the length of the vector, from the origin it moves to the right 2 and up 1. So, the curve is a parabolic curve which concave up to the positive y-axis at the height z = 2. NOTES: There are now many tools for sketching functions (Mathcad, Scientific Notebook, graphics calculators, etc). In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. We are given one vector paralell to the plane, namely the direction vector of the line; that is, the vector h2;¡1;3i. 1 - Vector functions and space curves In this section, we will I de ne vector valued functions I look at space curves of certain vector functions. I am completely lost on finding the length though. Course Description Sequences and series, multi-variable functions and their graphs, vector algebra and vector functions, partial differentiation. Define the limit of a vector-valued function. x t y t t= =2 3, ,0 2≤ ≤ 12. 50)y = cos x, π 4, 2 2 50) Identify the type of surface represented by the given equation. Write the general equation of a vector-valued function in component form and unit-vector form. A tangent is a line that just touches a curve at one point, without cutting across it. GeoGebra can create implicit curves defined by equations, if the equations are written using polynomials in \(x\) and \(y\). These are general guidelines for all curves, so each step may not always apply to all functions. The parametric curves traced by the solutions are sometimes also called their trajectories. Write down their equations and sketch their graphs. Remark: It is quite labor-intensive, but it is possible to sketch the phase. y 1 2 3 x 1 2 3 (b) Find an equation for the plane tangent to this surface at. At this point, you may notice a similarity between vector-valued functions and parameterized curves. 832 CHAPTER 12 Vector-Valued Functions Section 12. (a) Make a large sketch of the curve described by the vector function , , and draw the vectors r (1), r (1. As we will see below, the gradient vector points in the direction of greatest rate of increase of f(x,y) In three dimensions the level curves are level surfaces. b) find r'(t) c) Sketch the position vector r(t) and the tangent vector r'(t) for the tiven value of t. Find the derivative r '(t) of the vector function rt). 1 Pointsand Vectors Each point in two dimensions may be labeled by two coordinates (a,b) which specify the position of the point in some units with respect to some axes as in the ﬁgure on the left below. (b) Draw the vector r '(1) starting at (1, l) and compare it with the vector Explain why these vectors are so close to each other in length and direction. (10 points) Sketch the plane curve with the vector equation r(t) = e2ti+ etj; and sketch the position vector r(t) and the tangent vector r0(t) for t = 0. 1 Calculus of Vector-Functions Find a vector equation for the curve of intersection of two surfaces. They are a very simple thing, worth to study once and then feel comfortable in the world of vector graphics and advanced animations. r(t) = 2 \cos t i + 2 \sin t j + k. However, there is one idea, not mentioned in the book, that is very useful to sketching and analyzing phase. Get an answer for 'I need help to sketch the vector field F(x,y) = 2xi - yj by choosing four relevant vectors in each quadrant. Find two curves from (0;0;0) to (2; 1; 1). The curve C has a turning point P and crosses the x -axis at the point Q as shown in Figure 2. Parametric equations allow us to describe a wider class of curves. Question: Sketch the curve with the given vector equation r(t)=. Select the rectangular sketch you just made for the profile and the equation given curve sketch for the path. To obtain a tangent vector, you can either use the fact that this vector is parallel to lines of slope ( 5=4), or you could directly try to nd a tangent vector T by algebraically determining a. Be sure to indicate the orientation of the graph. 2, #14 Direction Fieldsand Euler’s Method Sketch the direction ﬁeld of the diﬀerential equation. A similar technique can be used to represent surfaces in a way that is more general than the equations for surfaces we have used so far. Problem 1(b) - Fall 2008 Find parametric equations for the line L of intersection of the planes x 2y + z = 10 and 2x + y z = 0: Solution: The vector part v of the line L of intersection is orthogonal to the normal vectors h1; 2;1iand h2;1; 1i. Find two curves lying on the sphere S: x2 + y2 + z2 = 1. 832 CHAPTER 12 Vector-Valued Functions Section 12. 1 Calculus of Vector-Functions Find a vector equation for the curve of intersection of two surfaces. Solution for Sketch the curve with the given vector equation. Then sketch a graph locating points at 0 ≤ t ≤ 3 and indicate the orientation of the curve. For example, a vector pointing in the northeasterly direction of the plane is {1,1}. A knot vector, , must be specified. The graph of any solution to this di erential equation passing through the point (x;y) = (2;1) has slope. Vector Operations The two basic vector operations are vector addition and scalar multiplication. This will be useful when finding vertical asymptotes and determining critical numbers. Parametric Equations for Curves (line segments, circle, ellipses). plane o) yz—plane O a;z— plane -axis X -axis 1 label the appropriate axes. x t y t 2 1 and 1 2. ENGI 4430 Parametric & Polar Curve Sketching Page 1-01 1. - 1282650. Parametric Equations of a Plane Curve. We have dealt extensively with vector equations for curves, ${\bf r}(t)=\langle x(t),y(t),z(t)\rangle$. Without eliminating the parameter, be able to nd dy dx and d2y dx2 at a given point on a parametric curve. Then write a second set of parametric equations that represent the same function, but with a faster speed and an opposite orientation. M Worksheet by Kuta Software LLC. To draw the curve, you have to specify the ( x , y ) coordinates of the points where the curve starts and ends. Indicate with an arrow the direction in which increases rt1 t, 3t, t 1. Which of the following is a tangent vector to the curve given by ˘ ˇ ˆ ˙ Describe in words the surface with cylindrical coordinate equation I A. The equations m 2 are called. Also, if one is given a point a on the line and a direction vector d for the line then the parametric form is r = a+td, t ∈ R. Sketch the curve with the given vector equation. r '(t) = i + j + k Find the unit tangent vector T(t) at the point with the given value of the parameter t. Vector functions can be difficult to understand, that is, difficult to picture. This required students to divide the region into two subregions, bounded by arcs determined by the given values of ,. Example 1: Find a set of parametric equations for the rectangular equation y = x 2 + 1, given t = 2 - x. Since we are not given a normal vector, we must find one. At this point on the curve, the unit normal vector is parallel to a 11,8, 9, the binormal vector is parallel to b 12, 12,4 and the unit tangent vector is parallel to c 1,2,3. The radius of curvature is opposite proportional to its arc measured from the origin. as shown by the blue curve, below. (b) Find rt (t). Find a vector equation and parametric equations for the line segment that joins P to Q. Find the equation of the line perpendicular to 5x−3y+z = 5 through the point Q(1,2,3). Vector Fields. equation (i) by x, but only equation (i) is exact. I want to talk about how to get a parametric equation for a line segment. So they can get the trends of it and confirm whether the given function is right. GeoGebra can create implicit curves defined by equations, if the equations are written using polynomials in \(x\) and \(y\). Show that the curve with parametric equations x = sin t, — sin2t is the curve of intersection of the Y = cos t, z surfaces z = x 2 and x 2 + y — 1. uk Functions (H) - Version 2 January 2016 (b) The diagram below on the left shows a sketch of the graph y = x2. 700) Sketch the curve with the given vector equation. If you, for instance, write y = sin(x) GeoGebra will classify this as function, a function that will be given a name by. (6 Points) Find a vector parallel to the line of intersection for the two planes x+ 2y+ 3z= 0 and x 3y+ 2z= 0: Solution: A vector which gives the direction of the line of intersection of these planes is. The calculus of vector functions and parametric surfaces. Notice that the curve in part h is just the sum of the curves in f and g. Calculus: Learn Calculus with examples, lessons, worked solutions and videos, Differential Calculus, Integral Calculus, Sequences and Series, Parametric Curves and Polar Coordinates, Multivariable Calculus, and Differential, AP Calculus AB and BC Past Papers and Solutions, Multiple choice, Free response, Calculus Calculator. On problems 11–12, a curve C is defined by the parametric equations given. The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0. x t y t t= =2 3, ,0 2≤ ≤ 12. Given a space curve, there are two natural geometric questions one might ask: how long is the curve and how much does it bend? In this section, we answer both questions by developing techniques for measuring the length of a space curve as well as its curvature. 1 Let R(t) = et cost~i+t~j +tet~k be position of a particle moving in space at time t. (a)Sketch the plane curve with the given vector equation. 832 CHAPTER 12 Vector-Valued Functions Section 12. 49)x = -4z2, no limit on y A)Cylinder B)Hyperboloid of two sheets C)Sphere D)Parabolic cylinder 49) Find the curvature κ and radius of curvature R for the curve at the given point. • Find a set of parametric equations to represent a curve. sketch the coordinate curves, and the corresponding unit vector basis for the given coordinate system. Observe that x and y satisfy the equation: x2 y2 1 So the curve traced out by r is a part of the unit circle. parametric equations that represent the same function, but with a slower speed 14) Write a set of parametric equations that represent y x. on the curve given by the parametric equations, as shown in Figure 12. Find points on the curve that correspond to u = 0, 0. How do you find a vector equation and parametric equations in t for the line through the point and perpendicular to the given plane. However, a two-dimensional coordinate system is insufficient for modeling many physical phenomena. Oblique Cartesian coordinate system ( u, v ) defined by: u = x + y, v = x Acˆ’ y. By signing. Define the limit of a vector-valued function.