1 Limits And Continuity 2 The Derivative 3 Topics In Differentiation 4 The Derivative In Graphing And Applications 5 Integration 6 Applications Of The Definite Integral In Geometry, Science, And Engineering 7 Principles Of Integral Evaluation 8 Mathematical Modeling With Differential Equations 9 Infinite Series 10 Parametric And Polar Curves; Conic Sections 11 Three-dimensional Space; Vectors 12 Vector-valued Functions 13 Partial Derivatives 14 Multiple Integrals 15 Topics In Vector Calculus expand_more
12.1 Introduction To Vector-valued Functions 12.2 Calculus Of Vector-valued Functions 12.3 Change Of Parameter; Arc Length 12.4 Unit Tangent, Normal, And Binormal Vectors 12.5 Curvature 12.6 Motion Along A Curve 12.7 Kepler’s Laws Of Planetary Motion Chapter Questions expand_more
Problem 1QCE: If C is a smooth curve parametrized by arc length, then the curvature is defined by s=. Problem 2QCE: Let rt be a smooth vector-valued function with curvature t. (a) The curvature may be expressed in... Problem 3QCE: Suppose that C is the graph of a smooth vector-valued function rs=xs,ys parametrized by arc length... Problem 4QCE: Suppose that C is a smooth curve and that x2+y2=4 is the osculating circle to CatP1,3. Then the... Problem 1ES: Use the osculating circle shown in the figure to estimate the curvature at the indicated point. Problem 2ES: Use the osculating circle shown in the figure to estimate the curvature at the indicated point. Problem 3ES: For a plane curve y=fx the curvature at x,fx is function x . In these exercise the graph of fx and x... Problem 4ES: For a plane curve y=fx the curvature at x,fx is function x . In these exercise the graph of fx and x... Problem 5ES: Use Formula (3) to find t. rt=t2i+t3j Problem 6ES: Use Formula (3) to find t. rt=4costi+sintj Problem 7ES: Use Formula (3) to find t. rt=e3ti+etj Problem 8ES: Use Formula (3) to find t. x=1t3,y=tt2 Problem 9ES: Use Formula (3) to find t. rt=4costi+4sintj+tk Problem 10ES Problem 11ES: Use Formula (3) to find t. x=cosht,y=sinht,z=t Problem 12ES: Use Formula (3) to find t. rt=i+tj+t3k Problem 13ES: Find the curvature and the radius of curvature at the stated point. rt=3costi+4sintj+tk;t=/2 Problem 14ES: Find the curvature and the radius of curvature at the stated point. rt=eti+etj+tk;t=0 Problem 15ES: Find the curvature and the radius of curvature at the stated point. x=etcost,y=etsint,z=et;t=0 Problem 16ES: Find the curvature and the radius of curvature at the stated point. x=sint,y=cost,z=12t2,t=0 Problem 17ES: Confirm that s is an arc length parameter by showing that dr/ds=1, and then apply Formula (1) to... Problem 18ES: Confirm that s is an arc length parameter by showing that dr/ds=1, and then apply Formula (1) to... Problem 19ES: Determine whether the statement is true or false. Explain your answer. A circle of radius 2 has... Problem 20ES: Determine whether the statement is true or false. Explain your answer. A vertical line in 2-space... Problem 21ES: Determine whether the statement is true or false. Explain your answer. If rs is parametrized by arc... Problem 22ES: Determine whether the statement is true or false. Explain your answer. If C is a curve in 2-space,... Problem 23ES: (a) Use Formula (3) to show that in 2-space the curvature of a smooth parametric curve is... Problem 24ES: Use part (b) of Exercise 23 to show that the curvature of y=fx can be expressed in term of the angle... Problem 25ES: Use the result in Exercise 23(b) to find the curvature at the state point. y=sinx;x=/2 Problem 26ES: Use the result in Exercise 23(b) to find the curvature at the state point. y=tanx;x=/4 Problem 27ES: Use the result in Exercise 23(b) to find the curvature at the state point. y=ex;x=1 Problem 28ES: Use the result in Exercise 23(b) to find the curvature at the state point. y24x2=9;2,5 Problem 29ES: Use the result in Exercise 23(a) to find the curvature at the stated point. x=t2,y=t3;t=12 Problem 30ES: Use the result in Exercise 23(a) to find the curvature at the stated point. x=e3t,y=et;t=0 Problem 31ES: Use the result in Exercise 23(a) to find the curvature at the stated point. x=t,y=1/t;t=1 Problem 32ES: Use the result in Exercise 23(a) to find the curvature at the stated point. x=2sin2t,y=3sint;t=/2 Problem 33ES: In each part, use the formulas in Exercise 23 to help find the radius of curvature at the stated... Problem 34ES Problem 35ES: Generate the graph of y=fx using a graphing utility, and then make a conjecture about the shape of... Problem 36ES: Generate the graph of y=fx using a graphing utility, and then make a conjecture about the shape of... Problem 37ES Problem 38ES: (a) Use a CAS to graph the parametric curve x=tcost,y=tsintfort0. (b) Make a conjecture about the... Problem 39ES: Use the formula in Exercise 23 (a) to show that for a curve in polar coordinates described by r=f... Problem 40ES: Use the result in Exercise 39 to show that a circle has constant curvature. Problem 41ES: Use the formula in Exercise 39 to find the curvature at the indicated point. r=1+cos;=/2 Problem 42ES: Use the formula in Exercise 39 to find the curvature at the indicated point. r=e2;=1 Problem 43ES: Use the formula in Exercise 39 to find the curvature at the indicated point. r=sin3;=0 Problem 44ES: Use the formula in Exercise 39 to find the curvature at the indicated point. r=;=1 Problem 45ES: Find the radius of curvature of the parabola y2=4pxat0,0 . Problem 46ES: At what point(s) does y=ex have maximum curvature ? Problem 47ES: At what point(s) does 4x2+9y2=36 have a minimum radius of curvature? Problem 48ES: Find the maximum and minimum values of the radius of curvature for the curve x=cost,y=sint,z=cost. Problem 49ES: Use the formula in Exercise 39 to show that the curvature of the polar curve r=ea is inversely... Problem 50ES: Use the formula in Exercise 39 and a CAS to show that the curvature of the lemniscate r=acos2 is... Problem 51ES Problem 52ES: The evolute of a smooth parametric curve C in 2-space is the curve formed from the centers of... Problem 53ES: These exercises are concerned with the problem of creating a single smooth curve by piecing together... Problem 54ES: These exercises are concerned with the problem of creating a single smooth curve by piecing together... Problem 55ES: These exercises are concerned with the problem of creating a single smooth curve by piecing together... Problem 56ES: These exercises are concerned with the problem of creating a single smooth curve by piecing together... Problem 57ES: These exercises are concerned with the problem of creating a single smooth curve by piecing together... Problem 58ES: Assume that s is an arc length parameter for a smooth vector-valued function r(s) in 3-space and... Problem 59ES: Assume that s is an arc length parameter for a smooth vector-valued function r(s) in 3-space and... Problem 60ES: Assume that s is an arc length parameter for a smooth vector-valued function r(s) in 3-space and... Problem 61ES Problem 62ES: (a) Use the chain rule and the first two Frenet-Serret formulas in Exercise 61 to show that T... Problem 63ES: Use the formula in Exercise 62(d) to find the torsion =t. The twisted cubic rt=2ti+t2j+13t3k Problem 64ES: Use the formula in Exercise 62(d) to find the torsion =t. The circular helix rt=acosti+asintj+ctk Problem 65ES: Use the formula in Exercise 62(d) to find the torsion =t. rt=eti+etj+2tk Problem 66ES: Use the formula in Exercise 62(d) to find the torsion =t. rt=tsinti+1costj+tk Problem 68ES: The accompanying figure is the graph of the radius of curvature versus in rectangular coordinates... format_list_bulleted