Calculus: Early Transcendentals (2nd Edition)
Calculus: Early Transcendentals (2nd Edition)
2nd Edition
ISBN: 9780321947345
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Chapter 11.9, Problem 50E

Special formula: Curvature for y = f(x) Assume that f is twice differentiable. Prove that the curve y = f(x) has curvature

κ ( x ) = | f ″ ( x ) | ( 1 + f ′ ( x ) 2 ) 3 / 2 .

(Hint: Use the parametric description x = t, y = f(t).)

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Chapter 11 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 11.1 - Interpret the following statement: Points have a...Ch. 11.1 - What is a position vector?Ch. 11.1 - Prob. 3ECh. 11.1 - Prob. 4ECh. 11.1 - Given a position vector v, why are there...Ch. 11.1 - Prob. 6ECh. 11.1 - Prob. 7ECh. 11.1 - Prob. 8ECh. 11.1 - If u = u1, u2 and v = v1, v2, how do you find u +...Ch. 11.1 - Prob. 10E
Ch. 11.1 - Prob. 11ECh. 11.1 - Express the vector v = v1, v2 in terms of the unit...Ch. 11.1 - How do you compute |PQ| from the coordinates of...Ch. 11.1 - Prob. 14ECh. 11.1 - How do you find a vector of length 10 in the...Ch. 11.1 - Prob. 16ECh. 11.1 - Vector operations Refer to the figure and carry...Ch. 11.1 - Vector operations Refer to the figure and carry...Ch. 11.1 - Vector operations Refer to the figure and carry...Ch. 11.1 - Vector operations Refer to the figure and carry...Ch. 11.1 - Prob. 21ECh. 11.1 - Vector operations Refer to the figure and carry...Ch. 11.1 - Components and magnitudes Define the points O(0,...Ch. 11.1 - Prob. 24ECh. 11.1 - Components and equality Define the points P(3, 1),...Ch. 11.1 - Components and equality Define the points P(3, 1),...Ch. 11.1 - Components and equality Define the points P(3, 1),...Ch. 11.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 11.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 11.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 11.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 11.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 11.1 - Vector operations Let u = 4, 2, v = 4, 6, and w =...Ch. 11.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 11.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 11.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 11.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 11.1 - Prob. 38ECh. 11.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 11.1 - Prob. 40ECh. 11.1 - Vector operations Let u = 3, 4, v = 1, 1, and w =...Ch. 11.1 - Prob. 42ECh. 11.1 - Prob. 43ECh. 11.1 - Prob. 44ECh. 11.1 - Unit vectors Define the points P(4, 1), Q(3, 4),...Ch. 11.1 - Prob. 46ECh. 11.1 - Prob. 47ECh. 11.1 - A boat in a current The water in a river moves...Ch. 11.1 - Another boat in a current The water in a river...Ch. 11.1 - Prob. 50ECh. 11.1 - Prob. 51ECh. 11.1 - Prob. 52ECh. 11.1 - Boat in a wind A sailboat floats in a current that...Ch. 11.1 - Prob. 54ECh. 11.1 - Prob. 55ECh. 11.1 - Prob. 56ECh. 11.1 - Prob. 57ECh. 11.1 - Prob. 58ECh. 11.1 - Explain why or why not Determine whether the...Ch. 11.1 - Prob. 60ECh. 11.1 - Unit vectors a. Find two unit vectors parallel to...Ch. 11.1 - Equal vectors For the points A(3, 4), B(6, 10),...Ch. 11.1 - Vector equations Use the properties of vectors to...Ch. 11.1 - Vector equations Use the properties of vectors to...Ch. 11.1 - Prob. 65ECh. 11.1 - Prob. 66ECh. 11.1 - Prob. 67ECh. 11.1 - Prob. 68ECh. 11.1 - Prob. 69ECh. 11.1 - Prob. 70ECh. 11.1 - Solving vector equations Solve the following pairs...Ch. 11.1 - Prob. 72ECh. 11.1 - Designer vectors Find the following vectors. 73....Ch. 11.1 - Designer vectors Find the following vectors. 74....Ch. 11.1 - Designer vectors Find the following vectors. 75....Ch. 11.1 - Ant on a page An ant walks due east at a constant...Ch. 11.1 - Clock vectors Consider the 12 vectors that have...Ch. 11.1 - Three-way tug-of-war Three people located at A, B,...Ch. 11.1 - Prob. 79ECh. 11.1 - Prob. 80ECh. 11.1 - Additional Exercises 8185. Vector properties Prove...Ch. 11.1 - Additional Exercises 8185. Vector properties Prove...Ch. 11.1 - Vector properties Prove the following vector...Ch. 11.1 - Vector properties Prove the following vector...Ch. 11.1 - Vector properties Prove the following vector...Ch. 11.1 - Prob. 86ECh. 11.1 - Magnitude of scalar multiple Prove that |cv| = |c|...Ch. 11.1 - Equality of vectors Assume PQ equals RS. Does it...Ch. 11.1 - Linear independence A pair of nonzero vectors in...Ch. 11.1 - Perpendicular vectors Show that two nonzero...Ch. 11.1 - Parallel and perpendicular vectors Let u = a, 5...Ch. 11.1 - The Triangle Inequality Suppose u and v are...Ch. 11.2 - Explain how to plot the point (3, 2, 1) in 3.Ch. 11.2 - What is the y-coordinate of all points in the...Ch. 11.2 - Describe the plane x = 4.Ch. 11.2 - Prob. 4ECh. 11.2 - Let u = 3, 5, 7 and v = 6, 5, 1. Evaluate u + v...Ch. 11.2 - What is the magnitude of a vector joining two...Ch. 11.2 - Which point is farther from the origin, (3, 1, 2)...Ch. 11.2 - Express the vector from P(1, 4, 6) to Q(1, 3, 6)...Ch. 11.2 - Points in 3 Find the coordinates of the vertices...Ch. 11.2 - Points in 3 Find the coordinates of the vertices...Ch. 11.2 - Points in 3 Find the coordinates of the vertices...Ch. 11.2 - Points in 3 Find the coordinates of the vertices...Ch. 11.2 - Plotting points in 3 For each point P(x, y, z)...Ch. 11.2 - Plotting points in 3 For each point P(x, y, z)...Ch. 11.2 - Sketching planes Sketch the following planes in...Ch. 11.2 - Sketching planes Sketch the following planes in...Ch. 11.2 - Sketching planes Sketch the following planes in...Ch. 11.2 - Sketching planes Sketch the following planes in...Ch. 11.2 - Sketching planes Sketch the following planes in...Ch. 11.2 - Sketching planes Sketch the following planes in...Ch. 11.2 - Planes Sketch the plane parallel to the xy-plane...Ch. 11.2 - Prob. 22ECh. 11.2 - Spheres and balls Find an equation or inequality...Ch. 11.2 - Spheres and balls Find an equation or inequality...Ch. 11.2 - Spheres and balls Find an equation or inequality...Ch. 11.2 - Spheres and balls Find an equation or inequality...Ch. 11.2 - Midpoints and spheres Find an equation of the...Ch. 11.2 - Midpoints and spheres Find an equation of the...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Prob. 34ECh. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Identifying sets Give a geometric description of...Ch. 11.2 - Prob. 39ECh. 11.2 - Prob. 40ECh. 11.2 - Prob. 41ECh. 11.2 - Prob. 42ECh. 11.2 - Prob. 43ECh. 11.2 - Prob. 44ECh. 11.2 - Unit vectors and magnitude Consider the following...Ch. 11.2 - Unit vectors and magnitude Consider the following...Ch. 11.2 - Unit vectors and magnitude Consider the following...Ch. 11.2 - Unit vectors and magnitude Consider the following...Ch. 11.2 - Prob. 49ECh. 11.2 - Unit vectors and magnitude Consider the following...Ch. 11.2 - Flight in crosswinds A model airplane is flying...Ch. 11.2 - Another crosswind flight A model airplane is...Ch. 11.2 - Crosswinds A small plane is flying horizontally...Ch. 11.2 - Prob. 54ECh. 11.2 - Prob. 55ECh. 11.2 - Maintaining equilibrium An object is acted upon by...Ch. 11.2 - Explain why or why not Determine whether the...Ch. 11.2 - Sets of points Describe with a sketch the sets of...Ch. 11.2 - Sets of points Describe with a sketch the sets of...Ch. 11.2 - Sets of points Describe with a sketch the sets of...Ch. 11.2 - Sets of points 61. Give a geometric description of...Ch. 11.2 - Sets of points 62. Give a geometric description of...Ch. 11.2 - Sets of points 63. Give a geometric description of...Ch. 11.2 - Sets of points 64. Give a geometric description of...Ch. 11.2 - Prob. 65ECh. 11.2 - Prob. 66ECh. 11.2 - Prob. 67ECh. 11.2 - Prob. 68ECh. 11.2 - Parallel vectors of varying lengths Find vectors...Ch. 11.2 - Parallel vectors of varying lengths Find vectors...Ch. 11.2 - Collinear points Determine whether the points P,...Ch. 11.2 - Collinear points Determine the values of x and y...Ch. 11.2 - Lengths of the diagonals of a box What is the...Ch. 11.2 - Prob. 74ECh. 11.2 - Three-cable load A 500-kg load hangs from three...Ch. 11.2 - Four-cable load A 500-lb load hangs from four...Ch. 11.2 - Possible parallelograms The points O(0, 0, 0),...Ch. 11.2 - Prob. 78ECh. 11.2 - Midpoint formula Prove that the midpoint of the...Ch. 11.2 - Equation of a sphere For constants a, b, c, and d,...Ch. 11.2 - Prob. 81ECh. 11.2 - Medians of a trianglewith coordinates In contrast...Ch. 11.2 - The amazing quadrilateral propertycoordinate free...Ch. 11.2 - Prob. 84ECh. 11.3 - Express the dot product of u and v in terms of...Ch. 11.3 - Express the dot product of u and v in terms of the...Ch. 11.3 - Compute 2, 3, 6 1, 8, 3.Ch. 11.3 - Prob. 4ECh. 11.3 - Prob. 5ECh. 11.3 - Prob. 6ECh. 11.3 - Prob. 7ECh. 11.3 - Prob. 8ECh. 11.3 - Prob. 9ECh. 11.3 - Prob. 10ECh. 11.3 - Prob. 11ECh. 11.3 - Prob. 12ECh. 11.3 - Prob. 13ECh. 11.3 - Prob. 14ECh. 11.3 - Prob. 15ECh. 11.3 - Prob. 16ECh. 11.3 - Prob. 17ECh. 11.3 - Prob. 18ECh. 11.3 - Prob. 19ECh. 11.3 - Prob. 20ECh. 11.3 - Prob. 21ECh. 11.3 - Prob. 22ECh. 11.3 - Prob. 23ECh. 11.3 - Prob. 24ECh. 11.3 - Sketching orthogonal projections Find projvu and...Ch. 11.3 - Sketching orthogonal projections Find projvu and...Ch. 11.3 - Sketching orthogonal projections Find projvu and...Ch. 11.3 - Sketching orthogonal projections Find projvu and...Ch. 11.3 - Calculating orthogonal projections For the given...Ch. 11.3 - Calculating orthogonal projections For the given...Ch. 11.3 - Prob. 31ECh. 11.3 - Prob. 32ECh. 11.3 - Calculating orthogonal projections For the given...Ch. 11.3 - Calculating orthogonal projections For the given...Ch. 11.3 - Prob. 35ECh. 11.3 - Calculating orthogonal projections For the given...Ch. 11.3 - Prob. 37ECh. 11.3 - Computing work Calculate the work done in the...Ch. 11.3 - Prob. 39ECh. 11.3 - Computing work Calculate the work done in the...Ch. 11.3 - Computing work Calculate the work done in the...Ch. 11.3 - Prob. 42ECh. 11.3 - Prob. 43ECh. 11.3 - Parallel and normal forces Find the components of...Ch. 11.3 - Parallel and normal forces Find the components of...Ch. 11.3 - Parallel and normal forces Find the components of...Ch. 11.3 - Prob. 47ECh. 11.3 - Prob. 48ECh. 11.3 - Orthogonal vectors Let a and b be real numbers....Ch. 11.3 - Prob. 50ECh. 11.3 - Prob. 51ECh. 11.3 - Orthogonal vectors Let a and b be real numbers....Ch. 11.3 - Prob. 53ECh. 11.3 - Vectors with equal projections Given a fixed...Ch. 11.3 - Vectors with equal projections Given a fixed...Ch. 11.3 - Vectors with equal projections Given a fixed...Ch. 11.3 - Vectors with equal projections Given a fixed...Ch. 11.3 - Decomposing vectors For the following vectors u...Ch. 11.3 - Decomposing vectors For the following vectors u...Ch. 11.3 - Decomposing vectors For the following vectors u...Ch. 11.3 - Decomposing vectors For the following vectors u...Ch. 11.3 - Prob. 62ECh. 11.3 - Prob. 63ECh. 11.3 - Prob. 64ECh. 11.3 - Prob. 65ECh. 11.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 11.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 11.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 11.3 - Orthogonal unit vectors in 3 Consider the vectors...Ch. 11.3 - Angles of a triangle For the given points P, Q,...Ch. 11.3 - Angles of a triangle For the given points P, Q,...Ch. 11.3 - Flow through a circle Suppose water flows in a...Ch. 11.3 - Heat flux Let D be a solid heat-conducting cube...Ch. 11.3 - Hexagonal circle packing The German mathematician...Ch. 11.3 - Hexagonal sphere packing Imagine three unit...Ch. 11.3 - Properties of dot products Let u = u1, u2, u3, v =...Ch. 11.3 - Prob. 77ECh. 11.3 - Prob. 78ECh. 11.3 - Prob. 79ECh. 11.3 - Properties of dot products Let u = u1, u2, u3, v =...Ch. 11.3 - Prob. 81ECh. 11.3 - Prob. 82ECh. 11.3 - Direction angles and cosines Let v = a, b, c and...Ch. 11.3 - Prob. 84ECh. 11.3 - Prob. 85ECh. 11.3 - CauchySchwarz Inequality The definition u v = |u|...Ch. 11.3 - CauchySchwarz Inequality The definition u v = |u|...Ch. 11.3 - CauchySchwarz Inequality The definition u v = |u|...Ch. 11.3 - Diagonals of a parallelogram Consider the...Ch. 11.3 - Prob. 90ECh. 11.4 - Prob. 1ECh. 11.4 - Prob. 2ECh. 11.4 - What is the magnitude of the cross product of two...Ch. 11.4 - Prob. 4ECh. 11.4 - Explain how to use a determinant to compute u v.Ch. 11.4 - Explain how to find the torque produced by a force...Ch. 11.4 - Cross products from the definition Find the cross...Ch. 11.4 - Cross products from the definition Find the cross...Ch. 11.4 - Cross products from the definition Sketch the...Ch. 11.4 - Prob. 10ECh. 11.4 - Prob. 11ECh. 11.4 - Prob. 12ECh. 11.4 - Prob. 13ECh. 11.4 - Prob. 14ECh. 11.4 - Coordinate unit vectors Compute the following...Ch. 11.4 - Prob. 16ECh. 11.4 - Prob. 17ECh. 11.4 - Coordinate unit vectors Compute the following...Ch. 11.4 - Prob. 19ECh. 11.4 - Prob. 20ECh. 11.4 - Area of a parallelogram Find the area of the...Ch. 11.4 - Area of a parallelogram Find the area of the...Ch. 11.4 - Area of a parallelogram Find the area of the...Ch. 11.4 - Area of a parallelogram Find the area of the...Ch. 11.4 - Area of a triangle For the given points A, B, and...Ch. 11.4 - Prob. 26ECh. 11.4 - Area of a triangle For the given points A, B, and...Ch. 11.4 - Area of a triangle For the given points A, B, and...Ch. 11.4 - Prob. 29ECh. 11.4 - Prob. 30ECh. 11.4 - Prob. 31ECh. 11.4 - Prob. 32ECh. 11.4 - Prob. 33ECh. 11.4 - Prob. 34ECh. 11.4 - Orthogonal vectors Find a vector orthogonal to the...Ch. 11.4 - Orthogonal vectors Find a vector orthogonal to the...Ch. 11.4 - Orthogonal vectors Find a vector orthogonal to the...Ch. 11.4 - Prob. 38ECh. 11.4 - Prob. 39ECh. 11.4 - Prob. 40ECh. 11.4 - Computing torque Answer the following questions...Ch. 11.4 - Computing torque Answer the following questions...Ch. 11.4 - Computing torque Answer the following questions...Ch. 11.4 - Computing torque Answer the following questions...Ch. 11.4 - Force on a moving charge Answer the following...Ch. 11.4 - Prob. 46ECh. 11.4 - Prob. 47ECh. 11.4 - Force on a moving charge Answer the following...Ch. 11.4 - Prob. 49ECh. 11.4 - Collinear points Use cross products to determine...Ch. 11.4 - Collinear points Use cross products to determine...Ch. 11.4 - Finding an unknown Find the value of a such that...Ch. 11.4 - Prob. 53ECh. 11.4 - Areas of triangles Find the area of the following...Ch. 11.4 - Areas of triangles Find the area of the following...Ch. 11.4 - Prob. 56ECh. 11.4 - Areas of triangles Find the area of the following...Ch. 11.4 - Prob. 58ECh. 11.4 - Prob. 59ECh. 11.4 - Prob. 60ECh. 11.4 - Prob. 61ECh. 11.4 - Express u, v, and w in terms of their components...Ch. 11.4 - Prob. 63ECh. 11.4 - Prob. 64ECh. 11.4 - Prob. 65ECh. 11.4 - Arm torque A horizontally outstretched arm...Ch. 11.4 - Prob. 67ECh. 11.4 - Three proofs Prove that u u = 0 in three ways. a....Ch. 11.4 - Associative property Prove in two ways that for...Ch. 11.4 - Prob. 70ECh. 11.4 - Prob. 71ECh. 11.4 - Prob. 72ECh. 11.4 - Identities Prove the following identities. Assume...Ch. 11.4 - Prob. 74ECh. 11.4 - Cross product equations Suppose u and v are known...Ch. 11.5 - How many independent variables does the function...Ch. 11.5 - How many dependent scalar variables does the...Ch. 11.5 - Prob. 3ECh. 11.5 - Explain how to find a vector in the direction of...Ch. 11.5 - What is an equation of the line through the points...Ch. 11.5 - Prob. 6ECh. 11.5 - How do you evaluate limtar(t), where r(t) = f(t),...Ch. 11.5 - How do you determine whether r(t) = f(t) i + g(t)...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Prob. 12ECh. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Prob. 14ECh. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Equations of lines Find equations of the following...Ch. 11.5 - Prob. 23ECh. 11.5 - Prob. 24ECh. 11.5 - Line segments Find an equation of the line segment...Ch. 11.5 - Line segments Find an equation of the line segment...Ch. 11.5 - Line segments Find an equation of the line segment...Ch. 11.5 - Line segments Find an equation of the line segment...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Curves in space Graph the curves described by the...Ch. 11.5 - Exotic curves Graph the curves described by the...Ch. 11.5 - Exotic curves Graph the curves described by the...Ch. 11.5 - Exotic curves Graph the curves described by the...Ch. 11.5 - Exotic curves Graph the curves described by the...Ch. 11.5 - Limits Evaluate the following limits. 41....Ch. 11.5 - Limits Evaluate the following limits. 42....Ch. 11.5 - Limits Evaluate the following limits. 43....Ch. 11.5 - Limits Evaluate the following limits. 44....Ch. 11.5 - Limits Evaluate the following limits. 45....Ch. 11.5 - Limits Evaluate the following limits. 46....Ch. 11.5 - Prob. 47ECh. 11.5 - Prob. 48ECh. 11.5 - Prob. 49ECh. 11.5 - Prob. 50ECh. 11.5 - Prob. 51ECh. 11.5 - Prob. 52ECh. 11.5 - Prob. 53ECh. 11.5 - Skew lines A pair of lines in 3 are said to be...Ch. 11.5 - Prob. 55ECh. 11.5 - Domains Find the domain of the following...Ch. 11.5 - Domains Find the domain of the following...Ch. 11.5 - Domains Find the domain of the following...Ch. 11.5 - Prob. 59ECh. 11.5 - Line-plane intersections Find the point (if it...Ch. 11.5 - Prob. 61ECh. 11.5 - Line-plane intersections Find the point (if it...Ch. 11.5 - Prob. 63ECh. 11.5 - Curve-plane intersections Find the points (if they...Ch. 11.5 - Curve-plane intersections Find the points (if they...Ch. 11.5 - Curve-plane intersections Find the points (if they...Ch. 11.5 - Matching functions with graphs Match functions af...Ch. 11.5 - Prob. 68ECh. 11.5 - Prob. 69ECh. 11.5 - Closed plane curves Consider the curve r(t) = (a...Ch. 11.5 - Closed plane curves Consider the curve r(t) = (a...Ch. 11.5 - Closed plane curves Consider the curve r(t) = (a...Ch. 11.5 - Closed plane curves Consider the curve r(t) = (a...Ch. 11.5 - Golf slice A golfer launches a tee shot down a...Ch. 11.5 - Curves on spheres 75. Graph the curve...Ch. 11.5 - Prob. 76ECh. 11.5 - Prob. 77ECh. 11.5 - Limits of vector functions Let r(t) = (f(t), g(t),...Ch. 11.5 - Prob. 79ECh. 11.5 - Prob. 80ECh. 11.5 - Prob. 81ECh. 11.5 - Prob. 82ECh. 11.6 - Prob. 1ECh. 11.6 - Explain the geometric meaning of r(t).Ch. 11.6 - Prob. 3ECh. 11.6 - Compute r(t) when r(t) = t10, 8t, cos t.Ch. 11.6 - How do you find the indefinite integral of r(t) =...Ch. 11.6 - How do you evaluate abr(t)dt?Ch. 11.6 - Derivatives of vector-valued functions...Ch. 11.6 - Prob. 8ECh. 11.6 - Prob. 9ECh. 11.6 - Derivatives of vector-valued functions...Ch. 11.6 - Prob. 11ECh. 11.6 - Derivatives of vector-valued functions...Ch. 11.6 - Prob. 13ECh. 11.6 - Prob. 14ECh. 11.6 - Prob. 15ECh. 11.6 - Prob. 16ECh. 11.6 - Prob. 17ECh. 11.6 - Prob. 18ECh. 11.6 - Prob. 19ECh. 11.6 - Prob. 20ECh. 11.6 - Prob. 21ECh. 11.6 - Prob. 22ECh. 11.6 - Prob. 23ECh. 11.6 - Prob. 24ECh. 11.6 - Prob. 25ECh. 11.6 - Prob. 26ECh. 11.6 - Prob. 27ECh. 11.6 - Prob. 28ECh. 11.6 - Prob. 29ECh. 11.6 - Prob. 30ECh. 11.6 - Derivative rules Let...Ch. 11.6 - Derivative rules Let...Ch. 11.6 - Derivative rules Let...Ch. 11.6 - Derivative rules Let...Ch. 11.6 - Derivative rules Let...Ch. 11.6 - Derivative rules Let...Ch. 11.6 - Derivative rules Compute the following...Ch. 11.6 - Derivative rules Compute the following...Ch. 11.6 - Derivative rules Compute the following...Ch. 11.6 - Derivative rules Compute the following...Ch. 11.6 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 11.6 - Prob. 42ECh. 11.6 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 11.6 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 11.6 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 11.6 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 11.6 - Indefinite integrals Compute the indefinite...Ch. 11.6 - Prob. 48ECh. 11.6 - Indefinite integrals Compute the indefinite...Ch. 11.6 - Indefinite integrals Compute the indefinite...Ch. 11.6 - Indefinite integrals Compute the indefinite...Ch. 11.6 - Indefinite integrals Compute the indefinite...Ch. 11.6 - Finding r from r Find the function r that...Ch. 11.6 - Prob. 54ECh. 11.6 - Prob. 55ECh. 11.6 - Finding r from r Find the function r that...Ch. 11.6 - Finding r from r Find the function r that...Ch. 11.6 - Finding r from r Find the function r that...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Definite integrals Evaluate the following definite...Ch. 11.6 - Prob. 67ECh. 11.6 - Prob. 68ECh. 11.6 - Prob. 69ECh. 11.6 - Prob. 70ECh. 11.6 - Prob. 71ECh. 11.6 - Prob. 72ECh. 11.6 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 11.6 - Prob. 74ECh. 11.6 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 11.6 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 11.6 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 11.6 - Relationship between r and r 78. Consider the...Ch. 11.6 - Relationship between r and r 79. Consider the...Ch. 11.6 - Prob. 80ECh. 11.6 - Relationship between r and r 81. Consider the...Ch. 11.6 - Relationship between r and r 82. Consider the...Ch. 11.6 - Relationship between r and r 83. Give two families...Ch. 11.6 - Prob. 84ECh. 11.6 - Vectors r and r for lines a. If r(t) = at, bt, ct...Ch. 11.6 - Proof of Sum Rule By expressing u and v in terms...Ch. 11.6 - Proof of Product Rule By expressing u in terms of...Ch. 11.6 - Prob. 88ECh. 11.6 - Cusps and noncusps a. Graph the curve r(t) = t3,...Ch. 11.6 - Motion on a sphere Prove that r describes a curve...Ch. 11.7 - Given the position function r of a moving object,...Ch. 11.7 - What is the relationship between the position and...Ch. 11.7 - Write Newtons Second Law of Motion in vector form.Ch. 11.7 - Write Newtons Second Law of Motion for...Ch. 11.7 - Given the acceleration of an object and its...Ch. 11.7 - Given the velocity of an object and its initial...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Velocity and acceleration from position Consider...Ch. 11.7 - Comparing trajectories Consider the following...Ch. 11.7 - Comparing trajectories Consider the following...Ch. 11.7 - Comparing trajectories Consider the following...Ch. 11.7 - Comparing trajectories Consider the following...Ch. 11.7 - Comparing trajectories Consider the following...Ch. 11.7 - Comparing trajectories Consider the following...Ch. 11.7 - Trajectories on circles and spheres Determine...Ch. 11.7 - Prob. 26ECh. 11.7 - Trajectories on circles and spheres Determine...Ch. 11.7 - Trajectories on circles and spheres Determine...Ch. 11.7 - Trajectories on circles and spheres Determine...Ch. 11.7 - Trajectories on circles and spheres Determine...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Two-dimensional motion Consider the motion of the...Ch. 11.7 - Two-dimensional motion Consider the motion of the...Ch. 11.7 - Two-dimensional motion Consider the motion of the...Ch. 11.7 - Two-dimensional motion Consider the motion of the...Ch. 11.7 - Two-dimensional motion Consider the motion of the...Ch. 11.7 - Two-dimensional motion Consider the motion of the...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Solving equations of motion Given an acceleration...Ch. 11.7 - Prob. 46ECh. 11.7 - Three-dimensional motion Consider the motion of...Ch. 11.7 - Three-dimensional motion Consider the motion of...Ch. 11.7 - Three-dimensional motion Consider the motion of...Ch. 11.7 - Three-dimensional motion Consider the motion of...Ch. 11.7 - Three-dimensional motion Consider the motion of...Ch. 11.7 - Prob. 52ECh. 11.7 - Prob. 53ECh. 11.7 - Trajectory properties Find the time of flight,...Ch. 11.7 - Trajectory properties Find the time of flight,...Ch. 11.7 - Trajectory properties Find the time of flight,...Ch. 11.7 - Trajectory properties Find the time of flight,...Ch. 11.7 - Motion on the moon The acceleration due to gravity...Ch. 11.7 - Firing angles A projectile is fired over...Ch. 11.7 - Prob. 60ECh. 11.7 - Nonuniform straight-line motion Consider the...Ch. 11.7 - A race Two people travel from P(4, 0) to Q(4, 0)...Ch. 11.7 - Circular motion Consider an object moving along...Ch. 11.7 - Prob. 64ECh. 11.7 - A circular trajectory An object moves clockwise...Ch. 11.7 - Prob. 66ECh. 11.7 - Speed on an ellipse An object moves along an...Ch. 11.7 - Travel on a cycloid Consider an object moving on a...Ch. 11.7 - Prob. 69ECh. 11.7 - Golf shot A golfer stands 390 ft (130 yd)...Ch. 11.7 - Another golf shot A golfer stands 420 ft (140 yd)...Ch. 11.7 - Prob. 72ECh. 11.7 - Initial velocity of a golf shot A golfer stands...Ch. 11.7 - Ski jump The lip of a ski jump is 8 m above the...Ch. 11.7 - Designing a baseball pitch A baseball leaves the...Ch. 11.7 - Prob. 76ECh. 11.7 - Prob. 77ECh. 11.7 - Parabolic trajectories Show that the...Ch. 11.7 - Tilted ellipse Consider the curve r(t) = cos t,...Ch. 11.7 - Equal area property Consider the ellipse r(t) = a...Ch. 11.7 - Another property of constant | r | motion Suppose...Ch. 11.7 - Prob. 82ECh. 11.7 - Prob. 83ECh. 11.8 - Find the length of the line given by r(t) = t, 2t,...Ch. 11.8 - Explain how to find the length of the curve r(t) =...Ch. 11.8 - Express the arc length of a curve in terms of the...Ch. 11.8 - Suppose an object moves in space with the position...Ch. 11.8 - An object moves on a trajectory given by r(t) = 10...Ch. 11.8 - Prob. 6ECh. 11.8 - Explain what it means for a curve to be...Ch. 11.8 - Is the curve r(t) = cos t, sin t parameterized by...Ch. 11.8 - Arc length calculations Find the length of he...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Prob. 13ECh. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Prob. 16ECh. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Arc length calculations Find the length of the...Ch. 11.8 - Speed and arc length For the following...Ch. 11.8 - Speed and arc length For the following...Ch. 11.8 - Speed and arc length For the following...Ch. 11.8 - Speed and arc length For the following...Ch. 11.8 - Arc length approximations Use a calculator to...Ch. 11.8 - Prob. 28ECh. 11.8 - Arc length approximations Use a calculator to...Ch. 11.8 - Prob. 30ECh. 11.8 - Prob. 31ECh. 11.8 - Prob. 32ECh. 11.8 - Prob. 33ECh. 11.8 - Prob. 34ECh. 11.8 - Prob. 35ECh. 11.8 - Prob. 36ECh. 11.8 - Arc length of polar curves Find the length of the...Ch. 11.8 - Arc length of polar curves Find the length of the...Ch. 11.8 - Arc length of polar curves Find the length of the...Ch. 11.8 - Prob. 40ECh. 11.8 - Prob. 41ECh. 11.8 - Arc length parameterization Determine whether the...Ch. 11.8 - Arc length parameterization Determine whether the...Ch. 11.8 - Arc length parameterization Determine whether the...Ch. 11.8 - Prob. 45ECh. 11.8 - Prob. 46ECh. 11.8 - Prob. 47ECh. 11.8 - Arc length parameterization Determine whether the...Ch. 11.8 - Arc length parameterization Determine whether the...Ch. 11.8 - Arc length parameterization Determine whether the...Ch. 11.8 - Explain why or why not Determine whether the...Ch. 11.8 - Length of a line segment Consider the line segment...Ch. 11.8 - Tilted circles Let the curve C be described by...Ch. 11.8 - Prob. 54ECh. 11.8 - Prob. 55ECh. 11.8 - Spiral arc length Consider the spiral r = 4, for ...Ch. 11.8 - Prob. 57ECh. 11.8 - Arc length using technology Use a calculator to...Ch. 11.8 - Prob. 59ECh. 11.8 - Prob. 60ECh. 11.8 - Prob. 61ECh. 11.8 - Prob. 62ECh. 11.8 - Projectile trajectories A projectile (such as a...Ch. 11.8 - Variable speed on a circle Consider a particle...Ch. 11.8 - Arc length parameterization Prove that the line...Ch. 11.8 - Arc length parameterization Prove that the curve...Ch. 11.8 - Prob. 67ECh. 11.8 - Prob. 68ECh. 11.8 - Prob. 69ECh. 11.8 - Change of variables Consider the parameterized...Ch. 11.9 - What is the curvature of a straight line?Ch. 11.9 - Explain the meaning of the curvature of a curve....Ch. 11.9 - Give a practical formula for computing the...Ch. 11.9 - Interpret the principal unit normal vector of a...Ch. 11.9 - Give a practical formula for computing the...Ch. 11.9 - Explain how to decompose the acceleration vector...Ch. 11.9 - Explain how the vectors T, N, and B are related...Ch. 11.9 - How do you compute B?Ch. 11.9 - Give a geometrical interpretation of the torsion.Ch. 11.9 - How do you compute the torsion?Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Curvature Find the unit tangent vector T and the...Ch. 11.9 - Prob. 20ECh. 11.9 - Alternative curvature formula Use the alternative...Ch. 11.9 - Alternative curvature formula Use the alternative...Ch. 11.9 - Alternative curvature formula Use the alternative...Ch. 11.9 - Alternative curvature formula Use the alternative...Ch. 11.9 - Alternative curvature formula Use the alternative...Ch. 11.9 - Alternative curvature formula Use the alternative...Ch. 11.9 - Prob. 27ECh. 11.9 - Prob. 28ECh. 11.9 - Prob. 29ECh. 11.9 - Prob. 30ECh. 11.9 - Prob. 31ECh. 11.9 - Prob. 32ECh. 11.9 - Prob. 33ECh. 11.9 - Prob. 34ECh. 11.9 - Components of the acceleration Consider the...Ch. 11.9 - Components of the acceleration Consider the...Ch. 11.9 - Components of the acceleration Consider the...Ch. 11.9 - Components of the acceleration Consider the...Ch. 11.9 - Prob. 39ECh. 11.9 - Prob. 40ECh. 11.9 - Computing the binormal vector and torsion In...Ch. 11.9 - Computing the binormal vector and torsion In...Ch. 11.9 - Prob. 43ECh. 11.9 - Prob. 44ECh. 11.9 - Prob. 45ECh. 11.9 - Computing the binormal vector and torsion Use the...Ch. 11.9 - Computing the binormal vector and torsion Use the...Ch. 11.9 - Prob. 48ECh. 11.9 - Explain why or why not Determine whether the...Ch. 11.9 - Special formula: Curvature for y = f(x) Assume...Ch. 11.9 - Curvature for y = f(x) Use the result of Exercise...Ch. 11.9 - Prob. 52ECh. 11.9 - Prob. 53ECh. 11.9 - Curvature for y = f(x) Use the result of Exercise...Ch. 11.9 - Prob. 55ECh. 11.9 - Curvature for plane curves Use the result of...Ch. 11.9 - Curvature for plane curves Use the result of...Ch. 11.9 - Curvature for plane curves Use the result of...Ch. 11.9 - Curvature for plane curves Use the result of...Ch. 11.9 - Same paths, different velocity The position...Ch. 11.9 - Same paths, different velocity The position...Ch. 11.9 - Same paths, different velocity The position...Ch. 11.9 - Same paths, different velocity The position...Ch. 11.9 - Graphs of the curvature Consider the following...Ch. 11.9 - Graphs of the curvature Consider the following...Ch. 11.9 - Graphs of the curvature Consider the following...Ch. 11.9 - Graphs of the curvature Consider the following...Ch. 11.9 - Curvature of ln x Find the curvature of f(x) = ln...Ch. 11.9 - Curvature of ex Find the curvature of f(x) = ex...Ch. 11.9 - Prob. 70ECh. 11.9 - Finding radii of curvature Find the radius of...Ch. 11.9 - Finding radii of curvature Find the radius of...Ch. 11.9 - Finding radii of curvature Find the radius of...Ch. 11.9 - Prob. 74ECh. 11.9 - Curvature of the sine curve The function f(x) =...Ch. 11.9 - Parabolic trajectory In Example 7 it was shown...Ch. 11.9 - Parabolic trajectory Consider the parabolic...Ch. 11.9 - Prob. 78ECh. 11.9 - Zero curvature Prove that the curve...Ch. 11.9 - Prob. 80ECh. 11.9 - Maximum curvature Consider the superparabolas...Ch. 11.9 - Alternative derivation of the curvature Derive the...Ch. 11.9 - Computational formula for B Use the result of part...Ch. 11.9 - Prob. 84ECh. 11.9 - Descartes four-circle solution Consider the four...Ch. 11 - Explain why or why not Determine whether the...Ch. 11 - Prob. 2RECh. 11 - Prob. 3RECh. 11 - Prob. 4RECh. 11 - Prob. 5RECh. 11 - Working with vectors Let u = 2, 4, 5 and v = 6,...Ch. 11 - Working with vectors Let u = 2, 4, 5 and v = 6,...Ch. 11 - Prob. 8RECh. 11 - Prob. 9RECh. 11 - Prob. 10RECh. 11 - Prob. 11RECh. 11 - Scalar multiples Find scalars a, b, and c such...Ch. 11 - Velocity vectors Assume the positive x-axis points...Ch. 11 - Prob. 14RECh. 11 - Spheres and balls Use set notation to describe the...Ch. 11 - Spheres and balls Use set notation to describe the...Ch. 11 - Spheres and balls Use set notation to describe the...Ch. 11 - Identifying sets. Give a geometric description of...Ch. 11 - Identifying sets. Give a geometric description of...Ch. 11 - Identifying sets. Give a geometric description of...Ch. 11 - Identifying sets. Give a geometric description of...Ch. 11 - Prob. 22RECh. 11 - Prob. 23RECh. 11 - Cross winds A small plane is flying north in calm...Ch. 11 - Sets of points Describe the set of points...Ch. 11 - Angles and projections a. Find the angle between u...Ch. 11 - Prob. 27RECh. 11 - Prob. 28RECh. 11 - Vectors normal to a plane Find a unit vector...Ch. 11 - Angle in two ways Find the angle between 2, 0, 2...Ch. 11 - Prob. 31RECh. 11 - Lines in space Find an equation of the following...Ch. 11 - Lines in space Find an equation of the following...Ch. 11 - Lines in space Find an equation of the following...Ch. 11 - Lines in space Find an equation of the following...Ch. 11 - Lines in space Find an equation of the following...Ch. 11 - Area of a parallelogram Find the area of the...Ch. 11 - Area of a triangle Find the area of the triangle...Ch. 11 - Curves in space Sketch the curves described by the...Ch. 11 - Curves in space Sketch the curves described by the...Ch. 11 - Prob. 41RECh. 11 - Prob. 42RECh. 11 - Prob. 43RECh. 11 - Prob. 44RECh. 11 - Prob. 45RECh. 11 - Orthogonal r and r Find all points on the ellipse...Ch. 11 - Prob. 47RECh. 11 - Baseball motion A toddler on level ground throws a...Ch. 11 - Prob. 49RECh. 11 - Prob. 50RECh. 11 - Prob. 51RECh. 11 - Prob. 52RECh. 11 - Velocity and trajectory length The acceleration of...Ch. 11 - Prob. 54RECh. 11 - Arc length of polar curves Find the approximate...Ch. 11 - Prob. 56RECh. 11 - Arc length parameterization Find the description...Ch. 11 - Tangents and normals for an ellipse Consider the...Ch. 11 - Prob. 59RECh. 11 - Prob. 60RECh. 11 - Properties of space curves Do the following...Ch. 11 - Prob. 62RECh. 11 - Analyzing motion Consider the position vector of...Ch. 11 - Analyzing motion Consider the position vector of...Ch. 11 - Analyzing motion Consider the position vector of...Ch. 11 - Analyzing motion Consider the position vector of...Ch. 11 - Prob. 67RECh. 11 - Prob. 68RECh. 11 - Prob. 69RECh. 11 - Curve analysis Carry out the following steps for...Ch. 11 - Prob. 71RECh. 11 - Prob. 72RECh. 11 - Prob. 73RE

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