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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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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Chapter 11.6, Problem 36E

Derivative rules Let

u ( t ) = 2 t 3 i + ( t 2 − 1 ) j − 8 k   a n d   v ( t ) = e t i + 2 e − t j − e 2 t   k .

Compute the derivative of the following functions.

36. u(t) × v(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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