Calculus: Early Transcendentals (3rd Edition)
3rd Edition
ISBN: 9780134763644
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Textbook Question
Chapter 14.1, Problem 12E
9–14. Lines and line segments Find a function r(t) that describes the given line or line segment.
The line of intersection of the planes 2x + 3y + 4z = 7 and 2x + 3y + 5z = 8
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Chapter 14 Solutions
Calculus: Early Transcendentals (3rd Edition)
Ch. 14.1 - Restrict the domain o f the vector function in...Ch. 14.1 - Explain why the curve in Example 5 lies on the...Ch. 14.1 - How many independent variables does the function...Ch. 14.1 - How many dependent scalar variables does the...Ch. 14.1 - Prob. 3ECh. 14.1 - Prob. 4ECh. 14.1 - How do you evaluate limtar(t), where r(t) = f(t),...Ch. 14.1 - How do you determine whether r(t) = f(t) i + g(t)...Ch. 14.1 - Find a function r(t) for the line passing through...Ch. 14.1 - Find a function r(t) whose graph is a circle of...
Ch. 14.1 - Prob. 9ECh. 14.1 - Prob. 10ECh. 14.1 - Lines and line segments Find a function r(t) that...Ch. 14.1 - 914. Lines and line segments Find a function r(t)...Ch. 14.1 - Prob. 13ECh. 14.1 - Prob. 14ECh. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Graphing curves Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Curves in space Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Exotic curves Graph the curves described by the...Ch. 14.1 - Limits Evaluate the following limits. 41....Ch. 14.1 - Limits Evaluate the following limits. 42....Ch. 14.1 - Limits Evaluate the following limits. 43....Ch. 14.1 - Limits Evaluate the following limits. 44....Ch. 14.1 - Limits Evaluate the following limits. 45....Ch. 14.1 - Limits Evaluate the following limits. 46....Ch. 14.1 - Prob. 37ECh. 14.1 - Domains Find the domain of the following...Ch. 14.1 - Domains Find the domain of the following...Ch. 14.1 - Domains Find the domain of the following...Ch. 14.1 - Prob. 41ECh. 14.1 - Curve-plane intersections Find the points (if they...Ch. 14.1 - Curve-plane intersections Find the points (if they...Ch. 14.1 - Curve-plane intersections Find the points (if they...Ch. 14.1 - Matching functions with graphs Match functions af...Ch. 14.1 - Prob. 46ECh. 14.1 - 4750. Curve of intersection Find a function r(t)...Ch. 14.1 - 4750. Curve of intersection Find a function r(t)...Ch. 14.1 - 4750. Curve of intersection Find a function r(t)...Ch. 14.1 - Curve of intersection Find a function r(t) that...Ch. 14.1 - Golf slice A golfer launches a tee shot down a...Ch. 14.1 - Curves on surfaces Verify that the curve r(t) lies...Ch. 14.1 - 5256. Curves on surfaces Verify that the curve...Ch. 14.1 - Curves on surfaces Verify that the curve r(t) lies...Ch. 14.1 - Curves on surfaces Verify that the curve r(t) lies...Ch. 14.1 - 5256. Curves on surfaces Verify that the curve...Ch. 14.1 - 5758. Closest point on a curve Find the point P on...Ch. 14.1 - 5758. Closest point on a curve Find the point P on...Ch. 14.1 - Curves on spheres 75. Graph the curve...Ch. 14.1 - Prob. 60ECh. 14.1 - Prob. 61ECh. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Closed plane curves Consider the curve r(t) = (a...Ch. 14.1 - Limits of vector functions Let r(t) = (f(t), g(t),...Ch. 14.2 - Prob. 1QCCh. 14.2 - Suppose r(t) has units of m/s. Explain why T(t) =...Ch. 14.2 - Let u(t)=t,t,t and v(t)=1,1,1 compute...Ch. 14.2 - Let r(t)=1,2t,3t2. Compute r(t)dt.Ch. 14.2 - Prob. 1ECh. 14.2 - Explain the geometric meaning of r(t).Ch. 14.2 - Prob. 3ECh. 14.2 - Compute r(t) when r(t) = t10, 8t, cos t.Ch. 14.2 - How do you find the indefinite integral of r(t) =...Ch. 14.2 - How do you evaluate abr(t)dt?Ch. 14.2 - Find C if r(t)=et,3cost,t+10+C and r(0)=0,0,0.Ch. 14.2 - Find the unit tangent vector at t = 0 for the...Ch. 14.2 - Derivatives of vector-valued functions...Ch. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Derivatives of vector-valued functions...Ch. 14.2 - Prob. 13ECh. 14.2 - Derivatives of vector-valued functions...Ch. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Prob. 17ECh. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Prob. 20ECh. 14.2 - Prob. 21ECh. 14.2 - Prob. 22ECh. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 14.2 - Prob. 26ECh. 14.2 - Prob. 27ECh. 14.2 - Prob. 28ECh. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Prob. 31ECh. 14.2 - Prob. 32ECh. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Derivative rules Let...Ch. 14.2 - Prob. 39ECh. 14.2 - Prob. 40ECh. 14.2 - Prob. 41ECh. 14.2 - Derivative rules Suppose u and v are...Ch. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Prob. 44ECh. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Prob. 46ECh. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Derivative rules Let u(t) = 1, t, t2, v(t) = t2,...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Derivative rules Compute the following...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Prob. 54ECh. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Higher-order derivatives Compute r(t) and r(t) for...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Prob. 60ECh. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Indefinite integrals Compute the indefinite...Ch. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Prob. 66ECh. 14.2 - Prob. 67ECh. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Finding r from r Find the function r that...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Definite integrals Evaluate the following definite...Ch. 14.2 - Prob. 79ECh. 14.2 - Prob. 80ECh. 14.2 - Prob. 81ECh. 14.2 - Prob. 82ECh. 14.2 - Prob. 83ECh. 14.2 - Relationship between r and r 78. Consider the...Ch. 14.2 - Relationship between r and r 79. Consider the...Ch. 14.2 - Prob. 86ECh. 14.2 - Relationship between r and r 81. Consider the...Ch. 14.2 - Relationship between r and r 82. Consider the...Ch. 14.2 - Relationship between r and r 83. Give two families...Ch. 14.2 - Motion on a sphere Prove that r describes a curve...Ch. 14.2 - Vectors r and r for lines a. If r(t) = at, bt, ct...Ch. 14.2 - Proof of Sum Rule By expressing u and v in terms...Ch. 14.2 - Proof of Product Rule By expressing u in terms of...Ch. 14.2 - Prob. 94ECh. 14.2 - Cusps and noncusps a. Graph the curve r(t) = t3,...Ch. 14.3 - Given r(t)=t,t2,t3, find v(t) and a(t).Ch. 14.3 - Find the functions that give the speed of the two...Ch. 14.3 - Prob. 3QCCh. 14.3 - Prob. 4QCCh. 14.3 - Prob. 5QCCh. 14.3 - Given the position function r of a moving object,...Ch. 14.3 - What is the relationship between the position and...Ch. 14.3 - Write Newtons Second Law of Motion in vector form.Ch. 14.3 - Write Newtons Second Law of Motion for...Ch. 14.3 - Given the acceleration of an object and its...Ch. 14.3 - Given the velocity of an object and its initial...Ch. 14.3 - The velocity of a moving object, for t 0, is...Ch. 14.3 - A baseball is hit 2 feet above home plate, and the...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Velocity and acceleration from position Consider...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Comparing trajectories Consider the following...Ch. 14.3 - Prob. 27ECh. 14.3 - Carnival rides 28. Suppose the carnival ride in...Ch. 14.3 - Trajectories on circles and spheres Determine...Ch. 14.3 - Prob. 30ECh. 14.3 - Trajectories on circles and spheres Determine...Ch. 14.3 - Trajectories on circles and spheres Determine...Ch. 14.3 - Path on a sphere show that the following...Ch. 14.3 - Path on a sphere show that the following...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Two-dimensional motion Consider the motion of the...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Solving equations of motion Given an acceleration...Ch. 14.3 - Prob. 50ECh. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Three-dimensional motion Consider the motion of...Ch. 14.3 - Prob. 56ECh. 14.3 - Prob. 57ECh. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Trajectory properties Find the time of flight,...Ch. 14.3 - Motion on the moon The acceleration due to gravity...Ch. 14.3 - Firing angles A projectile is fired over...Ch. 14.3 - Prob. 64ECh. 14.3 - Speed on an ellipse An object moves along an...Ch. 14.3 - Golf shot A golfer stands 390 ft (130 yd)...Ch. 14.3 - Another golf shot A golfer stands 420 ft (140 yd)...Ch. 14.3 - Prob. 68ECh. 14.3 - Initial speed of a golf shot A golfer stands 420...Ch. 14.3 - Ski jump The lip of a ski jump is 8 m above the...Ch. 14.3 - Designing a baseball pitch A baseball leaves the...Ch. 14.3 - Parabolic trajectories Show that the...Ch. 14.3 - Prob. 73ECh. 14.3 - A race Two people travel from P(4, 0) to Q(4, 0)...Ch. 14.3 - Circular motion Consider an object moving along...Ch. 14.3 - Prob. 76ECh. 14.3 - A circular trajectory An object moves clockwise...Ch. 14.3 - Prob. 78ECh. 14.3 - Tilted ellipse Consider the curve r(t) = cos t,...Ch. 14.3 - Equal area property Consider the ellipse r(t) = a...Ch. 14.3 - Another property of constant | r | motion Suppose...Ch. 14.3 - Prob. 82ECh. 14.3 - Nonuniform straight-line motion Consider the...Ch. 14.4 - What does the arc length formula give for the...Ch. 14.4 - Consider the portion of a circle r(t) = (cos t,...Ch. 14.4 - Prob. 3QCCh. 14.4 - Find the length of the line given by r(t) = t, 2t,...Ch. 14.4 - Explain how to find the length of the curve r(t) =...Ch. 14.4 - Express the arc length of a curve in terms of the...Ch. 14.4 - Suppose an object moves in space with the position...Ch. 14.4 - An object moves on a trajectory given by r(t) = 10...Ch. 14.4 - Use calculus to find the length of the line...Ch. 14.4 - Explain what it means for a curve to be...Ch. 14.4 - Is the curve r(t) = cos t, sin t parameterized by...Ch. 14.4 - Arc length calculations Find the length of he...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Prob. 13ECh. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Prob. 16ECh. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Arc length calculations Find the length of the...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed and arc length For the following...Ch. 14.4 - Speed of Earth Verify that the length of one orbit...Ch. 14.4 - Speed of Jupiter Verify that the length of one...Ch. 14.4 - Arc length approximations Use a calculator to...Ch. 14.4 - Prob. 30ECh. 14.4 - Arc length approximations Use a calculator to...Ch. 14.4 - Prob. 32ECh. 14.4 - Prob. 33ECh. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Prob. 37ECh. 14.4 - Prob. 38ECh. 14.4 - Prob. 39ECh. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Arc length parameterization Determine whether the...Ch. 14.4 - Explain why or why not Determine whether the...Ch. 14.4 - Length of a line segment Consider the line segment...Ch. 14.4 - Tilted circles Let the curve C be described by...Ch. 14.4 - Prob. 46ECh. 14.4 - Prob. 47ECh. 14.4 - Toroidal magnetic field A circle of radius a that...Ch. 14.4 - Projectile trajectories A projectile (such as a...Ch. 14.4 - Variable speed on a circle Consider a particle...Ch. 14.4 - Arc length parameterization Prove that the line...Ch. 14.4 - Arc length parameterization Prove that the curve...Ch. 14.4 - Prob. 53ECh. 14.4 - Change of variables Consider the parameterized...Ch. 14.5 - What is the curvature of the circle r() =...Ch. 14.5 - Use the alternative curvature formula to compute...Ch. 14.5 - Prob. 3QCCh. 14.5 - Prob. 4QCCh. 14.5 - Prob. 5QCCh. 14.5 - Prob. 6QCCh. 14.5 - Prob. 7QCCh. 14.5 - What is the curvature of a straight line?Ch. 14.5 - Explain the meaning of the curvature of a curve....Ch. 14.5 - Give a practical formula for computing the...Ch. 14.5 - Interpret the principal unit normal vector of a...Ch. 14.5 - Give a practical formula for computing the...Ch. 14.5 - Explain how to decompose the acceleration vector...Ch. 14.5 - Explain how the vectors T, N, and B are related...Ch. 14.5 - How do you compute B?Ch. 14.5 - Give a geometrical interpretation of the torsion.Ch. 14.5 - How do you compute the torsion?Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Curvature Find the unit tangent vector T and the...Ch. 14.5 - Prob. 20ECh. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Alternative curvature formula Use the alternative...Ch. 14.5 - Prob. 27ECh. 14.5 - Prob. 28ECh. 14.5 - Prob. 29ECh. 14.5 - Prob. 30ECh. 14.5 - Prob. 31ECh. 14.5 - Prob. 32ECh. 14.5 - Prob. 33ECh. 14.5 - Prob. 34ECh. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Components of the acceleration Consider the...Ch. 14.5 - Prob. 39ECh. 14.5 - Prob. 40ECh. 14.5 - Computing the binormal vector and torsion In...Ch. 14.5 - Computing the binormal vector and torsion In...Ch. 14.5 - Prob. 43ECh. 14.5 - Prob. 44ECh. 14.5 - Prob. 45ECh. 14.5 - Computing the binormal vector and torsion Use the...Ch. 14.5 - Computing the binormal vector and torsion Use the...Ch. 14.5 - Prob. 48ECh. 14.5 - Explain why or why not Determine whether the...Ch. 14.5 - Special formula: Curvature for y = f(x) Assume...Ch. 14.5 - Curvature for y = f(x) Use the result of Exercise...Ch. 14.5 - Prob. 52ECh. 14.5 - Prob. 53ECh. 14.5 - Curvature for y = f(x) Use the result of Exercise...Ch. 14.5 - Prob. 55ECh. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Curvature for plane curves Use the result of...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Same paths, different velocity The position...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Graphs of the curvature Consider the following...Ch. 14.5 - Curvature of ln x Find the curvature of f(x) = ln...Ch. 14.5 - Curvature of ex Find the curvature of f(x) = ex...Ch. 14.5 - Prob. 70ECh. 14.5 - Finding radii of curvature Find the radius of...Ch. 14.5 - Finding radii of curvature Find the radius of...Ch. 14.5 - Finding radii of curvature Find the radius of...Ch. 14.5 - Designing a highway curve The function
r(t) =...Ch. 14.5 - Curvature of the sine curve The function f(x) =...Ch. 14.5 - Parabolic trajectory In Example 7 it was shown...Ch. 14.5 - Parabolic trajectory Consider the parabolic...Ch. 14.5 - Prob. 78ECh. 14.5 - Zero curvature Prove that the curve...Ch. 14.5 - Prob. 80ECh. 14.5 - Maximum curvature Consider the superparabolas...Ch. 14.5 - Alternative derivation of the curvature Derive the...Ch. 14.5 - Computational formula for B Use the result of part...Ch. 14.5 - Prob. 84ECh. 14.5 - Descartes four-circle solution Consider the four...Ch. 14 - Prob. 1RECh. 14 - Sets of points Describe the set of points...Ch. 14 - Graphing curves Sketch the curves described by the...Ch. 14 - Prob. 4RECh. 14 - Curves in space Sketch the curves described by the...Ch. 14 - Curves in space Sketch the curves described by the...Ch. 14 - Intersection curve A sphere S and a plane P...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Vector-valued functions Find a function r(t) that...Ch. 14 - Prob. 13RECh. 14 - Intersection curve Find the curve r(t) where the...Ch. 14 - Intersection curve Find the curve r(t) where the...Ch. 14 - Prob. 16RECh. 14 - Prob. 17RECh. 14 - Prob. 18RECh. 14 - Prob. 19RECh. 14 - Prob. 20RECh. 14 - Prob. 21RECh. 14 - Prob. 22RECh. 14 - Prob. 23RECh. 14 - Prob. 24RECh. 14 - Finding r from r Find the function r that...Ch. 14 - Finding r from r Find the function r that...Ch. 14 - Prob. 27RECh. 14 - Prob. 28RECh. 14 - Prob. 29RECh. 14 - Velocity and acceleration from position consider...Ch. 14 - Velocity and acceleration from position Consider...Ch. 14 - Solving equations of motion Given an acceleration...Ch. 14 - Prob. 33RECh. 14 - Orthogonal r and r Find all points on the ellipse...Ch. 14 - Modeling motion Consider the motion of the...Ch. 14 - Prob. 36RECh. 14 - Prob. 37RECh. 14 - Firing angles A projectile is fired over...Ch. 14 - Prob. 39RECh. 14 - Baseball motion A toddler on level ground throws a...Ch. 14 - Prob. 41RECh. 14 - Prob. 42RECh. 14 - Prob. 43RECh. 14 - Prob. 44RECh. 14 - Arc length Find the arc length of the following...Ch. 14 - Prob. 46RECh. 14 - Velocity and trajectory length The acceleration of...Ch. 14 - Prob. 48RECh. 14 - Arc length parameterization Find the description...Ch. 14 - Tangents and normals for an ellipse Consider the...Ch. 14 - Prob. 51RECh. 14 - Prob. 52RECh. 14 - Properties of space curves Do the following...Ch. 14 - Prob. 54RECh. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Analyzing motion Consider the position vector of...Ch. 14 - Prob. 59RECh. 14 - Curve analysis Carry out the following steps for...Ch. 14 - Prob. 61RECh. 14 - Prob. 62RECh. 14 - Prob. 63RECh. 14 - Prob. 64RE
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- Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point (x, y, z) on the line x = 6 + t, y = 8 − t, z = 3 + t. Also, at time t, the second insect is at the point (x, y, z) on the line x = 1 + t, y = 2 + t, z = 2t. Assume that distances are given in inches. (a) Find the distance between the two insects at time t = 0. (b) Use a graphing utility to graph the distance between the insects from t = 0 to t = 10. (c) Using the graph from part (b), what can you conclude about the distance between the insects? (d) How close to each other do the insects get?arrow_forwardThe long run. A chair manufacturer hires its assembly-line labour for $18 an hour and calculates that the rental cost of its machinery is $6 per hour. Suppose that a chair can be produced using 4 hours of labour or machinery in any combination. The firm is currently using 1 hour of labour for every 3 hours of machine time. (Assume that labour is on the horizontal axis and capital is on the vertical axis). 3. Graphically illustrate your answer by drawing an isoquant, an isocost line for the current combination of labour and capital and an isocost line for the optimal combination of labour and capital. An isocost corresponding to the optimal combination of labour and capital is [a vertical line, a horizontal line, an upward sloping straight line, an upward sloping curve which is not a straight line, a downward sloping straight line, a downward sloping curve which is not a straight line, L-shaped] has slope [ ] at the optimal combination of inputs An isoquant…arrow_forward(a) Graph f(x) = 2x+1and g(x) = 2-x+2on the sameCartesian plane.(b) Shade the region bounded by the y-axis, f(x) = 2x+1,and g(x) = 2-x+2on the graph drawn in part (a).(c) Solve f(x) = g(x) and label the point of intersectionon the graph drawn in part (a).arrow_forward
- Using a directrix of y = −2 and a focus of (2, 6), what quadratic function is created? f(x) = − (x − 2)2 − 2 f(x) = (x − 2)2 + 2 f(x) = (x − 2)2 − 2 or f(x) = − (x + 2)2 − 2 (it can only be one)arrow_forwardThe graphs of y=1.1x and y=x have two points of intersection, while the graphs of y=2^x and y=x have no points of intersection. It follows that there is a real number 1 < p<2 for which the graphs of y=p^x and y=x have exactly one point of intersection and y=x is tangent to y=p^x. Find the value of p Determine the coordinates of the single point of intersection y=p^x and y=x. Type the values for the x- and y-coordinates or an approximation.arrow_forwarda. Show that h(x) = x3/4 and k(x) = (4x)1/3 are inverses of one another. b. Graph h and k over an x-interval large enough to show the graphs intersecting at (2, 2) and (-2, -2). Be sure the picture shows the required symmetry about the line y = x. c. Find the slopes of the tangent lines to the graphs at h and k at (2, 2) and (-2, -2). d. What lines are tangent to the curves at the origin?arrow_forward
- Consider the following. The x y-coordinate plane is given. A horizontal line, a curve, and a shaded region are graphed. The horizontal line labeled f(x) = 16 crosses the y-axis at y = 16. The curve g(x) = 8x − x2 enters the window at the origin, goes up and right becoming less steep, changes direction at (4, 16) touching the horizontal line, goes down and right becoming more steep, and exits the window at x = 8 on the positive x-axis. The area below the horizontal line, above the curve, right of the y-axis, and left of x = 4 is shaded. (a) Form the integral that represents the area of the shaded region. 0 dx (b) Find the area of the region. (Give an exact answer. Do not round.) Need Help? Read Itarrow_forwardThe equation of a curve is y = x + 2cos x. Find the x co-ordinates of the stationary points of the curve for 0arrow_forwardThe finite region bounded by x=y^2 and x=6-2y^2 rotated around the line x=8. The region bounded by the x-axis, y=x^2 and y=2-x rotated around the x-axis. The region bounded by the x-axis, y=x^2 and y=2-x rotated around the y-axis.arrow_forward
- Lines and line segments Find a function r(t) that describes the given line or line segment. The line through P(3, 4, 5) that is orthogonal to the plane 2x - z = 4arrow_forward(a) Graph f(x) = 2x and g(x) = 12 on the same Cartesian plane.(b) Shade the region bounded by the y-axis, f(x) = 2x , and g(x) = 12 on the graph drawn in part (a).(c) Solve f(x) = g(x) and label the point of intersection on the graph drawn in part (a).arrow_forward4.2-2 continuation of the problem Deduce that x is a regular mapping if and only if EG - F2 is never zero. (Thisis often the easiest way to check regularity. We will see, beginning in the nextchapter, that the functions E, F, G are fundamental to the geometry ofsurfaces.)arrow_forward
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