Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
3rd Edition
ISBN: 9780134995991
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Textbook Question
Chapter 14.3, Problem 52E
Three-dimensional motion Consider the motion of the following objects. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical and opposite g, the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated.
- a. Find the velocity and position
vectors , for t ≥ 0. - b. Make a sketch of the trajectory.
- c. Determine the time of flight and range of the object.
- d. Determine the maximum height of the object.
48. A golf ball is hit east down a fairway with an initial velocity of 〈50, 0, 30〉 m/s. A crosswind blowing to the south produces an acceleration of the ball of −0.8 m/s2.
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Chapter 14 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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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- A cable television company estimates that with x thousand subscribers, its monthly revenue and cost (in thousands of dollars) are given by the following equations. R(x) = 45x - 0.24x2 C(x) = 257 + 13xarrow_forwardx³-343 If k(x) = x-7 complete the table and use the results to find lim k(x). X-7 x 6.9 6.99 6.999 7.001 7.01 7.1 k(x) Complete the table. X 6.9 6.99 6.999 7.001 7.01 7.1 k(x) (Round to three decimal places as needed.)arrow_forward(3) (4 points) Given three vectors a, b, and c, suppose: |bx c = 2 |a|=√√8 • The angle between a and b xc is 0 = 135º. . Calculate the volume a (bxc) of the parallelepiped spanned by the three vectors.arrow_forward
- Calculate these limits. If the limit is ∞ or -∞, write infinity or-infinity. If the limit does not exist, write DNE: Hint: Remember the first thing you check when you are looking at a limit of a quotient is the limit value of the denominator. 1. If the denominator does not go to 0, you should be able to right down the answer immediately. 2. If the denominator goes to 0, but the numerator does not, you will have to check the sign (±) of the quotient, from both sides if the limit is not one-sided. 3. If both the numerator and the denominator go to 0, you have to do the algebraic trick of rationalizing. So, group your limits into these three forms and work with them one group at a time. (a) lim t-pi/2 sint-√ sin 2t+14cos ² t 7 2 2 2cos t (b) lim sint + sin 2t+14cos = ∞ t-pi/2 2 2cos t (c) lim cost-√sin 2t+14cos² t = t-pi/2 2cos t (d) lim t→pi/2 cost+√ sin t + 14cos 2cos ² t = ∞ (e) lim sint-v sin 2 t + 14cos = 0 t-pi/2 (f) lim t-pi/2 sin t +√ sin 2sin 2 t 2 t + 14cos t 2sin t cost- (g)…arrow_forwardThink of this sheet of paper as the plane containing the vectors a = (1,1,0) and b = (2,0,0). Sketch the parallelogram P spanned by a and b. Which diagonal of P represents the vector a--b geometrically?arrow_forward(1) (14 points) Let a = (-2, 10, -4) and b = (3, 1, 1). (a) (4 points) Using the dot product determine the angle between a and b. (b) (2 points) Determine the cross product vector axb. (c) (4 points) Calculate the area of the parallelogram spanned by a and b. Justify your answer. 1arrow_forward
- (d) (4 points) Think of this sheet of paper as the plane containing the vectors a = (1,1,0) and b = (2,0,0). Sketch the parallelogram P spanned by a and b. Which diagonal of P represents the vector ab geometrically? d be .dx adjarrow_forward(2) (4 points) Find all vectors v having length 1 that are perpendicular to both =(2,0,2) and j = (0,1,0). Show all work. a=arrow_forwardFor the following function, find the full power series centered at a of convergence. 0 and then give the first 5 nonzero terms of the power series and the open interval = f(2) Σ 8 1(x)--(-1)*(3)* n=0 ₤(x) = + + + ++... The open interval of convergence is: 1 1 3 f(x)= = 28 3x6 +1 (Give your answer in help (intervals) .)arrow_forward
- For the following function, find the full power series centered at x = 0 and then give the first 5 nonzero terms of the power series and the open interval of convergence. f(x) = Σ| n=0 9 f(x) = 6 + 4x f(x)− + + + ++··· The open interval of convergence is: ☐ (Give your answer in help (intervals) .)arrow_forwardLet X be a random variable with the standard normal distribution, i.e.,X has the probability density functionfX(x) = 1/√2π e^-(x^2/2)2 .Consider the random variablesXn = 20(3 + X6) ^1/2n e ^x^2/n+19 , x ∈ R, n ∈ N.Using the dominated convergence theorem, prove that the limit exists and find it limn→∞E(Xn)arrow_forwardLet X be a discrete random variable taking values in {0, 1, 2, . . . }with the probability generating function G(s) = E(sX). Prove thatVar(X) = G′′(1) + G′(1) − [G′(1)]2.[5 Marks](ii) Let X be a random variable taking values in [0,∞) with proba-bility density functionfX(u) = (5/4(1 − u^4, 0 ≤ u ≤ 1,0, otherwise. Let y =x^1/2 find the probability density function of Yarrow_forward
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