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All Textbook Solutions for Multivariable Calculus
27E28EFinding Intervals on Which a Curve Is Smooth In Exercises 2734, find the open interval(s) on which the curve given by the vector-valued function is smooth. r()=2cos3i+3sin3j,0230EFinding Intervals on Which a Curve Is Smooth In Exercises 2734, find the open interval(s) on which the curve given by the vector valued function is smooth. r(t)=2t8+t3i+2t28+t3j32E33E34E35E36EUsing Two MethodsIn Exercises 37 and 38, find (a) ddt[r(t)u(t)] and (b) ddt[r(t)u(t)] in two different ways. (i) Find the product first, then differentiate. (ii) Apply the properties of Theorem 12.2. r(t)=ti+2t2j+t3k,u(t)=t4k38EFinding an Indefinite Integral In Exercises 39-46, find the indefinite integral. (2ti+j+9k)dt40EFinding an Indefinite Integral In Exercises 39-46, find the indefinite integral. (1ti+jt3/2k)dt42E43E44E45E46E47E48E49E50E51EEvaluating a Definite Integral In Exercises 47-52, evaluate the definite integral. 03 ti+t2j dt53E54E55E56E57EFinding an Antiderivative In Exercises 53-58, find r(t) that satisfies the initial condition(s). r(t)=11+t2i+1t2j+1tk,r(1)=2i59EThink About It Find two vector-valued functions f(t) and g(t) such that ab[f(t)g(t)]dt[abf(t)dt][abg(t)dt].61E62E63E64E65E66E67E68E69EParticle MotionA particle moves in the yz-plane along the curve represented by the vector-valued function r(t)=(2cost)j+(3sint)k. (a) Describe the curve. (b) Find the minimum and maximum values of r and r.71E72E73ETrue or False? In Exercises 73-76, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The definite integral of a vector-valued function is a real number.75E76E1E2E3E4EFinding Velocity and Acceleration Along a Plane Curve In Exercises 3-10, the position vector r describes the path of an object moving in the x y-plane. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Vector Point r(t)=t2i+tj (4,2)6EFinding Velocity and Acceleration Along a Plane Curve In Exercises 3-10, the position vector r describes the path of an object moving in the x y-plane. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Vector Point r(t)=2costi+2sintj (2,2)8EFinding Velocity and Acceleration Along a Plane Curve In Exercises 3-10, the position vector r describes the path of an object moving in the x y-plane. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. Position Vector Point r(t)=tsint,1cost (,2)10EFinding Velocity and Acceleration Vectors in Space In Exercises 11-20, the position vector r describes the path or an object moving in space. (a) Kind the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time r(t)=ti+5tj+3tk t=112EFinding Velocity and Acceleration Vectors in Space In Exercises 11-20, the position vector r describes the path or an object moving in space. (a) Kind the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time r(t)=ti+t2j+12t2k t=414EFinding Velocity and Acceleration Vectors in Space In Exercises 11-20, the position vector r describes the path or an object moving in space. (a) Kind the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time r(t)=titj+9t2k t=016EFinding Velocity and Acceleration Vectors in Space In Exercises 11-20, the position vector r describes the path or an object moving in space. (a) Kind the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time r(t)= 4t,3cost,3sint t=18E19E20EFinding a Position Vector by Integration In Exercises 21-26, use the given acceleration vector and initial conditions to find the velocity and position vectors. Then find the position at time t=2. a(t)=i+j+k,v(0)=0,r(0)=022E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50ECircular Motion In Exercises 51 and 52, use the results of Exercises 47-50. A psychrometer (an instrument used to measure humidity) weighing 4 ounces is whirled horizontally using a 6-inch string (see figure). The string will break under a force of 2 pounds. Find the maximum speed the instrument can attain without breaking the string. (Use F=ma, where m=1/128.)52E53E54E55EParticle Motion Consider a particle moving on an elliptical path described by r(t)=acosti+bsintj where =d/dt is the constant angular speed. (a) Find the velocity vector. What is the speed of the particle? (b) Find the acceleration vector and show that its direction is always toward the center of the ellipse.57E58E59E60E61E62E63E1E2EFinding the Unit Tangent Vector In Exercises 3-8, find the unit tangent vector to the curve at the specified value of the parameter. r(t)=t2i=2tj,t=14EFinding the Unit Tangent Vector In Exercises 3-8, find the unit tangent vector to the curve at the specified value of the parameter. r(t)=5costi+5sintj,t=36EFinding the Unit Tangent Vector In Exercises 3-8, find the unit tangent vector to the curve at the specified value of the parameter. r(t)=3tilntj,t=e8E9E10E11E12E13E14EFinding the Principal Unit Normal Vector In Exercises 15-20, find the principal unit normal vector to the curve at the specified value of the parameter. r(t)=ti+12t2j,t=216EFinding the Principal Unit Normal Vector In Exercises 15-20, find the principal unit normal vector to the curve at the specified value of the parameter. r(t)=ti+t2j+lntk,t=118E19E20E21E22E23E24E25E26E27E28E29E30E31ECircular MotionIn Exercises 3134, consider an object moving according to the porition vector r(t)=acosti+asintj. Determine the direction of T and N relative to the position vector r.33E34E35E36E37E38EFinding Tangential and Normal Components of AccelerationIn Exercises 3540, find the tangential and normal components of acceleration at the given time t for the space curve r(t). r(t)=etsinti+etcostj+etk,t=040E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76ECurvature Consider points P and Q on a curve What does it mean for the curvature at P to be less than the curvature at Q?Arc Length Parameter Let r(t) be a space curse. How can you determine whether t is the arc length parameter?3E4E5E6E7E8EProjectile Motion The position of a baseball. is represented by r(t)=502ti+(3+502t16t2)j. Find the arc length of the trajectory of the baseball.10E11E12E13E14E15E16EInvestigation Consider the graph of the vector-valued function r(t)=ti+(4t2)j+t3k on the interval [0, 2]. (a) Approximate the length of the curve by finding the length of the line segment connecting its endpoints. (b) Approximate the length of the curve by summing the lengths of the line segments connecting the terminal points of the vectors r(0),r(0.5),r(1),r(1.5),andr(2). (c) Describe how you could obtain a more accurate approximation by continuing the processes in parts (a) and (b). (d) Use the integration capabilities of a graphing utility to approximate the length of the curve. Compare this result with the answers in parts (a) and (b).18E19E20E21E22E23E24E25E26EFinding CurvatureIn Exercises 2328, find the curvature of the plane curve at the given value of the parameter. r(t)=t,sint,t=228E29E30E31E32E33E34EFinding Curvature In Exercises 29-36, find the curvature of the curve. r(t)=4ti+3costj+3sintk36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65ESpeed The smaller the curvature of a bend in a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path y=13x3, where x and y are measured in miles, can safely go 30 miles per hour at (1,13). How fast can it go at (32,98)?67ECenter of Curvature Use the result of Exercise 67 to find the center of curvature for die curve at die given point, (a) y=ex,(0,1) (0.1) (b) y=x22,(1,12) (c) y=x3,(0,0)69E70E71E72E73E74E75E76ECurvature of a Cycloid Use the result of Exercise 75 to find the curvature K of the cycloid represented by the parametric equations x()=a(sin)andy()=a(1cos) What are the minimum and maximum values of K?Tangential and Normal Components of Acceleration Use Theorem 12.10 to find aT and aN for each curve given by the vector-valued function. (a) r(t)=3t2i+(3tt3)j (b) r(t)=ti+t2j+12t2k79E80ECurvatureVerify that the curvature at any point (x,y) on the graph of y=coshx is 1/y2.82E83E84E85E86E87E88E89E90E91E92E93E94EEvaluating a FunctionIn Exercises 1 and 2, evaluate the function at the given values of the independent variables. Simplify the results. f(x,y)=x2y3 (a) f(0,4) (b) f(2,1) (c) f(3,2) (d) f(x,7)2REFinding the Domain and Range of a FunctionIn Exercises 3 ad 4, find the domain and range of the function. f(x,y)=xyFinding the Domain and Range of a FunctionIn Exercises 3 ad 4, find the domain and range of the function. f(x,y)=36x2y2Sketching a SurfaceIn Exercises 5 and 6, describe and sketch the surface given by the function. f(x,y)=26RESketching a Contour MapIn Exercises 7 and 8, describe the level curves of the function. Sketch a contour map of the surface using level curves for the given c -values. z=32x+y,c=0,2,4,6,8Sketching a Contour MapIn Exercises 7 and 8, describe the level curves of the function. Sketch a contour map of the surface using level curves for the given c -values. z=2x2+y2,c=1,2,3,4,5ConjectureConsider the function f(x,y)=x2+y2. (a) Sketch the graph of the surface given by f. (b) Make a conjecture about the relationship between the graphs of f and g(x,y)=f(x,y)+2. Explain your reasoning. (c) Make a conjecture about the relationship between the graphs of f and g(x,y)=f(x,y2). Explain your reasoning. (d) On the surface in part (a), sketch the graphs of z=f(1,y) and z=f(x,1).Cobb-Douglas Production Function A manufacturer estimates that its production can be modeled by f(x,y)=1000.8y0.2 where x is the number of units of labor and y is the number of units of capital. (a) Find the production level when x=100andy=200. (b) Find the production level when x=500andy=1500.Sketching a Level Surface In Exercises 11 and 12, describe and sketch the graph of the level surface f(x,y,z)=c at the given value of c. f(x,y,z)=x2y2+z2,c=2Sketching a Level SurfaceIn Exercises 11 and 12, describe and sketch the graph of the level surface f(x,y,z)=c at the given value of c. f(x,y,z)=4x2y2+4z2,c=0Limit and ContinuityIn Exercises 1318, find the limit (if it exists) and discuss the continuity of the function. lim(x,y)(1,1)xyx2+y214RE15RELimit and ContinuityIn Exercises 1318, find the limit (if it exists) and discuss the continuity of the function. lim(x,y)(0,0)x2yx4+y2Limit and ContinuityIn Exercises 1318, find the limit (if it exists) and discuss the continuity of the function. lim(x,y,z)(3,1,2)lnzxyz