Math

CalculusCalculus (MindTap Course List)Suppose u and v are vector functions that possess limits as t → a and let c be a constant. Prove the following properties of limits. (a) lim t → a [ u ( t ) + v ( t ) ] = lim t → a u ( t ) + lim t → a v ( t ) (b) lim t → a c u ( t ) = c lim t → a u ( t ) (c) lim t → a [ u ( t ) ⋅ v ( t ) ] = lim t → a u ( t ) ⋅ lim t → a v ( t ) (d) lim t → a [ u ( t ) × v ( t ) ] = lim t → a u ( t ) × lim t → a v ( t )BuyFind*launch*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 13.1, Problem 53E

Textbook Problem

Suppose **u** and **v** are vector functions that possess limits as
*c* be a constant. Prove the following properties of limits.

(a)

(b)

(c)

(d)

Calculus (MindTap Course List)

Ch. 13.1 - Find the domain of the vector function....Ch. 13.1 - Find the domain of the vector function....Ch. 13.1 - Find the limit. limt0(e3ti+t2sin2tj+cos2tk)Ch. 13.1 - Find the limit. limt1(t2tt1i+t+8j+sintlntk)Ch. 13.1 - Find the limit. limt1+t21t2,tan1t,1e2ttCh. 13.1 - Find the limit. limttet,t3+t2t31,tsin1tCh. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....

Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Draw the projections of the curve on the three...Ch. 13.1 - Draw the projections of the curve on the three...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Find a vector equation and parametric equations...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Show that the curve with parametric equations...Ch. 13.1 - Show that the curve with parametric equations...Ch. 13.1 - Find three different surfaces that contain the...Ch. 13.1 - Find three different surfaces that contain the...Ch. 13.1 - At what points does the curve r(t)=ti+(2tt2)k...Ch. 13.1 - At what points does the helix r(t)=sint,cost,t...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Show that the curve with parametric equations...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Find a vector function that represents the curve...Ch. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - If two objects travel through space along two...Ch. 13.1 - Two particles travel along the space curves...Ch. 13.1 - a Graph the curve with parametric equations...Ch. 13.1 - The view of the trefoil knot shown in Figure 8 is...Ch. 13.1 - Suppose u and v are vector functions that possess...Ch. 13.1 - Show that limtar(t)=b if and only if for every 0...Ch. 13.2 - The figure shows a curve C given by a vector...Ch. 13.2 - a Make a large sketch of the curve described by...Ch. 13.2 - a Sketch the plane curve with the given vector...Ch. 13.2 - a Sketch the plane curve with the given vector...Ch. 13.2 - a Sketch the plane curve with the given vector...Ch. 13.2 - a Sketch the plane curve with the given vector...Ch. 13.2 - a Sketch the plane curve with the given vector...Ch. 13.2 - a Sketch the plane curve with the given vector...Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the derivative of the vector function....Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - Find the unit tangent vector T(t) at the point...Ch. 13.2 - If r(t)=t,t2,t3, find r(t),T(1),r(t), and...Ch. 13.2 - If r(t)=e2t,e2t,te2t, find T(0),r(0), and...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find a vector equation for the tangent line to the...Ch. 13.2 - Find the point on the curve...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - a Find the point of intersection of the tangent...Ch. 13.2 - The curves r1(t)=t,t2,t3 and r2(t)=sint,sin2t,t...Ch. 13.2 - At what point do the curves r1(t)=t,1t,3+t2 and...Ch. 13.2 - Evaluate the integral. 02(tit3j+3t5k)dtCh. 13.2 - Evaluate the integral. 14(2t3/2i+(t+1)tk)dtCh. 13.2 - Evaluate the integral. 01(1t+1i+1t2+1j+tt2+1k)dtCh. 13.2 - Evaluate the integral....Ch. 13.2 - Evaluate the integral. (sec2ti+t(t2+1)3j+t2lntk)dtCh. 13.2 - Evaluate the integral. (te2ti+t1tj+11t2k)dtCh. 13.2 - Find r(t) if r(t)=2ti+3t2j+tk and r(1)=i+j.Ch. 13.2 - Find r(t) if r(t)=ti+etj+tetk and r(0)=i+j+k.Ch. 13.2 - Prove Formula 1 of Theorem 3.Ch. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prove Formula 5 of Theorem 3.Ch. 13.2 - Prove Formula 6 of Theorem 3.Ch. 13.2 - If u(t)=sint,cost,t and v(t)=t,cost,sint, use...Ch. 13.2 - If u and v are the vector functions in Exercise...Ch. 13.2 - Find f(2), where f(t)=u(t)v(t),u(2)=1,2,1,...Ch. 13.2 - If r(t)=u(t)v(t), where u and v are the vector...Ch. 13.2 - If r(t)=acost+bsint, where a and b are constant...Ch. 13.2 - If r is the vector function in Exercise 51, show...Ch. 13.2 - Show that is r is a vector function such that r...Ch. 13.2 - Find an expression for ddt[u(t)(v(t)w(t))].Ch. 13.2 - If r(t)0, show that ddt|r(t)|=1|r(t)|r(t)r(t).Ch. 13.2 - If a curve has the property that the position...Ch. 13.2 - If u(t)=r(t)[r(t)r(t)], show that...Ch. 13.2 - Show that the tangent vector to a curve defined by...Ch. 13.3 - Find the length of the curve....Ch. 13.3 - Find the length of the curve. r(t)=2t,t2,13t3,0t1Ch. 13.3 - Find the length of the curve. r(t)=2ti+etj+etk,0t1Ch. 13.3 - Find the length of the curve....Ch. 13.3 - Find the length of the curve. r(t)=i+t2j+t3k,0t1Ch. 13.3 - Find the length of the curve....Ch. 13.3 - Find the length of the curve correct to four...Ch. 13.3 - Find the length of the curve correct of four...Ch. 13.3 - Find the length of the curve correct of four...Ch. 13.3 - Graph the curve with parametric equations...Ch. 13.3 - Let C be the curve of intersection of the...Ch. 13.3 - Find, correct to four decimal places, the length...Ch. 13.3 - a Find the arc length function for the curve...Ch. 13.3 - a Find the arc length function for the curve...Ch. 13.3 - Suppose you start at the point (0,0,3) and move 5...Ch. 13.3 - Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j...Ch. 13.3 - a Find the unit tangent and unit normal vectors...Ch. 13.3 - a Find the unit tangent and unit normal vectors...Ch. 13.3 - a Find the unit tangent and unit normal vectors...Ch. 13.3 - a Find the unit tangent and unit normal vectors...Ch. 13.3 - Use Theorem 10 to find the curvature. r(t)=t3j+t2kCh. 13.3 - Use Theorem 10 to find the curvature....Ch. 13.3 - Use Theorem 10 to find the curvature....Ch. 13.3 - Find the curvature of r(t)=t2,lnt,tlnt at the...Ch. 13.3 - Find the curvature of r(t)=t,t2,t3 at the point...Ch. 13.3 - Graph the curve with parametric equations...Ch. 13.3 - Use Formula 11 to find the curvature. y=x4Ch. 13.3 - Use Formula 11 to find the curvature. y=tanxCh. 13.3 - Use Formula 11 to find the curvature. y=xexCh. 13.3 - At what point does the curve have maximum...Ch. 13.3 - At what point does the curve have maximum...Ch. 13.3 - Find an equation of a parabola that has curvature...Ch. 13.3 - a Is the curvature of the curve C shown in the...Ch. 13.3 - Use a graphing calculator or computer to graph...Ch. 13.3 - Use a graphing calculator or computer to graph...Ch. 13.3 - Plot the space curve and its curvature function...Ch. 13.3 - Plot the space curve and its curvature function...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - a Graph the curve r(t)=sin3t,sin2t,sin3t. At how...Ch. 13.3 - The graph of r(t)=t32sint,132cost,t is shown in...Ch. 13.3 - Use Theorem 10 to show that the curvature of a...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Use the formula in Exercise 42 to find the...Ch. 13.3 - Consider the curvature at x=0 for each member of...Ch. 13.3 - Find the vectors T, N, and B at the given point....Ch. 13.3 - Find the vectors T, N, and B at the given point....Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the osculating circles of the...Ch. 13.3 - Find equations of the osculating circles of the...Ch. 13.3 - At what point on the curve x=t3,y=3t,z=t4 is the...Ch. 13.3 - Is there a point on the curve in Exercise 53 where...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Show that the osculating plane at every point on...Ch. 13.3 - Show that at every point on the curve...Ch. 13.3 - The rectifying plane of a curve at a point is the...Ch. 13.3 - Show that the curvature is related to the tangent...Ch. 13.3 - Show that the curvature of a plane curve is...Ch. 13.3 - a Show that dB/ds is perpendicular to B. b Show...Ch. 13.3 - The following formulas, called the Frenet-Serret...Ch. 13.3 - Use the Frenet-Serret formulas to prove each of...Ch. 13.3 - Show that the circular helix r(t)=acost,asint,bt,...Ch. 13.3 - Use the formula in Exercise 63d to find the...Ch. 13.3 - Find the curvature and torsion of the curve...Ch. 13.3 - The DNA molecule has the shape of a double helix...Ch. 13.3 - Lets consider the problem of designing a railroad...Ch. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - The figure shows the path of a particle that moves...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - Find the velocity and position vectors of a...Ch. 13.4 - a Find the position vector of a particle that has...Ch. 13.4 - a Find the position vector of a particle that has...Ch. 13.4 - The position function of a particle is given by...Ch. 13.4 - What force is required so that a particle of mass...Ch. 13.4 - A force with magnitude 20 N acts directly upward...Ch. 13.4 - Show that if a particle moves with constant speed,...Ch. 13.4 - A projectile is fired with an initial speed of 200...Ch. 13.4 - Rework Exercise 23 if the projectile is fired from...Ch. 13.4 - A ball is thrown at an angle of 45 to the ground....Ch. 13.4 - A projectile is fired from a tank with initial...Ch. 13.4 - A rifle is fired with angle of elevation 36. What...Ch. 13.4 - A batter hits a baseball 3 ft above the ground...Ch. 13.4 - A medieval city has the shape of a square and is...Ch. 13.4 - Show that a projectile reaches three-quarters of...Ch. 13.4 - A ball is thrown eastward into the air from the...Ch. 13.4 - A ball with mass 0.8 kg is thrown southward into...Ch. 13.4 - Water traveling along a straight portion of a...Ch. 13.4 - Another reasonable model for the water speed of...Ch. 13.4 - A particle has position function r(t). If...Ch. 13.4 - a If a particle moves along a straight line, what...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - The magnitude of the acceleration vector a is 10...Ch. 13.4 - If a particle with mass m moves with position...Ch. 13.4 - The position function of a spaceship is...Ch. 13.4 - A rocket burning its onboard fuel while moving...Ch. 13.R - What is a vector function? How do you find its...Ch. 13.R - What is the connection between vector functions...Ch. 13.R - How do you find the tangent vector to a smooth...Ch. 13.R - If u and v are differentiable vector functions, c...Ch. 13.R - How do you find the length of a space curve given...Ch. 13.R - a What is the definition of curvature? b Write a...Ch. 13.R - a Write formulas for the unit normal and binormal...Ch. 13.R - a How do you find the velocity, speed, and...Ch. 13.R - State Keplers Laws.Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - a Sketch the curve with vector function...Ch. 13.R - Let r(t)=2t,(et1)/t,ln(t+1). a Find the domain of...Ch. 13.R - Find a vector function that represents the curve...Ch. 13.R - Find parametric equations for the tangent line to...Ch. 13.R - If r(t)=t2i+tcostj+sintk, evaluate 01r(t)dt.Ch. 13.R - Let C be the curve with equations...Ch. 13.R - Use Simpsons Rule with n=6 to estimate the length...Ch. 13.R - Find the length of the curve...Ch. 13.R - The helix r1(t)=costi+sintj+tk intersects the...Ch. 13.R - Reparametrize the curve r(t)=eti+etsintj+etcostk...Ch. 13.R - For the curve given by r(t)=sin3t,cos3t,sin2t,...Ch. 13.R - Find the curvature of the ellipse x=3cost,y=4sint...Ch. 13.R - Find the curvature of the curve y=x4 at the point...Ch. 13.R - Find an equation of the osculating circle of the...Ch. 13.R - Find an equation of the osculating plane of the...Ch. 13.R - The figure shows the curve C traced by a particle...Ch. 13.R - A particle moves with position function...Ch. 13.R - Find the velocity, speed, and acceleration of a...Ch. 13.R - A particle starts at the origin with initial...Ch. 13.R - An athlete throws a shot at an angle of 45 to the...Ch. 13.R - A projectile is launched with an initial speed of...Ch. 13.R - Find the tangential and normal components of the...Ch. 13.R - A disk of radius 1 is rotating in the...Ch. 13.R - In designing transfer curves to connect sections...Ch. 13.P - A particle P moves with constant angular speed ...Ch. 13.P - A circular curve of radius R on a highway is...Ch. 13.P - A projectile is fired from the origin with angle...Ch. 13.P - a A projectile is fired from the origin down an...Ch. 13.P - A ball rolls off a table with a speed of 2 ft/s....Ch. 13.P - Find the curvature of the curve with parametric...Ch. 13.P - If a projectile is fired with angle of elevation ...Ch. 13.P - A cable has radius r and length L and is wound...Ch. 13.P - Show that the curve with vector equation...

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