Lafcalc.weebly.com



New 11.2VectorsThe vector in component form is a, b is the vector starting at (0, 0) and going through the point (a, b).The magnitude of the vector v is defined as a2+b2.The vector can also be written with magnitude and direction angle like this:(rcosθ, rsinθ) where r is the magnitude of the vector and θ is the directional angle which is the smallest angle formed by the x-axis and the vector.If the position at any time is (xt,yt), then the position vector is xt, ytthe velocity vector is dxdt,dydt the particle’s speed is the magnitude of v denoted by v=dxdt2+dydt2the particle’s acceleration vector is at=d2xdt2,d2ydt2 If the particle is moving along a path so that its velocity is vt=(v1t,v2t) the displacement from t=a to t=b is given by the vector abv1tdt, abv2tif the particle’s original position is (x(a),y(a)) the new position would be x(a)+abv1tdt, y(a)+abv2t the total distance traveled is abv(t)dt=ab(v1t)2+(v2t)2dtA particle moves in the xy-plane so that at any time t, the position of the particle is given by xt=t3+4t2, yt=t4-t3Find the velocity vector when t=1.Write the equation of the tangent line to the graph when t=1.Find the acceleration vector when t=1.A particle moves in the xy-plane so that at any time t, t≥0, the position of the particle is given by xt=t2+3t, yt=t3-3t2. Find the magnitude of the velocity vector when t=1. (Find the speed of the particle.)A particle moves in the xy-plane so that x=3-4cost and y=1-2sint, where 0≤t≤2π. The path of the particle intersects the x-axis twice. Write an expression that represents the distance traveled by the particle between the two x-intercepts. Do not evaluate.A particle moves in the xy-plane so that at any time t, the position of the particle is given by xt=2t3-15t2+36t+5 and yt=t3-3t2+1, where t≥0. For what value(s) of t is the particle at rest?No CalculatorA particle moves in the xy-plane in such a way that its velocity vector is 3t2-4t, 8t3+5. If the position vector at t=0 is 7, -4, find the position of the particle at t=1.CalculatorAn object moving along a curve in the xy-plane has position (xt, yt) at time twith dxdt=sin?(t3) and dydt=cost2. At time t=2, the object is at the position (7, 4).Write the equation of the tangent line to the curve at the point where t=4.Find the speed of the vector at t=2.For what value of t, 0<t<1, does the tangent line to the curve have a slope of 4? Find the acceleration vector at this time.Find the position of the particle at time t=1.1973No calculatorA particle moves on the curve y=lnx so that the x-component has velocity x't=t+1 for t≥0. At time t=0, the particle is at the point (1, 0). At time t=1, the particle is what point?1985A particle moves in the xy-plane so that at any time t its coordinates are x=t2-1 and y=t4-2t3. At t=1, its acceleration vector is 1985If the velocity of a particle moving along the x-axis is vt=2t-4 and if at t=0 its position is 4, then at any time t its position x(t) is 1988For any time t≥0, if the position of a particle in the xy-plane is given by x=t2+1 and y=ln?(2t+1), then the acceleration vector is 1993The position of the particle moving along the x-axis is xt=sin2t-cos?(3t) for time t≥0. When t=π, the acceleration of the particle is 1993 A particle moves along the x-axis so that at any time t≥0, the acceleration of the particle is at=e-2t. If at t=0, the velocity of theparticle is 52 and its position is174, then its position at any time t>0 is xt=1993If a particle is moves in the xy-plane so that at any time t>0, its position vector is (lnt2+2t, 2t2), then at time t=2, its velocity vector is 1997If x=e2t and y=sin?(2t), then dydx=1997The length of the path described by the parametric equations x=cos3t and y=sin3t, for 0≤t≤π2 is given by what integral?1997For what values of t does the curve given by parametric equations x=t3-t2-1 and y=t4+2t2-8t have a vertical tangent?1998In the xy-plane, the graph of the parametric equations x=5t+2 and y=3t, for -3≤t≤3, is a line segment with slope 1998A particle moves on a plane curve so that at any time t>0 its x-coordinate ist3-t and its y-coordinate is 2t-13. The acceleration vector of the particle at t=1 is 1998The length of the path described by the parametric equations x=13t3 and y=12t2, where 0≤t≤1 , is given by the integral 1998 If f is the vector-valued function defined by ft=(e-t,cost), then f''t=2003No CalculatorFor 0≤t≤13, an object travels along an elliptical path given by the parametric equations x=3cost and y=4sint . At the point where t=13, the object leaves that path and travels along the line tangent to the path at that point. What is the slope of the line on which the object travels?2003 The position of a particle moving in the xy-plane is given by the parametric equations x=t3-3t2 and y=2t3-3t2-12t. For what values of t is the particle at rest?2003A curve C is defined by the parametric equations x=t2-4t+1 and y=t3. What is the equation of the line tangent to the graph of C at the point (-3, 8)? ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download