Determine the arc-length of the space curve defined by the position vector r(t) = 20 sin(t)i + 21 sin(t)j + 29 cos(t)k, - between and t = π.
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- Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.Find parametric equations for the line tangent to the helix r(t) =(sqrt(2)cos t)i + (sqrt(2) sin t)j + t k at the point where t = pai/4.2. Consider the vector-valued function r(t) = (sin 2t + cos 2t)i + (sin 2t – cos 2t)j. (a) Find the arc-length of the portion of the curve from t 0 to tz = /4. (b) Find the parametric equations of its tangent lines at t = 0 and t2 /4. (c) If 0 is the angle formed by the two tangent lines above, find cos 0.
- The path r(t)=(5 sin t) i+(5 cos t) j describes motion on the circle x^2+y^2=25. Find the particle's velocity and acceleration vectors at t=π/3 and π/6, and sketch them as vectors on the curve.Find the unit tangent vector to the curve defined by F(t) = (3 cos(t), 3 sin(t), 5 sin²(t)) at t= π x(t) y(t) = Ch z(t) 3 Use the unit tangent vector to write the parametric equations of a tangent line to the curve at t= 7 3The position vector for a particle moving on a helix is c(t) = (5 cos(t), 5 sin(t), t² ) . Find the speed s(to) of the particle at time to 11r. (Express numbers in exact form. Use symbolic notation and fractions where needed.) s(to) Find parametrization for the tangent line at time to 11r. Use the equation of the tangent line such that the point of tangency occurs when t = to. (Write your solution using the form (*,*,*). Use t for the parameter that takes all real values. Simplify all trigonometric expressions by evaluating them. Express numbers in exact form. Use symbolic notation and fractions as needed.) 1(t) = Where will this line intersect the xy-plane? (Write your solution using the form (*,*,*). Express numbers in exact form. Use symbolic notation and fractions where needed.) point of intersection:
- A point starts at the location (x, y) = (6,0) and moves CCW along a circular path that is centered at (2, y) = (0, 0). The point moves at a constant speed. Imagine an angle with a vertex at (0, 0) subtending the object's path. The angle sweeps out 2 radians per second, and let t represent the varying number of seconds that have elapsed since the point started moving. (Draw a diagram to represent this!) %3D a. Write a formula that expresses the x-coordinate of the point in terms of t. (Hint: Start by expressing the angle measure, 0, in terms of t.) Preview b. How long does it take the point to complete one full lap around the circular path? (Hint: The angle must sweep out 2T radians for the point to complete one full lap, and the angle sweeps out 2 radians per second.) seconds PreviewFind parametric equations for line that is tangent to the curve x=cost, y=sint, z=t at the point(cos(4π/6),sin(4π/6),4π/6) Parametrize the line so that it passes through the given point at t=0.Graph the curve with parametric equations x = sin(t), y = 3 sin(2t), z = sin(3t). Find the total length of this curve correct to four decimal places.
