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- Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.At time t=0, a particle is located at the point (3,9,4). It travels in a straight line to the point (7,8,6), has speed 6 at (3,9,4) and constant acceleration 4i-j+2k. Find an equation for the position vector r(t) of the particle at time t -O+¹+* The equation for the position vector r(t) of the particle at time t is r(t) = (Type exact answers, using radicals as needed.)The motion of a point on the circumference of a rolling wheel of radius 3 feet is described by the vector function 7(t) = 3(10t – sin(10t))ỉ + 3(1 – cos(104))} Find the velocity vector of the point. v(t) = Find the acceleration vector of the point. a(t) Find the speed of the point. s(t) =
- (1n |t – 1], e', vî ) 1. Let 7(t) = (a) Express the vector valued function in parametric form. (b) Find the domain of the function. (c) Find the first derivative of the function. (d) Find T(2). (e) Find the vector equation of the tangent line to the curve when t=2. 2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = x? and z = x3 (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to 7 '(t) and 7"(t). In fact, an alternate formula for this vector is ア'(t) × ア"(t) ア(t) ×デ"(t)| B(t) =The path r(t) = (t) i + (3t2 +7) į describes motion on the parabola y = 3x + 7. Find the particle's velocity and acceleration vectors at t= - 4, and sketch them as vectors on the curve. The velocity vector at t= - 4 is v(- 4) = (D i+ (Di (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)Find the parametric equations of the tangent line to the curve represented by the vector function r(t) = (9t+9, -3t² - 5t-5, -t³-2t² - 1) at the point (27, -27, -17).
- Find the velocity and acceleration vectors and the equation of the tangent line for the curve r(t) = √√3ti + e4j + 6e¯¹k at t = 0. (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*).) v(0) : = a(0) = (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*). Use t for the parameter that takes all real values.) l(t): =At time t = 0, a particle is located at the point (1, 2, 3). (Vector Functions) It travels in a straight line to the point (4, 1, 4), has speed 2 at (1, 2,3) and constant acceleration 3i – j+k. Find equation for the position vector r(t) of the particle at time t.Find both the parametric and vector equation of the line segment between (−2, 3) and (5, −1) where 0 ≤ t ≤ 5. Please explain your steps. Thank you.
- (a) Find the directional derivative of z = x²y at (5,5) in the direction of π/2 with the positive x-axis. (b) In which direction is the directional derivative the largest at the point (5, 5)? Enter your answer as a vector whose length is the largest value of the directional derivative.Let u(t) = 9t'i+ (² -1)j-6k and v(t)= e'i+5 ej- e k Compute the derivative of the following function. u(t) - v(t) Select the correct choice below and fill in the answer box(es) to complete your choice. O A. The derivative is the vector-valued function i+ k. O B. The derivative is the scalar function Click to select and enter your answer(s) and then click Check Answer. All parts showing Clear All OK earchFind the derivative of the vector function r(t) = (200) r(t) = In (15-t²) i+√4+tj-6e² k
