Question function r(t) = (1ª In 1, ¹72³, −3e¯³¹). Find the integral / r(t)dt. Consider the curve which is described by the vector b) Find parametric equations for the tangent line to the given curve at the point (0,-1,-3e-³). Can you find parametric equations for the tangent line to the given curve at the point (0,-,-3)?
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- A car travels over the hill having the shape of a parabola. When the car is at point A, it is traveling at 9 m/sec and increasing its speed at 3 m/sec2 . Determine the tangential and normal components of acceleration of the car at point A labeled belowThe flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y. (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).This is a three part problem. Given: Find the line integral with respect to arc length (6x + 8y)ds, where C is the line segment in the xy-plane with endpoints P = (8,0) and Q = (0,9) A. Find a vector parametric equation r(t) for the line segment C so that points P and Q correspond to t = 0 and t = 1, respectively. B. Using the parametrization in part A, what is the line integral with respect to arc length? C. Evaluate the line integral with respect to arc length in part B.
- Write an integral that represents the arc length of the curve on the given interval. Do not evaluate the integral. Parametric Equations Interval x = 5ln(t), y = 4t + 1 2 ≤ t ≤ 6Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = sqrt(t2+15)y = ln(t2 + 15) z = t; (4, ln(16), 1) x(t), y(t), z(t)=?Determine the moment of inertia of the area about the y-axis of the curve y2 = 2x, x=2, and the x-axis. Show complete solution with graph including strips. Solve using calculus.
- Find the work done by the force F(x , y , z)=(xyz ,−cos (yz), xz)moving a particle along a line segment from a point P (1,1,1) to a point Q (−2,1,3) correctly Hint: find the parametric equation of a line connecting P and Q, then evaluate the integral correctly.Verify by differentiation that the potential functions found inthe given problem produce the corresponding vector fields. Finding potential functions Find a potential function for the conservativevector fields.a. F = ⟨ex cos y, -ex sin y⟩b. F = ⟨2xy - z2, x2 + 2z, 2y - 2xz⟩1. Find the centroid of the region bounded by y = x^3 , y = −x + 2, y = 0. 2. Find an equation of the tangent line to the curve whose parametric equation is x = tan θ, y = sec θ
- A particle at (1, 0, 0) starts moving in space in such a way that its position vector at any time t ≥ 0 is R~ (t) = (cost + tsin t)ˆi + (sin t − t cost)ˆj + t²ˆk, t ≥ 0. (a) Find parametric equations for the line tangent to the trajectory of the particle at the point where t = π/2. (b) Calculate the acceleration of the particle at time t = π/2. (c) Calculate the total distance traveled by the particle in the time interval 0 ≤ t ≤ π/21. Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = 4t i + (5 − 2t) j + (1 + 3t) k 2. Find the derivative, r'(t), of the vector function. r(t)= (e^t^6)i-j+ln(1+6t)k 3. Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = 4t i + (5 − 2t) j + (1 + 3t) kThe area described in question 1(b) is filled by a flat metal plate with a surfacedensity σ = 3x. If the plate is secured to the y-axis by massless hinges,calculate its moment of inertia about the y-axis.[Note: the moment of inertia of a point mass, m is I = md^2, where d is thedistance to the rotation axis under consideration].