2. Consider the vector-valued function R(t) = Show that is continuous at t = 1. (e¹-², 1-1², (2-1)), t (t,0, t - 1), if -1 < t < 1 if t > 1.
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- Let R be a differentiable vector-valued function such that R'(t) = (3t2,2tvt? + 1, 2t + 3) and R(1) = (4, *2, 3). Find R(0).Determine the interval (s) on which the vector-valued function r(t)=ti+(sqrtt+1j)+(t^2+1)k continous A [-infinity, -1) B [-infinity, infinity) C (-1, infinity) D [-1, infinity)2) r(1) = ti -t j-t'k, t20 Draw the graph of the vector-valued function, explaining it in detail.
- Find the domain of the vector function (in interval notation) r(t)= t-1/t+1i+sin(t)j+ln(9-t^2)kCalculate the derivative of the vector-valued function, r(t) (t')i + (sect)j + (2*)kSuppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect. Please Provide Unique Answer. Thank you!
- the linearization of (fi,f2) = (x²y+ 3xy², 3x³ + 2xy) at (1,0) to approximate the value at (0.9,-0.1).f(x, 3) In(x2 + Yof function P(3, 4) at the point of = (9, 12) Directional derivative in the direction of the vector to you.Evaluate along the curve y=x2 from (-1,1) to (2,4). First find the vector valued function r(t) defining the curve.
- Determine where the vector function r(t) = (141) i + (147) j is continuous. (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")","[" or "]" depending on whether the interval is open or closed.) t EVerify the Sum and Product Rules for derivatives of vector-valued functions.2. (4 points) Compute the directional derivative of f at the given point in the direction of the given vector: f(x, y) = In(3+ 2a? + y?), P(2,1), ū = (-3, 4)