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- What is special about the derivatives of vector functions of con-stant length? Give an example.Represent the line segment from P(−2, −3, 8), Q(5, 1, −2) by a vector-valued function and by a set of parametric equations.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 domain of the vector function r(t)=(t^2−4)/(t−2)i+sqrt(t+5)j+(e^t)* k Select one: a. [−5,2)∪(2,∞) b. [−5,∞) c. (−∞,∞) d. (−5,2)∪(2,∞) e. (−∞,2)∪(2,∞)Let vector r(t) = <5sin(t), 5cos(t), 2t^(3/2)> be the position function of an object.a. Find vector v(t) and vector a(t), the velocity and acceleration of the object.b. Find the speed of the object.c. Find the distance the object travels on the interval [0, 1]. Also find the average speed of theobject on this interval.consider the vector-valued function r(t) = <2t3 – 15t2 + 36t, ln(4t – t^2)> a) State the domain of r(t) b) Identify where r(t) is not smooth over its domain. c) Identify any places where r(t) has vertical tangent lines. d) Identify any places where r(t) has horizontal tangent lines.
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- Prove the Sum and Difference Rules for vector functions.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)Represent the line segment from P to Q by a vector-valued function. P(4, 0, 7), Q(3, -3, 4) r(t) = Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list. Use t for the variable of parameterizat