Show that t, e^t, and sin(t) are linearly independent.

2) Let 7(t) = (t, t², e'). Find an equation for the unit tangent vector T(t).

Sketch the curve with the vector equation r(t) = cos(t)i − cos(t)j + sin(t)k. Show the direction of increasing t with an arrow drawn on your curve.

Determine a vector equation for the tangent line to the curve:

Suppose that the vector-valued function F(t) < 3t, cos(5t), sin(5t) represents the position of an object at time t. Part a) Find a vector-valued function a(t) representing the object's acceleration at time t. Part b) What is the distance travelled by the object between time t = 0 and time t 1?

Let f(w)f(w)be a function of vector ww, i.e.   f(w)=1/(1+e−wTx). Determine the first derivative and matrix of second derivatives of ffwith respect to w ?

Find a vector equation for the line that goes through the points (5,9,0) and (0, 11, 1). F(1) = with -o < t<∞ %D

Determine the equation for the line with normal vector n = [3,-4] passing through the point P(-1,-2)

Let r(t) =< 3t, 5t^2, 2t >. Find the unit tangent vector T(t).