Question

Determine the equation of the tangent line to the given path at the specified value of t. (Enter your answer as a comma-separated list of equations in (x, y, z) coordinates.) (sin(7t), cos(7t), 2t9/2); t = 1

Answer 1

Answer:

$(x,y,z)=(0.66+5.28t\ ,\ 0.75-4.69t\ ,\ 2+9t)$

Step-by-step explanation:

Tangent Line of a Vector Function

We are given a vector function of one variable, let's call it H(t) and it's given by

$H(t)=(sin(7t), cos(7t), 2t^{9/2})$

To compute the equation of the tangent line, we first find the derivative of H

$H'(t)=(7cos(7t),-7 sin(7t), 2*(9/2)t^{7/2})$

$H'(t)=(7cos(7t),-7 sin(7t), 9t^{7/2})$

Evaluating in t=1

$H'(1)=(7cos(7),-7 sin(7), 9)=(5.28,-4.6,9)$

Evaluating H(1):

$H(1)=(sin(7), cos(7), 2)=(0.66,0.75,2)$

The equation of the tangent line is given in rectangular coordinates as

$x=x_o+x't=0.66+5.28t$

$y=y_o+y't=0.75-4.69t$

$z=z_o+z't=2+9t$

Or, equivalently

$(x,y,z)=(0.66+5.28t\ ,\ 0.75-4.69t\ ,\ 2+9t)$