Mathematics, 05.03.2021 21:40 Samuelmoreno4496

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. x=t, y=e^-t, z=2t-t^2; (0, 1, 0)

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Answer from: raiderries

x = t

y = 1 - t

z = 2t

Step-by-step explanation:

Given

x=t

$y=e^{-t}$

z=2t-t^2

(0, 1, 0)

The vector equation is given as:

r(t) = (x,y,z)

Substitute values for x, y and z

$r(t) = (t,\ e^{-t},\ 2t - t^2)$

Differentiate:

$r'(t) = (1,\ -e^{-t},\ 2 - 2t)$

The parametric value that corresponds to (0, 1, 0) is:

t = 0

Substitute 0 for t in r'(t)

$r'(t) = (1,\ -e^{-t},\ 2 - 2t)$

$r'(0) = (1,\ -e^{-0},\ 2 - 2*0)$

$r'(0) = (1,\ -1,\ 2 - 0)$

$r'(0) = (1,\ -1,\ 2)$

The tangent line passes through (0, 1, 0) and the tangent line is parallel to r'(0)

It should be noted that:

The equation of a line through position vector a and parallel to vector v is given as:

r(t) = a + tv

Such that:

a = (0,1,0) and v = r'(0) = (1,-1,2)

The equation becomes:

r(t) = (0,1,0) + t(1,-1,2)

r(t) = (0,1,0) + (t,-t,2t)

r(t) = (0+t,1-t,0+2t)

r(t) = (t,1-t,2t)

By comparison:

r(t) = (x,y,z) and r(t) = (t,1-t,2t)

The parametric equations for the tangent line are:

x = t

y = 1 - t

z = 2t

Find parametric equations for the tangent line to the curve with the given parametric equations at t

Answer from: Quest

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12

Answer from: Quest

In all honesty, there really is no “work” per say to show. mainly it’s just using the process of elimination. the question is asking which domain(s) would be the most appropriate for the function c(n). n is the number of observed vehicles so right away, 1, 2, and 3 can be eliminated.why? 1, 2 start in the negatives and therefore it is impossible to observe negative cars. so, #4 would be the best answer to the given problem.

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