A. Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P. b. Find the point 4 units along the curve (in the direction of increasing t) from P.

r(t) = et sin t i + √2 et k, P(0, l ,√2)

The correct question has r(t) = e^t sint i + e^t cost j + √2 e^t k

(a) 2e^t

(b) r(t) = (L/2 + 1)sin(ln|L/2 + 1|) i + (L/2 + 1)cos(ln|L/2 + 1|) j + √2(L/2 + 1) k

Step-by-step explanation:

Given r(t) = e^t sint i + e^t cost j + √2 e^t k

(a) To find the arc length of the function, we first differentiate r(t) to obtain r'(t). Doing this, we have

r'(t) = (cost + sint)e^t i + (cost - sint)e^t j + √2 e^t k

Next, we find the magnitude of r'(t). Doing this, we obtain

|r'(t)| = √{[(cost + sint)e^t]² + [(cost - sint)e^t]²+ (√2 e^t)² k}

= √(4e^(2t))

= 2e^t

(b) Now, because t = 0 corresponds with the (0, 1, √2), we have the arc length function, L to be the integral from 0 to t of |r'(v)|dv

L = integral of 2e^v dv from v = 0 to t.

Evaluation the integral, we have

L = 2e^t - 2e^0

= 2e^t - 2

L = 2(e^t - 1)

Let us make t the subject of the formula.

Divide both sides by 2

L/2 = e^t - 1

Adding 1 to both sides

L/2 + 1 = e^t

e^t = L/2 + 1

Taking natural logarithm of both sides

t = ln|L/2 + 1|

Finally, we put t = ln|L/2 + 1| in the original equation, r(t) we were given to have

r(t) = (L/2 + 1)sin(ln|L/2 + 1|) i + (L/2 + 1)cos(ln|L/2 + 1|) j + √2(L/2 + 1) k

its 50 miles away from the starting point