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

pentagon

step-by-step explanation:

the coordinates   a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.

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