R(t)=e^t sin(t)i+ e^t cos(t) + √(2e^t)k
a) Find the arc length function for the curve measured from the point 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.
I got
r'(t)=e^t (cos(t)+sin(t))I+e^t (cos(t)-sin(t))+ e^t /√(2e^t)
For lr'(t)l= √(5/2) Is this right? And don't know what to do next.
To find the arc length function for the curve measured from the point in the direction of increasing t, we need to integrate the magnitude of the derivative of the curve with respect to t.
The magnitude of the derivative, ||r'(t)||, is given by:
||r'(t)|| = √[(e^t (cos(t)+sin(t)))^2 + (e^t (cos(t)-sin(t)))^2 + (e^t /√(2e^t))^2]
Simplifying this expression:
||r'(t)|| = √[e^(2t) (cos^2(t) + 2cos(t)sin(t) + sin^2(t)) + e^(2t) (cos^2(t) - 2cos(t)sin(t) + sin^2(t)) + 1/2]
||r'(t)|| = √[2e^(2t) + 1/2] = √[4e^(2t) + 1]
Now, to find the arc length function, we integrate ||r'(t)|| with respect to t:
s(t) = ∫[0, t] √[4e^(2u) + 1] du
Note: Instead of integrating from 0 to t, we integrate from 0 to t because we want to reparametrize the curve with respect to arc length starting from P.
Next, to find the point 4 units along the curve (in the direction of increasing t) from P, we need to evaluate the reparametrized curve at the corresponding arc length value.
To do that, we need to solve the following equation:
s = ∫[0, t] √(4e^(2u) + 1) du = 4
Unfortunately, it is not possible to solve this equation analytically. However, you can make use of numerical methods like Newton's method or integrate the equation numerically to find an approximate value of t that satisfies the equation.
Once you find the value of t, substitute it back into the original curve equation to obtain the corresponding point.
To find the arc length function for the curve R(t), you first need to calculate the magnitude of the derivative vector ||dR/dt||, which represents the speed of the curve at each point.
Start by finding the derivative of R(t):
r'(t) = e^t cos(t) i + e^t sin(t) + (1/2) e^t / sqrt(2e^t) k
Next, calculate the magnitude of r'(t):
||r'(t)|| = sqrt((e^t cos(t))^2 + (e^t sin(t))^2 + ((1/2) e^t / sqrt(2e^t))^2)
= sqrt(e^2t cos^2(t) + e^2t sin^2(t) + (1/4)e^2t)
Simplifying further:
||r'(t)|| = sqrt(e^2t (cos^2(t) + sin^2(t)) + (1/4)e^2t)
= sqrt(e^2t + (1/4)e^2t)
= sqrt(5/4)e^t
= (sqrt(5)/2)e^t
Now, the arc length function S(t) can be found by integrating ||r'(t)|| with respect to t:
S(t) = ∫(sqrt(5)/2)e^t dt
= (sqrt(5)/2)∫e^t dt
= (sqrt(5)/2)e^t + C
To reparametrize the curve with respect to arc length starting from point P, you need to find the arc length from the initial point to any given point on the curve.
Since we know the arc length function S(t) = (sqrt(5)/2)e^t + C, we can use it to find the arc length from the initial point to the point P.
The arc length from the initial point to P, denoted as s, is given by:
s = (sqrt(5)/2)e^t + C
To find the value of C, substitute t = 0 into the equation:
s(0) = (sqrt(5)/2)e^0 + C
= (sqrt(5)/2) + C
Since the arc length at t = 0 is 0, we have:
0 = (sqrt(5)/2) + C
C = -sqrt(5)/2
Now we can rewrite the arc length equation as:
s = (sqrt(5)/2)e^t - sqrt(5)/2
To find the point 4 units along the curve in the direction of increasing t from point P, we substitute s = 4 into the arc length equation and solve for the corresponding t value:
4 = (sqrt(5)/2)e^t - sqrt(5)/2
4 + sqrt(5)/2 = (sqrt(5)/2)e^t
(8 + sqrt(5))/sqrt(5) = e^t
ln((8 + sqrt(5))/sqrt(5)) = t
Now that you have the corresponding t value, substitute it back into the original curve equation R(t) to find the corresponding coordinates of the point.