Check the solution to "Find the curvature K of the curve at..." - Plainmath

College
Advanced Math
Upper Level Math

illusiia

Answered question

2020-12-28

Find the curvature K of the curve at the point

r (t) = e^{t} \cos t \cdot i + e^{t} \sin t \cdot j + e^{t} k, P (1, 0, 1)

Answer & Explanation

Bella

Skilled2020-12-29Added 81 answers

r (t) = e^{t} \cos t \cdot i + e^{t} \sin t \cdot j + e^{t} k

Differentiate

r^{'} (t) = e^{t} (\cos t - \sin t) i + e^{t} (\sin t + \cos t) j + e^{t} k

Find Magnitude

| r^{'} (t) | = e^{t} \sqrt{(\cos t - \sin t)^{2} + (\sin t + \cos t)^{2} + 1}

| r^{'} (t) | = e^{t} \sqrt{(\cos^{2} t - \sin^{2} t - 2 \cos t \sin t)^{2} + (\sin^{2} t + \cos^{2} t + 2 \cos t \sin t)^{2} + 1}

| r^{'} (t) | = e^{t} \sqrt{3}

Recall that:

T (t) = \frac{r^{'} (t)}{| r^{'} (t) |}

Therefore

T (t) = \frac{(e^{t} (\cos t - \sin t) i + e^{t} (\sin t + \cos t) j + e^{t} k)}{(e^{t} \sqrt{3})}

T (t) = \frac{1}{\sqrt{3} [(\cos t - \sin t) i + (\sin t + \cos t) j + k]}

Differentiate

T^{'} (t) = \frac{1}{\sqrt{3} [(- \sin t - \cos t) i + (\cos t - \sin t) j + 0 k]}

Find Magnitude

| T^{'} (t) | = \frac{1}{\sqrt{3} \sqrt{(- \sin t - \cos t)^{2} + (\cos t - \sin t)^{2} + 0)}}

| T^{'} (t) | = \frac{1}{\sqrt{3} \sqrt{(\sin^{2} t + \cos^{2} t + 2 \sin t \cos t) + (\cos^{2} t + \sin^{2} t - 2 \cos t \sin t) + 0)}}

| T^{'} (t) | = \frac{\sqrt{2}}{\sqrt{3}}

Recall that

k (t) = \frac{\frac{\sqrt{2}}{\sqrt{3}}}{(e^{t} \sqrt{3})} = \frac{\sqrt{2}}{(3 e^{t})}

Result

k (t) = \frac{\sqrt{2}}{(3 e^{t})}

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Upper Level Math

If two angles of two triangles are congruent to each other, then the third angles will be congruent to each other. Using the above statement, choose which two congruency criteria eventually becomes same.
A)SAS and ASA;
B)AAS and SAS;
C)AAS and ASA;
D) No two congruency criteria are same
Evaluate $6 C_{3}$
Solve the following differential equation dx/dy=-[(4y2+6xy)/(3y2+2x)]
The velocity distribution for laminar flow between parallel plates is given by:
$\frac{u}{u_{max}} = 1 - {(\frac{2 y}{h})}^{2}$
where h is the distance separating the plates and the origin is placed midway between the plates. Consider a flow of water at $15^{\circ} C$ , with $u_{max} = 0.30 \frac{m}{s e c}$ and $h = 0.50 m m$ . Calculate the shear stress on the upper plate and give its direction.
In August 2013, E*TRADE Financial was offering only 0.05% interest on its online checking accounts, with interest reinvested monthly. Find the associated exponential model for the value of a $5000 deposit after "t" years. Assuming that this rate of return continued for 7 years, how much would a deposit of $5000 in August 2013 be worth in August 2020? (Answer to the nearest $1. )
Write an equivalent first-order differential equation and initial condition for y
y=−1+∫x1(t−y(t))dt
Is {8} ∈ {{8}, {8}}?
$f (t) = t e^{- t} \cos h 4 t$

Didn't find what you were looking for?