Question

Reparameterize the curve r(t)= cos(t)i without (t)j (t)k by the arc length.

94

likes
470 views

Answer to a math question Reparameterize the curve r(t)= cos(t)i without (t)j (t)k by the arc length.

Expert avatar
Maude
4.7
105 Answers
Para reparametrizar a curva \mathbf{r}(t) = \cos(t)\mathbf{i} + \sin(t)\mathbf{j} + t\mathbf{k} pelo comprimento de arco, primeiro precisamos calcular o comprimento de arco s(t) da curva.

O comprimento de arco s(t) é dado por:
s(t) = \int_{a}^{t} |\mathbf{r}'(\tau)| \ d\tau

Onde \mathbf{r}'(\tau) é a derivada de \mathbf{r}(t) em relação a \tau, e a é uma constante de integração que podemos escolher. Vamos escolher a = 0 para simplificar o cálculo.

A derivada de \mathbf{r}(t) em relação a t é dada por:
\mathbf{r}'(t) = -\sin(t)\mathbf{i} + \cos(t)\mathbf{j} + \mathbf{k}

Então, podemos calcular |\mathbf{r}'(t)|:
|\mathbf{r}'(t)| = \sqrt{(-\sin(t))^2 + (\cos(t))^2 + 1} = \sqrt{2}

Agora, podemos escrever a expressão para s(t):
s(t) = \int_{0}^{t} \sqrt{2} \ d\tau = \sqrt{2}t

Para reparametrizar a curva pelo comprimento de arco, precisamos inverter a fórmula anterior para resolver em termos de t. Temos:
t = \frac{s}{\sqrt{2}}

Agora, substituímos t novamente na expressão original da curva \mathbf{r}(t):
\mathbf{r}(s) = \cos\left(\frac{s}{\sqrt{2}}\right)\mathbf{i} + \sin\left(\frac{s}{\sqrt{2}}\right)\mathbf{j} + \frac{s}{\sqrt{2}}\mathbf{k}

Portanto, a curva \mathbf{r}(t) = \cos(t)\mathbf{i} + \sin(t)\mathbf{j} + t\mathbf{k} reparametrizada pelo comprimento de arco é:
\mathbf{r}(s) = \cos\left(\frac{s}{\sqrt{2}}\right)\mathbf{i} + \sin\left(\frac{s}{\sqrt{2}}\right)\mathbf{j} + \frac{s}{\sqrt{2}}\mathbf{k}

Resposta: \mathbf{r}(s) = \cos\left(\frac{s}{\sqrt{2}}\right)\mathbf{i} + \sin\left(\frac{s}{\sqrt{2}}\right)\mathbf{j} + \frac{s}{\sqrt{2}}\mathbf{k}

Frequently asked questions (FAQs)
Math question: What is the quotient when 2/5 is divided by 1/3?
+
What is the standard equation of a hyperbola with center (3, -2), vertical transverse axis, vertex (3, -4), and foci at (3, 0) and (3, -8)?
+
Math question: Find the derivative of f(x) = 5x^3 + 2x^2 - 7x + 9.
+
New questions in Mathematics
A circle with a 12-inch diameter is folded in half and then folded in half again. What is the area of the resulting shape?
3(4×-1)-2(×+3)=7(×-1)+2
2/3+5/6×1/2
Find the equation of the line perpendicular to −5𝑥−3𝑦+5=0 passing through the point (0,−2)
The average number of babies born at a hospital is 6 per hour. What is the probability that three babies are born during a particular 1 hour period?
Solve : 15/16 divide 12/8 =x/y
It is known that the content of milk that is actually in a bag distributes normally with an average of 900 grams and variance 25 square grams. Suppose that the cost in pesos of a bag of milk is given by 𝐶(𝑥) = { 3800 𝑠𝑖 𝑥 ≤ 890 4500 𝑠𝑖 𝑥 > 890 Find the expected cost.
Convert 5/9 to a decimal
Use linear approximation to estimate the value of the sine of 31o.
X³-27
Solve equations by equalization method X-8=-2y 2x+y=7
Next%C3%B3n%2C+we+are+given+a+series+of+Tri%C3%A1angles+Right%C3%A1angles+%3Cbr%2F%3Ey+in+each+one+of+them+ are+known+2%28two%29+measurements+of+sides.+%3Cbr%2F%3Elet's+determine+all+trigonom%C3%A9tric+ratios.
cube root of 56
17. A loan for $104259 is taken out for 10 years with an annual interest rate of 9.4%, compounded quarterly. What quarterly payment is required to pay the loan off in 10 years? Enter to the nearest cent (two decimals). Do not use $ signs or commas in the answer.
What is the value of f(-3) for the function X squared+5x-8=
A 20,000 kg school bus is moving at 30 km per hour on a straight road. At that moment, it applies the brakes until it comes to a complete stop after 15 seconds. Calculate the acceleration and the force acting on the body.
00 piece jigsaw puzzle. the completed puzzle is 10x10. each piech connects to at least 2 other pieces. i plan to assemble by taking pieces out of box one by one. if i've already taken out 2 pieces that dont directly connect, what is the minimum number of additional pieces that i need to draw to in order to guarentee that the original 2 pieces connect?
A 20-year old hopes to retire by age 65. To help with future expenses, they invest $6 500 today at an interest rate of 6.4% compounded annually. At age 65, what is the difference between the exact accumulated value and the approximate accumulated value (using the Rule of 72)?
simplify w+[6+(-5)]
Question 3 A square has a perimeter given by the algebraic expression 24x – 16. Write the algebraic expression that represents one of its sides.