A space curve is given by the vector function r(t) = (3t, t² + 6t, t³ ). Then the unit tangent vector at t = O equals

Q: 1. Use the parameter r =t to find a vector-valued function for the space curve represented by the…

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Q: Let r(t) = t³ i + t² j + k be a space curve. Which of the following vectors is perpendicular to the…

A: The given curve is: r(t)=t3,t2,1tr'(t)=3t2,2t2,-1t2 Now, the given point is r(1). So, r'(1)=3,2,-1…

Q: Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The…

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Q: Let C be the curve of intersection of the paraboloid a² + y² = z and the cylinder +(y-1)² = 1 (a)…

A: Given that C is the curve of intersection of the paraboloid x2 + y2 = z and the cylinder x24 + y -…

Q: Sketch the space curve represented by the vectorvalued function                  r(t) = ti + (2t −…

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Q: Given the space curve r(t) = (t, 2e", e²²), find the tangential and normal components of…

A: The tangential component of acceleration can be found by using the formula r'(t).r''(t)r'(t) The…

Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder + (y − 1)² = 1 (a)…

A: We will use the basic knowledge of vector calculus and differential calculus to answer all three…

Q: The length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the…

A: Given information: Parametric equation of curve is : x=x(t)y=y(t)z=z(t) We need to find the length…

Q: Find the unit tangent vector for the vector function : r(t) = (t2 + 1,3 t, t3> %3D (1+4t2+9t)3…

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Q: Use the parameter x = t to find a vector-valued function for the space curve represented by the…

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Q: If is a smooth curve given by a vector function r(t) a < t < b and is a constant vector, show…

A: So we have;

Q: Let r(t) = sin t i + et j + In (t + 1) k %3D be a space curve. Which of the following vectors is…

A: See the attachment.

Q: Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the…

A: Given the curve     rt=2sin(t),2cos(t),4sin2(t) and the point P1,3,1.

Q: Let R be a vector function whose graph is smooth curve such that, R(2) =, R'(2) = , and R"(2) =…

A: Radius of curvature of teh function is given by formula, R = (1+y'(x)2)32y''(x)

Q: If Ū is a unit vector and Di f(x, y) = 0 for all points (x, y) then f is a constant function..

A: We have to determine whether the following statement is true or false : If U→ is a unit vector  and…

Q: Find an equation of the tangent plane to the surface represented by the vector-valued function at…

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Q: Find the unit tangent vector of the given curve. r(t) = (7- 2t)i + (2t - 3)j + (6 + t)k 2 1 O A.…

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Q: Let C be the curve of intersection of the paraboloid x? + y² = z and the cylinder + (y – 1)² = 1. 4…

A: We will find equation of common curve and parametrize it to find vector equation of C and the use…

Q: Find a vector function that represents the curve of intersection of the two surfaces. The paraboloid…

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Q: Match the vector-valued functions (a)–(f) with the space curves (1)–(vi) in Figure 10. (a) r(t) = (t…

A: (a) Identify the graph of the vector function rt=t+15, e0.08tcos t, e0.08t sin t. From the equations…

Q: Consider the vector function      r(t) =〈t^2 - t, t^2 + t, t^3〉 Find equation of three different…

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Q: Find a vector-valued function, using the given parameter, to represent the intersection of the…

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Q: Sketch the space curve. Vector-Valued Function r(t) = (5 sin t, 2t, 5 cos t) Interval [0, π]

A: Given rt=5sint,2t,5cost So, r't=5cost,2,-5sint So,…

Q: 6. Given the space curve r(t) = (t, 2et, e2t), find the tangential and normal components of…

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Q: Find the vector r' (to); then sketch the graph of r (t) in 2-space and draw the tangent vector r'…

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Q: Let f(r.y) = r'y- Find a unit vector tangent to the level curve f(r, y) = 3 at its point (1,2).

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Q: Find the Tangent vector, the Normal vector, and the Binormal vector (T, N and B) for the curve r(t)…

A: This question can be solved using the concept of vectors and derivatives.

Q: Consider the curve C in R3 whose parameterization is given by: eq. in image If is there a point on…

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Q: Give the length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in…

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

Q: Match the space curves in the figure below with the following vector-valued functions: O A. r(t) =…

A: See that options ,  A )  r ( t ) = < 1 , t , t >  If we consider the first two coordinates ,…

Q: Show that if r(t) is perpendicular to the tangent vector r′(t) for all t, then the curve r(t) lies…

A: If r(t) is perpendicular to the tangent vector r′(t) for all t, then the curve r(t) lies on a sphere…

Q: Find the value of a at which the plane curve traced out by the vector function E + bt> is not…

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Q: Find a vector function, r(t), that represents the curve of intersection of the The cylinder x² + y2…

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Q: Let C be the curve of intersection of the surfaces with equations   x2 + y  =  5   and   24x + 4y…

A: Given: Let C be the curve of intersection of the surfaces with equations x2+y=5  and   24x + 4y + z…

Q: Suppose r(t) is a vector function in R3 Show that if r(t) has a constant speed, then r′(t) is…

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Q: Given r(t) be the vector function that describes the curve of intersection between the cylinder…

A: First, parametrize the cylinder equation to get x(t) and y(t) and then find find z(t) from the…

Q: Match the vector-valued functions (a)–(f) with the space curves (1)–(vi) in Figure 10. (a) r(t) = (t…

A: (a) r(t)=t+15, e0.08tcos t, e0.08tsin t From the function we have x=t+15, y=e0.08tcos t and…

Q: Let r(t) = (x(t), y(t), z(t)) be a space curve that lies entirely on the surface of the sphere 1.…

A: The space curve is rt=xt, yt, zt. We know, rt2=x2t+y2t+z2t On comparing with the sphere equation,…

Q: Let r(t ) = ti + cost j + t? k be a space curve. Which of the following vectors is parallel to the…

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Q: Find a vector-valued function whose graph is the indicated surface. The plane z = 4y r(u, v) =

A: Solution ) the plane      z = 4y   Let's make substitution  x =u , y = v  Hence  z = 4v   r ( u, v)…

Q: Given a curve C: r (t), find a tangent vector r (f), a unit tangent vector u'(t), and the tangent of…

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

Q: Compute the unit binormal vector B for the following parameterized curve: F(t) 4 2 3t, 1

A: Given:    r(t) = ⟨ t2/2 , 4-3t , 1 ⟩

Q: Let r (t) = ti + e2t j + t³ k be a space curve. Which of the following vectors is perpendicular to…

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Q: Let r(t ) = t³i + t? j + k be a space curve. Which of the following vectors is perpendicular to the…

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Q: Find a vector-valued function whose graph is the indicated surface: the part of the paraboloid z =…

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Q: A smooth curve C is defined by some vector function R(t) with R (7) 2 = = (π,0,−2) and R' (t) = (2,…

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Q: Consider the vector function r: R → R³ given by r(t) = (e – 1, t, 3t + 1), and let C be the…

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Q: Show that the vector-valued function r(t) = e−t cos ti + e−t sin tj + e−t k  lies on the cone z2 =…

A: Given vector valued function is r(t)=e-tcosti+e-tsintj+e-tk.

Q: Let r (t ) = t i + e2t j + t³ k be a space curve. Which of the following vectors is perpendicular to…

A: Solve the following