1. Find the tangent vector to the curve 6t r(t) = [t3 – t, (2t + 1)*] t+1 at t = 1.

Q: Let r(t) = (-5t5 - 3, 4e-4t, 5 sin(2t)) - Find the unit tangent vector T (t) at the point t = 0.…

A: Given vector  r(t)→=<-5t5-3,4e-4t,-5sin(2t)>

Q: a. Find a parametrization of the curve x = (y* – 3)e2y.

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Show that z=tan-1(2xy/x^2-y^2) satisfies the Laplace equation

A: The given equation is, z=tan-1(2xyx2-y2). Obtain the partial derivative of z with respect to x as…

Q: 1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal…

A: TRUE Slope of the curve, when in parametric form = dy/dx = g'(t) / f'(t)

Q: 1. Write the tangential and normal components of acceleration for = (tsint)i + (tcost)j + tk, at t-0

A:

Q: Use Theorem 13.9 to find the directional derivative of the function at P in the direction of the…

A: Find your answer below

Q: Show that the function f(x,y)=ln√x2+y2satistes the Laplace equation.

A:

Q: 1. Let f(x, y) = -x2 – xy³ + 7. Find an equation for the plane tangent to f at (3, 2).

A:

Q: Compute the unit tangent vector to the curve parametrized by r(t)= (t,t^2,t^3) at the point r(1)

A:

Q: Sketch the graph of 7(t) = sin(t)i – 2cos(t)j–tk in 3-space; then find and draw the tangent vector…

A: Given, r→t=sinti^-2costj^-tk^

Q: Let 7(t) = (t + 4, – e*, 3 sin(3t)) - Find the unit tangent vector T (t) at the point t = :0 T(0) =

A: We have to find

Q: The tangent line to the curve r(t) =(t°,t“, t³ ) at the point (1, 1, 1) intersects the æy-plane at a…

A:

Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Consider the given information rt=eti+1-tj ,  t=0   Find the first derivative of rt…

Q: find a parametrization c(t) of the curve satisfying the given condition. y = 3x − 4, c(0) = (2, 2)

A: It is given that,

Q: Find the unit tangent vector . r(t) = 3ti − ln tj to the curve at the specified value of the…

A: given: a curve whose equation is rt=3ti-lntj. we have to find the unit tangent vector at the point…

Q: x-1 y-2 z-1 2 -6 7 1 F(t) = ( 21 + ¹ )i + ¹/ (m² − 3)} j + t√21³-1k at t=1 1 4 (c) show that the…

A:

Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Given: r(t)=(t,cost,sint)t=π2

Q: 2. Let F(t) = (e' cos t) î+ (e' sin t)j+ek. a. Find symmetric equations of the line tangent to the…

A: Given:  F→t = et costi^ +et sint j^ +et k^  To Find:  Symmetric equations of the line tangent to the…

Q: Find the derivative of the vector function r(t) = In(13 – t²)i+ y13+tj– 9e 3 k r'(t) = {

A: The given vector function is                                   r(t)=ln13-t2i+13+tj-9e-3tk

Q: 6. Verify that the function f(x, y) = ln (x² +2²) is a solution of the Laplace equation Zxx+Zzz = 0

A: Topic:- application of derivatives

Q: find a parametrization c(t) of the curve satisfying the given condition.  y = 3x − 4, c(3) = (2, 2)

A: To find the parametrization

Q: Find the work done by F = xyi+yj - yzk over the curve r(t) = ti + t²j + tk, 0≤t≤1

A:

Q: Find the derivative of the vector function r(t) = In(13 – ) i+ v3 +ij+ le- k r' (t) = (

A:

Q: 6. (a) Show that F = (3a²y+2, x3 + 3) is conservative. (b) Evaluate Sc(3x²y +2)dx + (x³ + 3)dy where…

A: According to the problem, we have

Q: Show that the function f(x, y) = e-2y cos 2.r satisfies the Laplace equation.

A: use partial derivative to check whether it  satisfy the equation or not

Q: find a parametrization c(t) of the curve satisfying the given condition. y = x2, c(0) = (3, 9)

A:

Q: Consider the vector-valued function r(t) = (t2− 2)i + sin-1(t)j +1/(1 + t)k Find parametric…

A: Here, the given vector-valued function is: r(t)=(t2-2) i+sin-1(t) j+11+tk To find the parametric…

Q: Find the tangential and normal components of acceleration at the given time t for the space curve…

A: Brief explanation of the answer has been provided in the file attached below. If you have any doubts…

Q: Find the unit tangent vector to the curve defined by F(t) = (5t, 2t, V16 – t) at t = – 1. T(– 1) =

A:

Q: Find the unit tangent vector to the curve defined by =(- 4t, 4t, V49 - t) at t = - 6. T(- 6)

A:

Q: x-1_y-2_z-1 -6 (c) show that the line 2 7 curve F(t) = (21+²) 1+ ²+ (in ²²-3) 1+1√21³²-1k at 1=1 i+…

A:

Q: Let 7(t) = (2t3 + 5, 2e-t, 4 sin(t)) - Find the unit tangent vector T (t) at the point t = 0 T(0) =…

A: Differentiation application

Q: Find the equation of the tangent line to the space curve r(t) = 2cos (t)i + 2 sin(t)j + tk at the…

A: The curve is given by: r(t)=2cos(t)i+2sin(t)j+tk

Q: Find the unit tangent vector to the curve defined by F(t) = ( 5t, V36 – t2) at t = 1. T(1)

A: Given: r→t=-4t,-5t,36-t2 at t=1 We have to find the value of T→1.

Q: Find the unit tangent vector T (t) at th

A: Given that : r(t)→ =- 3t4 - 2 , e5t , - 2 sin (3t)

Q: Find a parametrization of the tangent line to r(t) = (ln(t))i + t^-5j + 5tk at the point t = 1

A:

Q: 3. Find the derivative of the function f(r, y, z) = In (z2 +y + 4z + 5) in the direction of the…

A:

Q: b/ Show that the function satisfy Laplace equation f(x, y) In /9xy+3y2

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: 1. Find the unit tangent vector to r(t) = at t = n.

A:

Q: Consider the curve r (t) = (t, t°, tº); t > 0 Find the derivative vector r' (t)

A: Given : r(t) = ( t , t3 , t6 ) ; t ≥ 0

Q: 1. Find the tangent vector to the curve 6t r(t) = [t³ – t, t+1 , (2t + 1)*] at t = 1.

A: As per our guidelines we are supposed to answer only one question. If you want solution of any…

Q: Find the unit tangent vector to the curve defined by r(t) = /16 – t² ) at t = (- 4t, – 3t, T(- 2) =

A:

Q: Consider the vector-valued function In(2t + 1) sin t R(t) = î+ t t Redefine (t) such that it is…

A: we say the R(t)  is continuous at t=0  if limt→0 R→(t)= R(0)

Q: 10. Show that the function f(x, y, z) = 2z³ – 3 (x² + y²) z satisfies the Laplace equation.

A:

Q: Show that the function satisfy Laplace equation f(x,y) = In /9xy+3y² %3D

A: We need to show that the function f(x, y)=ln9xy+3y2 satisfies the Laplace equation. Now,…

Q: Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3…

A:

Q: Let 7 (t) = (3t3 + 4, – 4ešt, – 5 sin(4t)) - Find the unit tangent vector T (t) at the point t = 0…

A:

Q: Let 7(t) = (t2 – 3, 5e-*, – 3 sin(t)) - 3t Find the unit tangent vector T (t) at the point t = 0

A: Given vector is  r→(t) = t2-3, 5e-3t, -3sin(t) For find the unit tangent vector, First we need to…

Q: 1. Find the tangent vector to the curve 6t r(t) = [t° – t, (2t + 1)*] at t = 1.

A:

Q: 2. Find the parametrization of the tangent line to the space curve r(t) = (-t3, In(t), e² ) at t =…

A: