Consider the path r(t) = (4t, 2t², 2 In t) defined for t > 0. Find the length of the curve between the points (4, 2, 0) and (8, 8, 2 lIn(2).

Q: ) Find the orthogonal trajectory of the family of curve given by y = In(r³ + b)

A: First formulate a differential equation to eliminate the parameter b . Then replace (dy/dx) by…

Q: Sketch the plane curve r(t) = 2ti − 3tj,  and find its length over the given interval [0, 5]

A: Given plane curve: The parametric form of vector valued function is It is the equation of line in…

Q: Let C C R? be the path from (3, –2) to (3, 2) along the graph of the curve x – 5x?y? – 7x2 = 0.…

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Q: (b) Find a , b and c if the curvey = x + a x + b and y =c x + x have tangent lines paralle to each…

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Q: 5. Prove that whole length of the curve x3 + y2/3 = a/3 is 6a.

A: Given that x23+y23=a23 To find the length of the curve:…

Q: Sketch the plane curve r(t) = 3ti − tj and find its length over the given interval [0, 3] .

A:

Q: Consider the path r(t) = (4t,2t 2 ,2lnt) defined for t > 0. Find the length of the curve between the…

A: Given the path r(t) = (4t, 2t2, 2ln t), t>0. Find the length of the curve (4, 2, 0) and (20, 50,…

Q: (B) The length of the curve r(t)=(1+1)i+(1-1j+r/2k from the point t=- to the point r= is 2. -IN

A: The length of the curve means the arc of the length of the curve . The arc of the length of a curve…

Q: . Find the length of the curve defined by R(t) = (3t+1, 4t3/2, 4-3+2) from the point (1,0, 4) to…

A: Note : The length L of a curve r(t)→=(x(t) , y(t)  ,z(t) )  from t=a  , t=b is given by…

Q: Sketch the plane curve r(t) = t2i − 4tj and find its length over the given interval [0, 5] .

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Q: Find the exact length of the curve. = 플Vy (y-3), 4sys 16 3

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Q: Q4) Compute f z'dz, where C is the path formed by the line segment from -3 to 3 and the semicircle…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: Find the length of the curve given by ř () = (6t, 12/T, 3Int) for 1<tse. O 8e-4 O 2e-1 O 4e - 2 O…

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Q: Sketch the plane curve. r(t) = 2ti – tj, [0, 2] y y 2. 1 1 -1 1 4 -1 1 4 5 -1 y -1 2 3 4 1 2 3 4 5…

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Q: s Compute yz dz + zzdy + zydz along the curve (t, t2, t³), 0 st<1

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Q: Find the exact length af te of the curve, 2. x-

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Q: Consider the curve r (t) = (t, tº, tº) ; t > 0 Does the point (1, 1, 1) belong to the curve and if…

A: Please refer the attached image for complete solution.

Q: 2) Consider the oriented curve C given by C = C1uC2, where C2 is obtained by intersecting the…

A: We will first parametrize the curves C1 &amp; C2 then substitute the x,y,z values into the line…

Q: 7. Given that the line y = r -b is an asymptote to the ar curve y a,b €R,rz 1, then b = A -1-a 1-a C…

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Q: Prove that the curve y: (-,-)→ R³ is regular with factor H and determine whether the image y (t) =…

A: If γ't not equal to 0 for all t∈-π2,π2then it is regular. If γ''t=kγ't for all t∈-π2,π2 then the…

Q: Find the orthogonal trajectory of the family of curves x² + y² Cx

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Q: 19. Evaluate Scx ds, where C is a. the straight-line segment x = t, y = t/2, from (0, 0) to (4, 2).…

A: Part (a)

Q: compute the length of the curve over the given interval.  r(t) = (2t2 + 1, 2t2 − 1, t3), 0 ≤ t ≤ 2

A: Length of a curve is also like computing distance between two points but with the restriction that…

Q: If lx + my = 1 touches the curve (ax)" + (by)" = 1, show that %3D )+) = 1.

A:

Q: Sketch the plane curve r(t) = ti + t2j and find its length over the given interval [0, 4] .

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Find the orthogonal trajectory to the family of curves: y(x²+1) = Cx

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Q: Compute the length of the curve given parametrically by x=1/3t^3 and y=1/2t^2 for 0 less or equal to…

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Q: Verify that the following pairs of curves meet orthogonally x = 1 - y2, x = 1/3y2

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Q: Compute the length of the curve r(t)= (-5t, 4t – 3, – 5t + 5) over the interval 0 <t< 7

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

Q: Evaluate ∫C x ds, where C is the parabolic curve x = t, y = t2, from (0, 0) to (2, 4).

A: Given: Parabolic curve x=t, y=t2 from (0,0) to (2,4)

Q: Mike found the orthogonal line to the curve f (x) =- 3sin( zx) +1 at x=-1. In which step did he…

A: The objective is to state the step in which he made an error while calculating orthogonal line to…

Q: Find the equation of the plane tangent to the grapl UI z = f(x, y) = x² + 2y³. + 1 at the points in…

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Q: Find the length of the curve y = V- from z =1 to r = e. 3

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Q: Let C be the curve y = x³ – 3x2 + 4x – 1 joining the points (1, 1) and (2, 3). Find the value of…

A: We need to find value of integral.

Q: Find the exact length of the curve. y = In(1 - x²), o s x s

A: Solve the problem using definite integral

Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: Consider the path r(t) (14t, 7t, 7 In t) defined for t > 0. Find the length of the curve between the…

A: Given - r(t) = ( 14t, 7t2, 7lnt )  defined for t&gt;0 To find - The length of the curve between the…

Q: (b) Verify that Green's Theorem is true for S,3(y² – x) dx + (x² – y)dy given that the curve C is…

A:

Q: C is the curve y = e® from (2, e²) to (0,1) and F = (x², -y), compute ſ, F-dr.

A: Given that, C is the curve y=ex from 2,e2 to 0,1 and F=x2,-y Then, F=x2,-y=x2,-ex So, F→=x2i^-exj^

Q: Show that the curve In (2X-1) , n> i has no stationary point

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Q: Find the exact length of the loop of the curve = 3t – t°, y = 3t2 -

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Q: Find the length of the curve defined by R(t) = (3t+1, 4t³/2, 4 - 3t²) from the point (1,0, 4) to (4,…

A: Given…

Q: Draw the isoclines with their direction markers and sketch several solution curves, including the…

A: According to the given information, it is required to draw the isoclines with direction makers and…

Q: Consider the intersection curve C in the first octant, between the surfaces s1: 2 = 1- r? S2 : 2 +…

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Q: Find the family of curves orthogonal to the family 4y2 + 3x2 = k when x > 0 and y > 0. (A) 3y2 – 4x2…

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Q: ind the orthogonal trajectory of the curve y? = ax²(1- cx); with a held fixed.

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Q: Find the diffrential equation of for each of the curves determined by the condition that at each…

A: length of subtangent =ydydx

Q: Let C be the curve connecting (0,0) to (9,3) to (-3,3) to (0,0) using straight lines. Evaluate (sec…

A:

Q: Sketch the plane curve r(t) = t2i + 2tk, and find its length over the given interval [0, 3]

A: We have, r(t)=t2i+2tk  we have to determine the length of the curve over the interval [0,3]

Q: Use Green's Theorem to find work done by F = 2°yi + (y – 3)3 along the curve c, C is the closed path…

A: We will find out the required solution.

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.