Show that the following set is a curve: K = {(x, y) E R² | xy – x = 1}. Briefly explain your reasoning.

Q: (c) Sketch some typical members of the family of curves y = ax? + b.

A: Given the curves y=ax2+b is parabola  For a=1, b=0 y=x2

Q: Evaluate |F. dR, where F(x,y, z) = (x, – 2, y) and C' is the curve given by R(t) = (2t, 3t, –t2), 0…

A:

Q: Among the following curves, select ALL that is simple closed. fine's a. b. с. my

A: Closed curve: A closed curve does not have end points. Simple closed curve: A closed curve which…

Q: (b) Investigate where the curve y = xy4 - x² is concave up and concave down?

A: Given curve is y=x4-x2 find the concavity of the given curve

Q: Find T, N, and B for the curve r(t) = (10t, 10t, e') at (0,0, 1).

A: Given: we have ,

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…

A:

Q: Let C be the curve y = 8/a for 2.1 < x < 3.8.

A:

Q: Find T,N and k for the curve:

A:

Q: Fit a curve of the form y = aebx to the following data. Hence find f (8). 1 2 3 4 5 3 15 24

A: The curve here has to fit is, y=aebx Now note that the above equation can be written as,log y=log(…

Q: 13.3. Let y be the curve given by z(t) = t + it?, 0<t < 2T. Evaluate |z|²dz.

A:

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

A:

Q: Let C be the curve in space given by the following parameterization. Then d + 2dy +adz is equal to…

A: Given ∫xdx+2dy+zdz = ∫(xi+2j+zk).(dx i + dy j + dz k) = ∫F.dr Here F = (xi+2j+zk)…

Q: Evaluate [(x +2y) ds, where C is the curve defined by y = √√4 – æ², for x = [0, 1

A: Here ds is arc length.

Q: Find the exact length of the curve. x = et + e-t, y = 5 – 2t, 0 st< 4

A: Given curve: x=et+e-t, y=5-2t We have to find exact length of the curve at 0≤t≤4. Length of the…

Q: / Fit a curve of the form y = ax + bx² to the data

A: Given,    x 1 2 3 4 5 y 1.8 5.1 8.9 14.1 19.8 To find             y=ax+bx2…

Q: Given that the tangent and normal to a curve at point P(x, y) meet the x-axis in A and B…

A: It meets the x axis in A and B respectively and y axis in D and E respectively.To Find: Family of…

Q: If a curve C is given by r(t) = ⟨t, t2⟩ , what is | r'(t) | ?

A:

Q: Sketch the two curves 7(t) = tî + t²j; t e [1, 5] and B(T) = e"î + e²r3; T E [0, In 5]. How are they…

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Q: a) Given a curve y = px* + 2x has turning point at (–1,q). Find the value of p and q.

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Q: I Discuss the properties of the curve and sketch fully: a) y = x? e-x

A: (a) Given- y=x2e-x To Find- The properties of the curve and sketch the curve.

Q: Use the guidelines of this section to sketch the curve.

A:

Q: Find the length of the curve given by S(x) = (a -) defined on the interval (1, 2].

A: Consider the given curve:

Q: At how many points do the curves r₁ (t) = ((t+1)², -t2- 2t, t+2) and r₂ (t) = (1 − t², t, t)…

A: This is the problem of parametric equations and the theory of curves.

Q: Is the given statement true or false? Justify your answer.   If f''(c)=0, then  (c,f(c)) is an…

A:

Q: Find the length of the curve. x=(y - 1) 3/2 from y = 1 to y = 4 O 7 O 10 21 2 O 1

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Q: Find the length of the curve defined by from = 9 to z = 10. 2 y-4lm (()²-1) =

A: find derivative

Q: OShow that every point (b, c) in the plane lies on a tangent line to the graph of y = x³.

A: Consider the given information. y=x3 Find the derivative of the function. dydx=ddxx3y'=3x2

Q: Evaluate ∫с (x+y^1/2) ds, curve is given by figure 2

A:

Q: Sketch a curve with the following properties. 0, for x> 12

A:

Q: Evaluate (2 - xy) ds where C is the curve given below. -2 -1 -2

A: Given   ∫C2-xyds Curve C

Q: Trace the curve r = sinocos'o. 2)

A: Given: r=12sin3θcos3θ Explanation: r=12sinθcos3θat θ=0r=12sin0cos30=0⇒r=0at…

Q: The length of the curve defined by x =(v² + 2)E, y € [0,1]. Select one: none O 12

A: Arc length of the curve x=g(y), y∈[a, b] is, S=∫ab1+(g'(y))2 dy

Q: Explain why a cubic Bézier curve is completely determined by x(0), x (0), х(1), and x'(1).

A:

Q: Let C2 be the curve defined by R(t) = (t² +t + 3, 4t – 5, 2+ t – t2), t e [0, 1]. %3D | (x + y+ 2)…

A:

Q: Let C be the curve y = 2/r for 1.3 <x < 3.2.

A: Given that: y=2x

Q: Consider the implicitly defined curve xy? +3x2 = 4. %3D Verify that the point (1, 1) is on the…

A: Consider the following curve: xy2+3x2=4 Substitute x=1 and y=1 in the above curve:…

Q: find and sketch the point of shortest (or longest) distance from the origin of curve…

A: We need to find and sketch the point of the shortest (or longest) distance from the origin of curve…

Q: The point of inflexion of the curve r = b0" are given by {a} r = b[¬n(n+1)]"¹² (b) r= b[n(n+1) ² (c)…

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Q: Which of the following is not true about the curve y x

A:

Q: Is the given statement true or false? Justify your answer. "If f"(c) = 0, then (c, f(c)) is an…

A: The given statement is " If f"c=0, then c,fc is an inflection point of the curve y=fx.

Q: )مه و و ي (x2 + y?, y* + 4x) · dr (2,1) )هد0( )ه,(

A:

Q: Find the length of the curve y = a +; from a = 0 to x = 1. O None 5/2 O6/2 3 O4/2 3. 3

A:

Q: Explain how to find the length of the curve                         r(t) = ⟨ƒ(t), g(t), h(t)⟩, for a…

A: To Determine: Explain how to find the length of the curve Given: we have rt=&lt;f(t),g(t),h(t)&gt;…

Q: IFind the exact length of the curve x= 3t -t, y=3t² ,0<t<2 ||

A: We have to find the exact length of the curve. The length of the curve is given by

Q: a) y = x? e-X

A: We have to discuss properties of the graph

Q: Draw a curve y = f(x) for which f' and f" have signs as indicated in Figure 2. FIGURE 2

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Q: Which of the following curves pass through the point (1, 4)? (a) c(t) = (t2, t + 3) (b) c(t) = (t2,…

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Q: analyze and trace the given curve y = x / (1 - x)2

A: To solve the following problem

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.