At what point do the curves r¡(t) = (t, 1 – t, 3 + t²) and r2(s) = (3 – s, s – 2, s²) intersect? Find their angle of intersection correct to the nearest degree. -

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: In what plane does the curve r(t) = t i + t2 k lie?

A: r(t) = t i + t2 k We need to find the plane of the curve located in?

Q: - 6y Find the tangent plane to the equation z at the point (12,24,4) z =

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Q: The cartesian equation of the curve x = vt ,y= 1– t is : (a)y = 1 – x², x 2 0 (b)y = 1 – x ,x 2 0…

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

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Q: Find the length of the curve. r(t) = ( 8t, t², ), 0 <t< 1 %3D 12

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Q: An insect is moving on the surface of a hot plate, the temperature of which at the point x units to…

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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: utwards from the point (0 nd S, the top half of the

A: Given, the vector field                        E(x, y, z)=x, y, z-3this vector field can be…

Q: Do the curves ri(t) = (t² + 3, t+1, t² – 5) and rž(s) = (4s, 2s – 2, s + 1) intersect? If so, at whi…

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Q: The cartesian equation of the curve x = 1-t,y= t is : (a)y = 1– x², x20 (b)y =1-x ,x 2 0 (c) x = 1 –…

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Q: At what points does the curve  r(t) = ti + (5t − t2)k  intersect the paraboloid  z = x2 + y2

A: We have to find the intersection point

Q: (a) Prove that the lines with parametric equations: x = 1+t, y = 1+ 6t, z = 2t and x = 1+ 2s, y = 5…

A: aLet the L1 be the line:x=1+t, y=1+6t,z=2tand L2 be the line:x=1+2s, y=5+15s, z=-2+6sThe direction…

Q: Consider the curve C shown in the attached figure, which is the intersection between surfaces S, and…

A: To parameterize the curve which is the intersection of the below curves in the first octant.…

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

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Q: Express the curve by an equation in x and y given x(t) = e' and y(t) = 7 – e2'. a) O-x2 + y = 7, x 2…

A: simplify the equation and choose correct equation.

Q: Where does the line x = 1 + t, y = 3−t, z = 2t intersect the cylinder x² + y² = 16?

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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: The line X=t, y+1=t, Z=t intersects the plane. x-2y+Z=2 at the point

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Q: The cartesian equation of the curve x= vt ,y =1-t is : (a)y = 1– x², x 2 0 (b)y =1-x ,x20 (c) x = 1…

A: Eliminate t in given equation.

Q: 3. z = (c

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Q: r(t) = 6ti + 3/2ťj+2¢k, 0<t<1

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Q: where Projuct of in tegra tion SÃr dR A-zi-xj-YK and =dxi +arj +dzK dR and %3D the seguent of the…

A: Use line integral

Q: Find the tangent plane to the equation z 5a2 - 4y +5y at the point (5, 5, - 200)

A: The equation of the tangent plane to a surface  is given by fx(x-a)+fy(y-b)+fz(z-c)=0where (a,b,c) =…

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: Find the length of the curve r(t) = (1,t2, t³). 0<t < 1.

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Q: Find the lengths of the curves in Exercises 1–4 1. y = (1/3)(x² + 2)³/2 from x = 0 to x = 3

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Q: Evaluate [ f (x,y,z) ds given that f(x,y,z) =, and the curve C is ř(t) = 2ti +t²j+÷tk (1<t<∞) %3D…

A: Given: f(x,y,z)=x2xy+6yz r→(t)=2t i→+t2 j→+13t3 k→   1≤t&lt;∞

Q: For curve r = (t, t2,t-³) %3D Does this curve go through the point (-1, 1, -1)?

A: We have to find if the given curve passes through the point (-1,1,-1).

Q: Find the coordinates of the point(s) on the curve 2y = 10 – a² that are closest to the origin. а) О…

A: Given that

Q: An insect is moving on the surface of a hot plate, the temperature of which at the point x units to…

A: Definition used-   To keep the temperature constant, the insect has to move in a direction that is…

Q: Show that the curve a(t) = (t, , ) lies in a plane and find the equation of this plane.

A: For all 't' , we have α'(t) = 1 , -1t2 ,-1 - 1t2α"(t) = 0 , 2t3 , 2t3 Now  α'(t) × α"(t) = 2t3 ,…

Q: Find the point at which the line F (t) = +t intersects the yz plane.

A: Given: An equation of a line in the parametric form is, r→t=&lt;2,-7,7&gt;+t&lt;-5,-6,6&gt; 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…

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Q: The cartesian equation of the curve x= f ,y = 1-t is : (a)y = 1– x², x 20 (b)y =1-x ,x 2 0 (c) x = 1…

A: The given curve is in paramatric form as following- x=√t y=1-t Where t is a parameter. To find the…

Q: (b) Find the points on the line r(t) = (t, 1+2t, 1-2t) whose distance from the plane 2x - 2y+z=1 is…

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Q: Find the length of the curve r(t) (1,t²,), 0sts1 %3D «3 Ost<1

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Q: (a) Prove that curves a, ß: I → R² are congruent if k̟ = Kg and they have the same speed.

A: Part (a):   If any two continuous function defined on the interval I, and the two curves are…

Q: Find the tangent plane to the equation z 4e- 4y at the point (8, 16, 4)

A: Let x0,y0,z0=8,16,4 be a point on the given surface. Then, the equation of tangent is…

Q: Find the angle between the curve 2y = e – x/2 and y – axis

A: Given the curve 2y =e^(-x/2) and y-axis We have find angle between curve and y-axis

Q: The length of the curve y = r from r = 0 to r= 2 is A. V1+1®dr B. V1+ 3r² dr C. V1+*dr D. V1+ 9r* dr

A: Here I am attaching image so that you understand each and every step.

Q: The cartesian equation of the curve x = 1-t,y= vt is : (a)y = 1- x2, x 2 0 (b)y = 1 - x ,x 2 0 (c) x…

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Q: The point on the curve x? + 2y = 0 that is nearest to the point (0,-;) occurs where y'is A) 0 B)…

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Q: The line 2x + 5y = 1 meets the curve x² + 5xy - 4y² + 10 = 0 at the points A and B. Find the…

A: The midpoint of the line joining the points: x1, y1 and x2, y2 is x1+x22, y1+y22. The roots of the…

Q: Let Q = 4x2 + 6xy+ 4y? . The points on the curve Q = 7 which are closest to the origin are: %3D %3D…

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Q: a) Prove that the lines with parametric equations: x = 1+ t, y = 1 + 6t, z = 2t and x = 1+ 2s, y = 5…

A: Given that: Line L1:  x=1+t, y=1+6t, z=2tLine L2:  x=1+2s, y=5+15s, z=-2+6s

Q: At which points does the curve r(t)=ti+(2t -t² )k intersect the paraboloid z = x² +y? ?

A: Given: r→t=ti^+2t-t2k^ and z=x2+y2

Q: An insect is moving on the surface of a hot plate, the temperature of which at the point x units to…

A:

Q: At what point do the curves r;(t) = (t, 4 – t, 24 + t2) and r2(s) = (6 – s, s – 2, s2) intersect?…

A: The curves are r1(t)=t, 4-t, 24+t2 and r2(s)=6-s, s-2, s2

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.