Show that the length of the perpendicular from the foot of the ordi- nate of any point on the tangent at the same point to the Catenary y = c cosh(x/c) is constant.

Q: Find Equatien of the tavgnt lin tu th cuve at tk giver pointe y= sec X ,

A:

Q: Find the point on the curver(t) = (12 sin t)i - (12 cos t)j + 5t k at a distance 13p units along the…

A:

Q: Find the points on the cardioid r = 1−cosθ at which there is a horizontal tangent line, a vertical…

A: Consider the given: r=1−cosθWrite the cartesian…

Q: Find the tangent to the folium of Descartes  x  3+ y 3= 6xy  at the point (3,3)

A: Consider the given curve: Here the objective is to find the tangent at point p (3, 3).

Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx2 + y?, (6,…

A:

Q: Set up an intergal of the Curve for the length 3T Y%3DZE* sin (x), O4 xE 2

A:

Q: Let f(x) = x³/2. Determine the length of the arc swept over x E la, b).

A:

Q: Let f(x,y)=x^3-cosy. Find the tangent plan for f(x,y) based at (1,pi).

A:

Q: Find the length of the curve over the given interval. r = 4 sin θ, [0, π]

A:

Q: Find the length L of the arc formed by y=e^4x/4,0≤x≤3

A: To find the length L of the arc formed by y=e^4x/4,0≤x≤3

Q: Find the points on (t + sin t, t − 2 sin t) where the tangent is vertical or horizontal

A:

Q: Find the arclength of y = 2x + 5 on 0 < x < 4

A:

Q: Find an equation for the tangent plane to z = cos(x2 + y2) at the point /等,0).

A:

Q: Show that the tangent at P(x,,y,) on the curve -1 Vx+Jy=Ja is yy, ,=a

A:

Q: Find the mass of a thin wire shaped in the form of a helix, r(t) = ( 3 cos t, 3 sin t, 4t) with 0 <…

A: The curve is given by  rt=⟨3cost,3sint,4t⟩ , 0≤t≤3π2                   (i) Then,…

Q: Find the arclength of the graph of the function over the indicated interval. 1 (а) у — 6 -,1 < x <…

A:

Q: gment JK and segment HK are tangents to rcle G. J

A: Given that:

Q: e) With respect to y = cosx, is there a vertical stretch or compression? f) With respect to y =…

A: To answer parts e and f

Q: Find the arclength of r(t)= on the interval from t equals zero tot equals 2 times pi.

A: Given:  r(t)=&lt;5cot(t), 3 sin(t), 4 sin(t)&gt;To find:  Arc length form t=0 to t=2π

Q: Find the speed of the cycloid c(t) = (2t – 2 sin t, 2 – 2 cos t) at points where the tangent line is…

A:

Q: Find the speed of the cycloid c(t) = (4t −4 sin t, 4−4 cos t) at points where the tangent line is…

A:

Q: Find the length of one arch of the cycloid 2(t – sin(t)), y = 2(1 – cos(t). х

A:

Q: Suppose that y = f(x) is a smooth curve on the interval [a, b]. Define and find a formula for the…

A: y=f(x) is a smooth curve on the interval [a, b]. Formula for the arc length L of the curve y = f(x)…

Q: Find the length of the arc of the curve y = In x over the interval: [1, 9| . Round the answer to…

A: This is a question related to derivative of the function. Based on the general formula we will find…

Q: Find the speed of the cycloid c(t) = (4t − 4 sin t, 4 − 4 cost) atpoints where the tangent line is…

A: Given: c(t)=4t-4sin(t), 4-4cos(t)

Q: Find the exact length of the curve. y = In(sec(x)), OsxSa/4

A: Length of a curve f(x) in interval [a,b] is given by ∫ab1+(f'(x))2dx

Q: Find the point on the cycloid c(t) = (t − sin t, 1 − cos t) where the tangent line has slope 1/2.

A:

Q: 28. Show that the graphs of y = tan x and y = cot x have nohorizontal tangents.

A: A horizontal tangent line is a mathematical feature on a graph located where a functions' s…

Q: 4. Use the arc length formula to show that the length of a linear function f(x) = mx +c on the…

A:

Q: Find the point on the curve r(t) = (12 sin t)i - (12 cos t)j + 5t k at a distance 13pai units along…

A: Given,

Q: A satellite (S) in orbit around the Earth is sending two signals that are tangent to Earth at points…

A: Given that , mACB⏜=208° mAB⏜=152° ∠S=?

Q: Find the points on the cardioid r = 4(1 – sin0) at which there is a horizontal tangent line, a…

A: According to the given information, it is required to determine the horizontal tangent line,…

Q: Find the length of one arch of the cycloid = 5(t-sin(t)),y- 5(1-cos(t)).

A:

Q: Find the exact length of the curve. y = In(sec(x)), 0<x< 1/3 In (2 + v 2 ) – - In (1)

A: Since you have posted 2 questions we will solve one for you please resubmit the question. Given:…

Q: Find the point On the curve R(1) = 5sint i+5cost j+12t at adistance (267) units along the curve from…

A: R(t)= 5sint i + 5 cost j + 12t k V=dR/dt =5sint i + 5 cost j + 12t k V= |v|= [(5cost)2 + (-5sint)2 +…

Q: Prove that all normal to the curve x =acid+at sin t, Y=a sin t-atcost.Are at a distance a from the…

A: We need to prove that normal to the given curve are at a distance of a from origin.

Q: ind the length of arc of the curve 6xy = x⁴ + 3, from the point where x = 1 to the point where x =…

A: Here given equation as 6xy=x4+3ory=16x3+3x Differentiate with respect to x…

Q: ind all the tangents to the curve y cos (x %3D

A: Here we have to find the tangents on the curve y=cos(x+y)...........(1)     , -2π≤x≤2π parallel to…

Q: e, Find at the giren point on eauation Of the tangent live. to the corve. X=46",y=f'+|,(4,0)

A:

Q: From the vertex (0, c) of the catenary y = c cosh(x/c) a line L is drawn perpendicular to the…

A:

Q: Find the length of one arch of the cycloid x= 5(t-sin(t)), y= 5(1-cos(t)).

A:

Q: a) In exercise 1.1, describe the erro statement of the tangent ratio.

A: Given : The statement, Figure :

Q: Find the length of the catenary y = a cosh from x = 0 and x = a X1.

A:

Q: What does the arc length formula give for the lengthof the line r(t) = ⟨2t, t, -2t⟩, for 0 ≤ t ≤ 3?

A: Given:          r(t)=⟨2t,t,−2t⟩  for 0≤t≤3.

Q: Find the length of one arch of the cycloid x = 9(t – sin(t), y = 9(1 – cos(t)).

A:

Q: Find the length of the arc of the parabola y sq = x from  ( 0, 0 )  to ( 1,1)

A:

Q: Show that the arc length of the graph of y = sin x on the interval [0, 2π] is equal to the…

A: Given, Equation of Ellipse is x2+2y2=2 Now divide the equation of ellipse by "2" we get,…

Q: Find the length of the arc of the parabola y sq = x cu .  from  ( 0, 0 )  to ( 1,1)

A:

Q: Find the length of one arch of the cycloid x= 6(t-sin(t)) y= 6(1-cos(t))

A:

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.