3:27 12.0 X. 49l ..l D Calculus College 3 of 10 (3) Find the equation of tangent to the curve given by x-d cos'e y-d' sin'e at point - a) Vi(x+ y)-d b) 3x -y- y-i-20 d) (x-y)-3a

Q: (1 )  (a)  Find two positive numbers x and y whose product is maximum and 2x+y=100. ( b )  Find…

A: a. Given The product is given by P=xy Plugging the values: For P to be maximum, the derivative…

Q: 4) Consider the curve given by: cos(xy) = x(y – 7) – 1 a) dy Find y' xp dy dx b) Find an equation of…

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Q: 10. The slope of the tangent line to the curve y(cos x)+e' =5 at the point where x= is (5+In 5) In5…

A: The slope of a  tangent line of a curve at any point will be equal to the slope of the curve at that…

Q: 4. Sketch the graph of the curve y = Vx and the tangent line in the Cartesian Plane using a scale of…

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Q: Find fe F dr for F = 7yi - (sin y)j on the curve counterclockwise around the unit circle C starting…

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Q: Consider the curve described by the relation y + eV= x* - x² + 1. Find an equation of the tangent…

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Q: 1. Find the length of the curve given by e' (sin t+cos t) , yY = e'(cos t – sin t), x = 1<t< 4 .

A: The length of a parametric curve is given by L=∫abx't2+y't2dt Now, we have…

Q: Let C be the graph of the equation In(3y – 2e") = sinh“(Tx) – tan(y – 1). | dy (a) Find dr (b) Find…

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Q: 1. Consider the curve C: r(t) = (t - sin t)i + (1– cos t)j, 0StS 2. %3| | (a) Find an equation of…

A: Since solve the first question for you. If you wanting any specific question to be solved then…

Q: 7. The velocity of an object traveling along a straight line is v(t) = cos(t) + 4t + 1 meters per…

A: Solved below

Q: 3. Consider the parameteric equations for a curve C given by x(t) = cos t(1 – cost), y(t) = sin t(1…

A: Given that: xt=cost1−costyt=sint1−cost

Q: 15. Suppose you start at the point (0, 0, 3) and move 5 units along the curve x = 3 sin t, y = 4t, z…

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Q: Find the point on the ellipse x = 2 cos t, y = sin t, 0 … t … 2pi closest to the point (3/4, 0).…

A: To Find: Point on the ellipse x=2cost, y=sint, 0≤t≤2π, closet to the point 34,0. Concept: Distance…

Q: Find the point on the curve r(t) = (5 sin t)i + (5 cos t)j + 12t k at a distance 26p units along the…

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Q: find y'' for the curve cosx-siny=4

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Q: |3| 7/7 2) Find the arc length of y = ln (sec x) from x = 0 to r 11

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Q: Find T and N for the curve 7(t) = (3 cos(5t), 3 sin(5t), 2t) at the point t = 0 COS Give your…

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Q: re poinit Cur 8X° + 22X 24X e tangent line is parallel to the x-axis? . (1,0) (2,1) (3,0)

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Q: #15) Determine y' a.y=sin 2x b. y = x² sinx co y = sin(-x) Determine #16.> d.) y = cos x sin x e-> y…

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Q: 2 4. Find the length of arc of the curve x3 + y3 = 1 in the first quadrant from the point where x =…

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Q: -1 The tangent line to the graph of y = tanh (4x) is parallel to z + y = 5 at point(s) where

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Q: Q.18 A curve 'C'is defined as x= a cos³0, y = a sin°0 in [O, t/2]. What will be the point P on curve…

A: Explanation of the answer is as follows

Q: The slope of the tangent line to the graph of z- y? = In(zy) at the point (1,1) is Select one: O a.…

A: Given the equation of the curve x2 -y2 =ln(xy) We have to find the slope at (1,1) So, the slope is…

Q: 2. Find the equation of the line tangent to the curve at the point indicated. Express your answer in…

A: As we know that; The slope of the tangent line to the curve at a given point Px0,y0 is equal to the…

Q: The Tschirnhausen cubic is the curve given by the equation x^2=y^3+3y^2 At which point(s)…

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Q: Find the exact length of the curve. y = ln(cos x),   0≤ x ≤ π/3

A: Formula used: L=∫ab1+dydx2dx

Q: 3 4. Find the arc length of the graph of the function f(x) =;(x² + 1)ī over the interval [0, 1]. =…

A: y =f(x) =23(x2+1)32f(x)=23(x2+1)32f'(x)=23*32(x2+1)12(2x)f'(x)=(2x)(x2+1)12 The Formula for arc…

Q: V. The tangent line at the point (t, y) = (- to 4'2 1 Then, the curve y = cos t is y = bt - - - a a…

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Q: 1. Find the length of the curve r(t) = (sin 2t, cos! cos 2t) from t =0 to t 2n. %3D II

A: 1)- Given: r→t=sin 2t, cos 2t from t=0 to t=2π. The objective is to determine the length of the…

Q: (11) Find the tangent line(s) to the parametric curve given by * = t5 - 4t3 y = t2 at (0, 4)

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Q: 5. Express the curve by an equation in x and y. x(t) = 2- sin(t) y(t) = cos(t) a. (x-2)² + y² =1 b.…

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Q: A b. C find an equation of the tangent to the curve at the given point (T/6,1) 4 sin ² x y = + y =…

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Q: 3 6. Find the arc length of f(x) = G (x2 + 2)z on the interval [0,9]

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Q: 1. Find the length of the curve r(t) = (sin 2t, cos 2t) from t 0 to t= 2n. CoS %3D

A: Since you have posted multiple question in this question which are not interlinked so i have solved…

Q: 3. Consider the curve given by r = 2+2 cos (0) where 0<0<2n. Find the slope of the tangent IT line…

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Q: u(0) = 4 and u'(0) = -4. Find the tangent line to the curve cos(Ty) = 5u(x)y² – 21. at the point (0,…

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Q: 8. Consider the curve given implicitly by -x2 + 3xy+2y² = 2 y. 8.1. Verify that (1, –1) lies on the…

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Q: Find the equation of the tangent line to the curve 1 y = 3 sin z + 4 cos a at the point (0, 1/4).…

A: Given:     y = 1 / ( 3 sinx + 4 cosx ) Point: ( 0 , 1/4 )

Q: 2. A curve C with equation y = sin2x 0 < x <5, has a stationary point, P. e3x dy (a) Find an…

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Q: Find T, N and B for the curve F(t) = (2 cos(5t), 2 sin(5t), 3t) at the point t -0 Give your answers…

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Q: Q.2. (3 points) Find the exact length of the curve y In(cos x),0 < x <

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Q: An equation of the tangent line to the curve y = sin(z) at x = 1/6 is y = -(z – 7/6) . An equation…

A: a.

Q: Let C be the graph of the equation In(3y – 2e") = sinh (ma) - tan(y- 1). dy (a) Find da (b) Find an…

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Q: Find the distance traveled by a particle with position (x, y) as tvaries in the given time interval.…

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Q: (3) Find the equation of tangent to the curve given by x=d cos' o y=a' sin'e at point o- VE (x+ y)…

A: Given the equation of curvesx=a3cos3ϕy=a3sin3ϕ

Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: 3) Find the exact length of the curve. y = In(cos r), 0 < x<

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Q: 11. Find an equation of the tangent to the curve y = In(4 + x") at the point %3D 12. i) Find the…

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Q: 1. Find the arc length of the curve f(x) = 3 4 (x + 2)2 on the interval [0, 4]. 3

A: Find the arc length

Q: Q1. Find the length of the of the curve y = In(cos x) from x = 0 to x = . %3D

A: Explanation of the solution is given below...

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.