3. If the position function of a particle is x(t)=t^3+8t^2-2t+4, t>0 Find when the particle is changing direction

Q: 3. (4x³y5 sin 3x + 3x*y$ cos 3x) dx + 5x*y* sin 3x dy = 0 y(0) = 1

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Q: 1. Find the orthogonal trajectories of the family of curves x? + y2 = cy.

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Q: Find an equation of the line tangent to f(x) =3 at x 3. %3D OA. y=xln 3+ (1– 3ln 3) OB. y=27(xln 3 +…

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Q: The linear approximation at æ = 0 to sin(2z) is A + Bx where A is:| Previev and where B is: |…

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Q: Question 1 Find the total differential of the function z = 7x³y8 Odz = 7x¹y³dx + 7x³y+dy dz =…

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Q: 7-The tangent of the curve x^2+y^2- 2xy3D0 at the point (1,1), is * O a. y=2-x. O b. y=x-2. C.…

A: We find equation of tangent to the curve

Q: Find the orthogonal trajectories to the family of curves: x² = C-v'(ln y² –1).

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Q: usk 2: Find the derivative of 4x2 + 6y3 = 50 by filling the blanks. %3D (2 xD(x-)+x 6)(y³4)_=__ dy >…

A: We have, 4x2+6y3=50            ⋯⋯1 We know that, from basic derivative formula ddxkxn=n·kxn-1ddxk=0…

Q: i) Find the orthogonal trajectories of the family of curves: y : 1+ x?

A: By Bartleby policy I have to solve only first one as these 2 are unrelated problems

Q: y = x – 2 + Ce* X-

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Q: Find the error propagated from the function y = 3x* if an error of 0.06 occurs in the independent…

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Q: find particular solution of (16x+5y)dx+(3x+y)dy=0. If the curve to pass thru the point (1,-3)

A: Given: 16x+5ydx=-3x+ydydydx=-16x+5y3x+y               It is a homogenous differential equation.…

Q: 7. what is the durivative oF q= arc cSc V1-3X %3D 3 Cz -Gx)I- 7 X 3. C2-6x)1-3x (-3x)-3x 3 C1- 3x)-3

A: First make it in simple trigonometric functions then diffrenciate it and just put all the given…

Q: Is ƒ(x, y) = x2 - xy + y2 - 3 more sensitive to changes in x or to changes in y when it is near the…

A: Partial derivative of f w.r.t a variable is the indicator of the sensitivity of f towards that…

Q: 25. Find x, y, eR from 2coshy - 7sinhx = 3 and coshy - 3 sinh² = 2 %3D

A: The given system of equation: 2cosh(y)-7sinh(x)=3                    ......  (1)cosh(y)-3sinh(x)=2…

Q: . Find P.S. of (16x +5y) dx + (3x+y) dy = 0 of the curre to pass thru the point (1,-3) 2. Find P.S.…

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Q: (6) Find the orthogonal trajectories of the family of curves r = 3C sin 0, where C being parameter.

A: Orthogonal trajectory curve of a family of curve is defined as a curve which intersects every member…

Q: 1 A point is moving along the graph of the functiony = dy 9x2 + 6 such that dt = 3 centimeters per…

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Q: fCx)=(1+x)* at X=0 is LUX)=1+ KX + "७) %3

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Q: Find the orthogonal trajectories of the one parameter family of curves ex+e-y=c

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Q: Compute the differential dy  for  y=(6x+sinx)^3

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Q: The derivative of y = 3cos(8x+6) is: 1. y' = 24 sin (8x+6) 2. y' = -24 sin (8x+6) 3. y' = -24 cos…

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Q: 2 35) Find a curve through the point (-4, 1) whose length integral, 1 s y s 2, is L = 1 +

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Q: 16. Let y = 4x² + 7x + 4. If Ax = 0.2 at x = 2, use linear approximation to estimate 4y Ay 2

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Q: A. = 2x - y + sin2t dt = 4x + 2y + 2cos2t (variation parameter). dt

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Q: 11. Find the length of the curve x = on the interval (2, 4] (integrate with respect to y) 12y3 un

A: Given that the function x =y55+112y3 Find the arc length of the graph of the given curve on the…

Q: Find the orthogonal trajectories for the family of curves. (Enter your solution in the form F(x, y)…

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Q: Compute the differential dz of the function. 3,,10 x°Y %3| dz

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Q: The

A: Below is sketch of  the given function.    Rotation about x axis will give a volume in 3D plane.

Q: 1. Find the directional derivative of the scalar function f = yzx^2 + 4xz^2 at the point P(1,-2,-1)…

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Q: (5) Find the curve y = f(x) in the xy-plane that passes through the point (9, 4) and whose slope at…

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Q: (4) Find the orthogonal trajectories of the family of curves z +y 2C?, where C being %3D a…

A: Note:-    as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: The derivative of y= sech1(Inx)+esinh(3x) equals Select one: 1 3e sinh"(3x) O y' = - xInx/1-Inx…

A: Proceed as shown below...

Q: 2. Find the orthogonal trajectories of the family of curves x2 + y² = 2my where m is a constant.

A: As given equation of curve is: x2+y2=2my Differentiate it with respect to x:…

Q: A heat-seeking particle is located at the point (2, 3) on a flat metal plate whose temperature at a…

A: A heat-seeking particle is located at the point 2,3 on a flat metal plate whose temperature at a…

Q: 6. Determine a, b, c and d so that the curve y = ax3 + bx2 + cx + d so that the curve will have a…

A: Note: Hi and thanks for your question! Unfortunately, as per our guideline, only one question should…

Q: 3. (2х - у - 4) dx - (х-3)dy -0 when x %3 1, y 3D 4.

A: We try to identify and group terms and integrate.

Q: Find the equation of the line tangent to f(x) = cos² x – sin²| %3D at x = 6

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Q: Evaluate the integrals

A: Given integral is:

Q: 2. Find two positive numbers whose product is 750 and for which the sum of one and 10 times the…

A: Product of two numbers:= 750; Sum of one and 10times the other is minimum;

Q: 2. Suppose that a particle is moving along the curve 4x2 + 16y? = 32. When it reaches the point (2,…

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Q: Find the linearization of ƒ(x) = 2x at x = 0. Then round its coefficients to two decimal places.

A: Given function is,

Q: End fz(3) in terms of the joine pdf of X and Y if

A: From the given information, Let, f(x) be the density function of X. f(y) be the density function of…

Q: (6) Find the orthogonal trajectories of the family of curves r= 2C sin 0, where C being parameter.

A: See the detailed solution below.

Q: Make the curve y = ax4 + bx³ + cx² + dx + e pass through the point (1, -8) be tangent to the line y=…

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Q: Once again, a particle travels at velocity t – 1 at time t. Divide the interval 0, 2 into 2k…

A: A particle travels at the velocity t-1 at time t.  We now divide the interval 0, 2 into 2k intervals…

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: