What are the critical points for the plane curve defined by the equations æ(t) = sint + cos(t), y(t) = sint – cos(t), and 0

Q: 3. Evaluate the given line integral. Follow the direction of C as given in the problem statement. 2.…

A: Line integral is integral of some function along a curve or path. Any field for which the line…

Q: 7. Find a single initial vahue problem that admits the family of solutions y = e and then find an…

A:

Q: 1. Find the orthogonal trajectories of the given family of curves: a. y = Ce-mx; with m held fixed

A:

Q: 3. Determine the coefficients a, b, c, d that will make the curve y = ax³ + bx? + cx +d have…

A: We will find the value of a, b, c, and d as following .

Q: 4. Let f(x, y) = 2TY x = In 2 g and y = In s. . a. Find the equation of the tangent plane to f(x, y)…

A: Let f(x,y)= 2xy , x=t/ln2 and y= lns Here we have to find the equation of the tangent plane to…

Q: 12 sin°(z) dr: C is any closed curve -dz enclosing 7/6 (a) Ic 6z – T -

A:

Q: 6. Set up but do NOT evaluate Evaluate the line integral 2 curve (t) - (t", t), where 0sts0. L ds,…

A: Consider the given integral as ∫Cyxds, where C is the curve r⇀t=t4,t3 and 0≤t≤1. Note that, the line…

Q: 2. Determine the coefficients a, b, c, and d so that the curve will satisfy the stipulated…

A: y=ax3+bx2+cx+d-1Critical points 1,0-2,27We know that at critical point dydx=0 or does not exist…

Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A:

Q: Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many…

A: Consider the given path rt = cost , cos2t , sint  Then, xt=cost , yt=cos2t , zt=sint

Q: 1. Give an equation of the tangent plane to the surface z = In(4y – x) at the point (3,1,0). 2. Use…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: directional derivative

A:

Q: Let z(t) = 5 cos (t) and y(t) = 3 sin(t) ELLIPSE 2- At t = 3, x(3) = -495 -1- y(3) = 0.42 -2- dz…

A:

Q: What are the control points of the splines Bezier curve from the two equations given below: a(x) =…

A: Given, ax=1+5x2+2x3...1bx=1-x+x3...2 The control points are lie on both the curves. So, to get…

Q: 2. Let F(t) = (e' cos t) î+ (e' sin t)j+ek. a. Find symmetric equations of the line tangent to the…

A: Given:  F→t = et costi^ +et sint j^ +et k^  To Find:  Symmetric equations of the line tangent to the…

Q: Express the curve by an equation in x and y given x(t) = sin(t) and y(t) = 8 + cos² (t). a) O-x² +y…

A:

Q: Compute I = (2x²132²y+ 4xy)ax +(x'+2x²-y') dy+zdz along The curve C C4U Cz UG given in the figure…

A: Given To compute I=∫C2x3+3x2y+4xy.dx+x3+2x2-y2dy+zdz along the curve C=C1∪C2∪C3 given in the figure.

Q: 6. At time t, a particle moving in the xy – plane is at position (x(t), y(t)) where the x(t)and y(t)…

A:

Q: 3. Find the family of oblique trajectories that intersect the family of the paraboles y =cx at angle…

A: given family of parabolas y2=cx at angle θ=60° claim- find the family of oblique trajectories that…

Q: (6) Find the orthogonal trajectories of the family of curves y = ax? + 4ax + 4a, where a being…

A:

Q: Consider curve C at R parameterized by: r(t) = cos(2t) î + tf+2 sin(2t)k, with 0 stsT An expression…

A:

Q: 1. Sketch the direction field and solution curve for the given DE and condition. Sketch the…

A: The posted question has multiple questions. But according to guidelines we will solve the first…

Q: 1.6 Evaluate the line integral f x*dx + xydx where C is the triangle with vertices at (0, 0), (0,…

A:

Q: Given the one-parameter family of curves y² = Cx³ + 5. Find the family of orthogonal trajectories.…

A:

Q: B. Evaluate the given line integral. Follow the direction of C as given in the problem statement. 4.…

A: Given ∫Cy2-3z+2 dsC is the line segment from 1,0,4 to 2,-1,1

Q: 17. Determine the equation of the family of orthogonal trajectories of the the family of curves…

A:

Q: Find an equation for the tangent plane to the graph of z = xe' cos z – 2 - at the point (2,0,0). 1.…

A:

Q: 9. Demonstrate that the functions Y1 = x2, y2 = sin(x), y3 = xe are linearly independent on R.

A: Recall: If y1,y2 and y3 are the given function and         Wy1,y2,y3x = y1y2y3y1'y2'y3'y1''y2''y3''…

Q: Let w = x°e4y cos(9z). Find the value of dwldt at the point (3, ln 5, 0) on the curve x = 3 cos(t),…

A: Given: w=x6e4ycos(9z)x=3 cos(t)y=ln(t+5)z=6tDifferentiating we…

Q: 2. Find an equation for the plane tangent to the surface 3x2 - 4xy+ z2 = 0 at the point (1, 1, 1).…

A:

Q: 11. Compute the length of the curve r(t) = (2t, In(t), t²) over the interval 1 <t < 4. (cos(t),…

A:

Q: A bug is crawling through the ry-plane in such a way that its coordinates (r(t), y(t)) at time t are…

A:

Q: (A) Evaluate z dz along from z= 0 to z-4+i along curve of c given by the line from z-0 to z= 2i and…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: What is the equation of the curve y = ax* + bx + cx² + dx + e tangent to y = 25x (2,14), to y = 2x +…

A: By verifying options and substitution of point (-1,11) it is verifying option C

Q: 2. Show that cos(wt – B), coswt, sinwt are linearly dependent functions of t. Show Complete Solution

A: Given that: cos(wt-B), coswt, sinwt

Q: 1. Sketch the direction field and solution curve for the given DE and condition. Sketch the…

A:

Q: Use Green's theorem to evaluate S,(5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: The posted question has multiple questions but according to guidelines we will solve the first…

Q: Express x = In(3t), y = 5 – t in the form y = f(x) by eliminating the parameter. %3D y =

A: Given that x=ln(3t) and y= 5-t we need to write y=f(x) by eliminating the parameter  Now , x= ln(3t)…

Q: 15 The curve y=xr'+px² +qx+r passes through the point (0, 5) and has stationary points at x= 4 and…

A: The given curve is  f(x) = x3 + px2 +qx + r The stationary points are that points where the curve…

Q: 23. Give a formula that gives the length of the outer loop for the curve: r= 4-8 cos(0) (Hint: Graph…

A: r=4-8cos(θ)For loops crossedr=04-8cos(θ)=08cos(θ)=4cos(θ)=12θ=π3,5π3(Have symmetry from 0 to π and π…

Q: Riemann sits on the line normal to the curve y=x2 -3x+2 at x = 1

A: Given           y=x²-3x+2 and x=1  Slope of the curve is given by =dy/dx

Q: (6) Find the orthogonal trajectories of the family of curves r= 5C cos 8, where C being parameter.

A: we have to find the orthogonal trajectory for the given function

Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: 1. Find the orthogonal trajectories of the family of curve z + 3y = cy,where cis an arbitrary…

A: We will first find the differential equation of the of the given family of curves

Q: 7. Let F = (2x sin(y)e", x² cos(y)e*, z² sin(y)e*). Find along the path c:T = (sin(t), t, cos(t))…

A: F=2xsinyez, x2cosyez, x2sinyez curlF=ijk∂∂x∂∂y∂∂z2xysinyezx2cosyezx2sinyez⇒curlF=0 So, F is a…

Q: 9. Find the orthogonal trajectories to the following families of curves: x = ke¯v, where k is a…

A:

Q: Question 4 Evaluate the line integral: (i) of T(x) = 4x^3 along the line segment from (-2,1) to…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.