At time t, the position of an ant moving in the xy-plane is given by parametric functions (x(t),y(t)), where dx/dt = 3^t +2 and dy/dt =t^2 − 4. At t=1, the position of the ant is at (2,3). a. Is the curve representing the path of the ant concave up or down at t=6? Justify your answer. b. Find the y coordinate of the ant at t=4

Q: find dy/dx

A:

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A: Given: y=xy ⇒y=x12y12 ⇒y12=x12

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A: We will use parameterization to find line integral

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A: To apply the shifting property of Laplace in the given equations.

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A:

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A: Given: z=ey+x+x2+12 Point (6,0,55)

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A: First we will plot the given parameters.

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A: f(x,y) = 5k+3yt point (0,2)in direction θ= 5π4 radiusto find directional derivativesolution1)…

Q: Find dy/dx: y = In [(1 + t^2)/(1 - t^2)]

A:

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Q: (d) Find the directional derivative of f at (-2,1), in the direction of (-3,5).

A: Given f(x,y)=9-x2-y2 fx= -2x fy=-2y ∇f(x,y)=<fx,fy>=<-2x,-2y>…

Q: apply linearity property of laplace in the equation f(t)= t cos 2t and f(t)=t^2e^-3t

A: f(t)=t cos 2tf(t)=t2 e-3t

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A:

Q: The equation of the line tangent to the curve F (x) = S e dt at a = %3D V1- O y-V2 = (=- ) O %3D y -…

A:

Q: ) Find the slope of the curve x +y3- 6x²y = 0 at x = 1

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Q: 4) Find the total derivative of the function: a.f(x, y) = ex²+y² 1 c. T(l,g) = 2₁ 2TT. b. z = x siny…

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Q: 2) r(1) = ti –t j-tk, t20 Draw the graph of the vector-valued function, explaining it in detail.

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Q: 15 /(х. у) 2гу-Зх, Р % (2, 1), - (3,2)

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.