let r1(t) = (t2, 1, 2t) , r2(t) = (1, 2, et) Compute d/dt r1(t) × r2(t)t=1 in two ways:(a) Calculate r1(t) × r2(t) and differentiate.(b) Use the Product Rule.

Q: Use the Chain Rule to calculate f(r(t)). dt f(x, y) = x³ - 5xy, r(t) = (cos(4t), sin(5t)), t = 0…

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Q: (a) If w = e"yz, where x = and compute dw/dt when t = 0. (b) If f(x, y, z) function of s and t and…

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Q: Q5/ Determine the total differential If Z = f(x,y) = x²y², x=2rcose, y = 2rsine

A: Here, the total derivative rule used is: z=f(x,y)=x2y2 & x=2rcos θ , y=2rsin θ

Q: z = (x +y) e°, x = 5t, y = 4 – t2, find dz/dt using the chain rule. Assume the variables are…

A: Given functions are : z =x+yex,    x = 5t ,   y = 4 - t2 By using Chain Rule, dzdt=dzdx·dxdt +…

Q: dw Use an appropriate form of the chain rule to find dt w = In(10x² – 4y + 5z³); c = t', y = tỉ, z =…

A: From the given function: w=ln10x2-4y+5z3 ; x=t12 ; y=t23 ; z=t-2 As we know that: The Power law:…

Q: B: Expressand as functions of u and v both by using the Chain Rule and by expressing z directly in…

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Q: Q5/ Determine the total differential If Z = f(x,y) = x²y², x=2rcose, y=2rsinė

A: this is a problem of differential. Based on the general formula and concepts we will answer the…

Q: Evaluate fSS 6e(²+y²+z²)/² dV over B where B= {(x, y, 2) : x² + y² + z² < 1}. Take T = 3.

A: Since we have    x2 + y2 + z2  expression in the integral and region B is the sphere of radius 1…

Q: If g(x) J.J S. 1 -dt , find g'(2).

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Q: a. Express dw/ dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

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Q: Find the linearization of g(x) = 3 + sec (t - 1) dt at x = -1.

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Q: Find the linearization L(x,y) of the function f(x,y) = e 3× cos (9y) at the points (0,0) and 0,

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Q: Assume x and y are functions of t. Evaluate dt for 2xy - 2x + 4y3 = - 56, with the conditions = -…

A: Given      2xy-2x+4y3=-56 →           1      dxdt=-16, x=4, y=-2

Q: Find the linearization z = = L(x, y) of the function z = f(x, y) = 3.x2 – y² + 5y at the point (xo,…

A: z=fx, y=3x2-y2+5y and the point is x0, y0=2, 1. Now, f2, 1=322-12+51=12-1+5=16 Now partially…

Q: (a) express dw/ dt as a function of t, both by using the Chain Rule and by expressing w in terms of…

A: Given, 1. w = x2 + y2 , x = cos t,  y = sin t;  t = π  2. w = x2 + y2 , x = cos t + sin t,  y = cost…

Q: Find the linearization of the function f (z) =23 at z = 1. L(z) = Using this linearlization, f…

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Q: Using the second fundamental theorem of calculus and chain rule, compute dz t dt. t+1 2x

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Q: (a) express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

A: Express dw/dt as a function of t by using the Chain Rule. Use the sum rule and split the bracket.…

Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = x3y4 at a. (1, 1), b. (0, 0)

A: Consider the function: fx,y=x3y4 To calculate the derivatives of the given function with respect to…

Q: If f* f(t) dt O 6 1 O -4 divergent = 20x √4x²+21 ∞ -4, then f(t)dt is X

A: Given

Q: Let f(x, y) = x?y? and e(t) = (t, t³) (a) Calculate: Vf•d (t) = d (b) Use the Chain Rule for Paths…

A: Solve by chain rule

Q: Use the Chain Rule to calculate f(c(t)). dt f(x, y, z) = xºy®e4z, c(t) = (t², t³, t – 1), t = 1…

A: To Determine: Use the chain rule to calculate:                                       When…

Q: Let f(x, y) xy, x = 1- Vt, and y = 1+ Vt. Find df using the chain rule. dt

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Q: Compute d/dt f (r(t)) at the given value of t for f (x, y, z) = xz − y^2, r(t) =〈t, t^3, 1 − t〉at t…

A: Given: The function fx,y,z=xz-y2and rt=t,t2,1-t, To find: ddtfrt at t=-2,

Q: Let g(x) V1+ 8t dt. Use the Fundamental Theorem of Calculus to find g'(x). g'(z) =

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Q: Let r(t) = (t2, 1 - t, 4t). Calculate the derivative of r(t) a(t) at t = 4, assuming that a(4) = (9,…

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Q: Let Dy be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

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Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = ex cos y at a. (0, 0), b. (0,…

A: Consider the following function: fx,y=excosy Differentiate fx,y with respect to x taking y as…

Q: Assume that z = ƒ(x, y), x = g(t), y = h(t), ƒx(2, -1) = 3, and ƒy(2, -1) = -2. If g(0) = 2, h(0) =…

A: We use the partial derivatives to find dz/dt  Then plug x=g(t) and y=h(t)

Q: Let f (x,y, z) = xyz2 + 1 and suppose that x(t) = 2 – t, y(t) = In(t), and z(t) = 3t. df Use the…

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Q: (a) express dw/ dt as a function of t, both by usingthe Chain Rule and by expressing w in terms of t…

A: w=x2+y2, x=cost, y=sint, t=π, We have to a) express dWdt as a function of t, both by using the Chain…

Q: FIND THE LINEARIZATION OF f(x) = 2+ [,* dt at z= 1. %3D 1+ Lütfen birini seçin: a. L(x) = 2x – 4 b.…

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Q: - Find the total differential df of f (x, y) = (x² + y²) 3/2.

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Q: Let D, be the linear transformation from C'(a, b] into C[a, b]. Find the preimage of the function.…

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Q: Let Dx be the linear transformation from C'[a, b] into Cla, b]. Find the preimage of the function.…

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Q: Find the linearization of the function f () = z7 at z = 1. L (z) = 固助 Using this linearlization, f…

A: Given function is

Q: Let Dx be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A: Solution: Consider the given linear transformation : Dxf = 2x + 3 Since D is the linear…

Q: Find the orthogonal trajectories of Vy = cx = CX

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Q: Find the linearization of the function at the given point. f(x,y,z) = 10xy + 7yz+ 9zx at (1,1,1) O…

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Q: Find the lineanzation LlXiy) of f (iy) =x*-5Xy at (2,1). Then use the linearization to approximate f…

A: Here in this question we need to linearize the given function and then approximate the value of…

Q: Find the linearization L(x,y) of the function f(x,y) = e* cos (9y) at the points (0,0) and ...

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Q: Let {yk } be the DFT of {x; }, where x = [0, –1, 0, 1] y3 = O i -1 -i 1

A: Here, x0=0, x1=−1, x2=0 and x3=1 We have to find y3 Now, y3 is given by…

Q: Let g(x) 8 In(t) dt. Use the Fundamental Theorem of Calculus tc g'(x) =

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Q: Using the second fundamental theorem of calculus and chain rule, compute t dt. t+1 dx 2x

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Q: find the linearization L(x, y) of the function ateach point. ƒ(x, y) = 3x - 4y + 5 at a. (0, 0), b.…

A: fx, y=3x-4y+5 Partially differentiating with respect to x, we get fxx, y=31-0+0=3 Partially…

Q: Assume x and y are functions of t. Evaluate dy / dt for 3xy - 6x + 4y 3 = -80, with the

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Q: Use the Quotient Rule to compute the following: t2 + 1 at t = -2 dt \ t2

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Q: Q2/ Express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t…

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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.