1. (a) Let z be a function of two variables given by z = f(r, y) = 1- r- 2r – y?. i. Find the domain and the range of the function. ii. Sketch the level curves for z = 0, 1,2.

Q: What does it mean for a function of two variables, f (x, y),to be differentiable at a point (a, b)?

A: A function differentiable at a point has a derivative that is defined at that point. This is defined…

Q: 1. Differentiate G(x) = Ax+Bx+C 2. At what point on the curve y= 1 + 3e* – 3x is the tangent line…

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Q: ,2 For function f (x, y) x2+1 Evaluate the gradient at the point (1,2)

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Q: 3) (a) Find the oirectional derivative of f(x, y) = xe-y x at (-2,1) in the direction u= i-j. O…

A: We need to find the directional derivative in the direction of vector u

Q: 1. The temperature at the point (r, y) on an unevenly heated plate is given by T = f(r, y) = 100+…

A: Given f=100+5x2+15x+10y2gradient of f∇f=∂f∂xi+∂f∂yj     =10x+15i+20yjAt (2, 1)∇f=10x+15i+20yj…

Q: The gradient of the tangent to the curve y=flx) at the point P(2, (2)) is given by A 12) – h f(2+k)…

A: To find the gradient of the tangent to the curve y=fx at the point P2, f2.

Q: Find an equation of the tangent line at x= a. x- y%3D a = 2 4 10 A. y=x+1 *B y=x-1 y3x-1 Xe y=x+2…

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Q: Q.1. Justify why the level curves very close to the origin (0, 0) for the function f(r, y) =…

A: To find the level curves, equate the function fx,y with a constant C.

Q: 18,8grt(10)) 10 f(x) = V2x – 2 compute the instantaneous slope at: (6, v10) Start with two points A:…

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Q: 4) The position function of a particle is given by F(t) = (t², 5t, tť² – 2t). When is the speed a…

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Q: 3. (a). For a curve defined by an equation f(x), satisfy dy + (1 – 2x)y = 4x for x > 0 and - dx f(1)…

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Q: (c) The tangent plane to the graph of z. = f(x, y) at the point (1, 0, 1) can be described by the…

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Q: Ex 4. Find a function of the form f(x)= ax* +bx² + cx+d with a local maximum at (0,–6) and a local…

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Q: Ex 4. Find a function of the form f(x) = ax* + bx² + cx +d with a local maximum at (0,–6) and a…

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Q: For each of the following functions, find the maximum and minimum values of the function on the…

A: To find the  maximum and minimum values of  a function on a region D , first find the critical…

Q: (2) Let f(x, y) = x² – y². Find the tangent line to the level curve f(x, y) = -3 at (1,2).

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Q: 2/6 Consider the upper half of the ellipsoid f(x.y) = 1--n and the point P-- 3 on the level curve…

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Q: 1. Calculate the derivative dy/dx for the plane curve defined by the equations…

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Q: Let f(x, y) = xe*Y. Find the maximum rate of change of f at the point (2, 1). хе

A: We will find the maximum rate of change.

Q: Let f'(z) = tangent line to the graph of the function h(z) = g(13x+3) at e and g(x) = f(/#). Let L…

A: Given, f'x = ex2  gx = fx L is the slope of the tangent line to the graph of function hx = g13x+3 at…

Q: 1. Find the domain and two level curves of f(x,y) = /25 – x² – y2

A: As per guidelines, we will solve the first question only. 1. Given function is   f(x,y)=25-x2-y2

Q: 1. Consider the curve F(x, y) = 0. Show that if F has continuous second derivatives, then a formula…

A: We will use chain rule of partial derivative  If F is a function of  two independent variables x and…

Q: Find the equation of a line normal to the curve of y= cos at x= 1. Select one: A. -33 x + 9y - 9 +…

A: Equation of Normal

Q: Find the coordinates of the point(s) at which the curve y = x-2/3 has gradient – 1/6. %3D -

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Q: 1. Given the function z = f(x, y) = Vry - I. (a) Find the domain and range of f(x, y). (b) Sketch…

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Q: The value of b makes the line y = 2x +b be tangent to the function f (x) = x² is O -1 none О 1 O 2

A: Tangent line related problem......

Q: Let  y = f(x) = x ⋅ cos(πx) Find the equation of the line tangent to the graph of  y = f (x) when…

A: Given Let  y = f(x) = x ⋅ cos(πx) equation of the line tangent to the graph of  y = f (x) when x =…

Q: 1. Given z = f(x, y) = √x² + y² + 4y. (a) Find its domain and range. (b) Sketch the level curves for…

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Q: Sketch the level curve of g(x,y) = y / 4 ln x corresponding to g(x,y) = -1/2

A: g(x,y)=y4ln x=-12⇒2y=-4ln x⇒y=-2ln x

Q: Find an equation for the level surface of the function f(x,y,z) = Vx+ y - Inz through the given…

A: Solve the given

Q: For the implicit curve y + xy + x2 = 3, find a point on the curve where the tangent line is…

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Q: 1. Given: f(x, y) = y a. Find and sketch the domain of the given function b. Find and sketch the…

A: Directional derivative of a function f(x,y,) in the direction of unit vector…

Q: Consider the function f(x,y) = [x² + y² if x # (0,0) if x = (0,0) 0 Evaluate f(x, y) over the…

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Q: Sketch the graph of function f(x, v)- J4-x - and draw the level curves at k- 0, 1,4

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Q: (c) Find the equation of the tangent plane to the graph of f(x, y) at (1,1,3).

A: Suppose surface S defined by a differentiable function  z=fx,y and point Px0,y0,z0 lie on the…

Q: (d) Suppose that y(x) is a function and that y(2) = 1. Let y² + y = x. Using implicit differentian…

A: The objective is to find y'(2)and find the equation of tangent line to y(x) at x=2.

Q: Draw the level curves for the function z = f(x, y) given by z = /4 – x² – y² when z = 0, 1, 2 and…

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Q: 5) Determine the slope of the tangent line to the function with the given information: a. f(x,y) =…

A: Given, the function                            We have to find the slope of the tangent line to…

Q: The tangent line to the curve y=vx-5 * for x=16 is a. y=1/8 (x+16). O b. y-4=1/6 (x-16). O c.…

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Q: #2. Consider the function f (x, y,z) = xy+3xz² + 5y at the point (1,0, 2) (a) Find the direction in…

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Q: Find an equation for the tangent plane to the graph of f(x, y) V3+ r2 – 3y2 at the point P(2, 1,…

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Q: Find a function f(x, y, z) whose level surface f =1 is the graph of the function g(x, y) = 5x + 3y.…

A: Here is the solution of the given problem. Level surface:

Q: Q. 2 For an F-curve with df = (6,10), find: a) F o.01 b) Fe.os c) F se Fatables,

A: Solution: Given information: The degrees of freedom for the F curve is df(6,10) Here we need to find…

Q: 1. Consider the function f(x, y) = {√x² + y² if x # (0,0) 0 if x = (0,0) Evaluate f(x, y) over the…

A: Thank you for the question. According to our guidelines we can answer only  first question and…

Q: 1. Consider the curve F(x, y) = 0. Show that if F has continuous second derivatives, then a formula…

A: Consider the curve F(x, y)=0. The objective is to show that if F has continuous second derivatives,…

Q: (2) Find an equation for the tangent plane to the graph of z = where z = f(x, y) is implicitly…

A: Given: The function is fx,y,z=xy+xz+yz-lnz2+1-4. Calculation: Obtain thethe value of  fxx,y, fyx,y…

Q: Find the gradient for the curve defined by y + xy = 3 at the point (1,2)

A: Gradient to given curve

Q: 3. Find the equation of the tangent line to the curve at (1,2) * y = 5 - x O A. x-6y + 22 = 0 O B.…

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Q: Consider the curve in the xy-plane represented by the two equations below. The slope of the line…

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.