Sketch the graph of a continuous function y = h(x) such that a. h(0) = 0, -2 < h(x) < 2 for all x, h'(x)→∞ as x → 0¯, and h'(x) →∞ as x →0*; b. h(0) = 0, –-2 < h(x) < 0 for all x, h'(x) →∞ as x→0", and h'(x) →-∞ as x →0*.

Q: az Find if yz – In z = x + y ax

A: To find: The partial derivative ∂z∂x, if yz-ln z=x+y. Calculation: The partial derivative ∂z∂x can…

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Q: Find the global maximum and minimum of the continuous function f(x) = x²-8 ln x on [1,4]. %3D

A: Given:F(x)=x2-8lnx        1,4

Q: Let f : [1, ∞) → R be a continuous function such that lim|f(x)| = 0. For n ≥ 1, let gn : [1, ∞) →…

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Q: For a continuous function f(x) on the set 2 [- 5,2], the net area Tx)dx must be -5 always positive…

A: The given question has been solved in detail

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Q: 1 Let f (x, y) =x²y*sin (- x²+y* when (x, y) ± (0,0) and f (0,0) =0. determine if f is continuous in…

A: We have to show given function is continuous at origin

Q: 2. Let f(x, y,2) = z-sin(xy), x = t, y = In t, z = e. Find derivative of f directly and by using…

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Q: (6) Are all continuous functions on ID differentiable on D?

A: we have to identify the given statement is true or not

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Q: The value of limit x→0 of {sin (a + x) – sin (a – x)}/x is

A: we have to find the value of  limx→0sin(a+x)-sin(a-x)x

Q: prove the identities assuming that the appropriate partial derivatives exist and are continuous.…

A: Let  F→=F1i→+F2j→+F3k→G→=G1i→+G2j→+G3k→F→+G→=F1+G1i→+F2+G2j→+F3+G3k→To prove :…

Q: Sketch the graph of a continuous function on [0,4] satisfying the given properties. f'(x) = 0 for x…

A: f'(x)=0 for x=1,2 and 3

Q: Express the limit as a definite integral on the given interval. lim n - 00 cos(x}) Дх, [2л, 5л] Xi i…

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Q: The limit represents the derivative of some function f at some number a. State such an f and a. lim…

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Q: If z = f (x, y) is a function that admits second continuous partial derivatives such that image1 A…

A: Given z=fx,y such that ∇fx,y=300−12x2−12y2,18y2−24xy. Here, ∇fx,y=fx,fy=300−12x2−12y2,18y2−24xy.…

Q: Sketch the graph of a continuous function y = h(x) such that a. h(0) = 0,–2 < h(x) < 2 for all x,…

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Q: (i) Find all first and second order partial derivatives of f(x, y) = x²y + cos(xy) (ii) Find lim xy²…

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Q: Let z = z(x, y) be a continuous function of x and y defined implicitly by In(3æ* + 4y* + zyz*) = =…

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Q: Suppose that the partial derivatives of a function ƒ(x, y, z) at points on the helix x = cos t, y =…

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Q: 3) Examine the limit of 1 (x² +y²).sin- x*+ (x, y) (0,0) f(x, y) = (0, (x, y)=(0,0)

A: Solution:Givenf(x,y)=x2+y2 sin 1x2+y2To Eamine the limit of a functionFormula:limx→0 sin xx=1

Q: The partial derivative of f(x, y) with respect to y at the point (x,y) is f(x+h,y)-f(x,y) h B f(x…

A: We have to Write the Partial derivative of f(x, y) with respect to y at the point (x0,y0).

Q: Find the partial derivative of f(x,y) with respect to y at the point (3,2) f(x,y) = ln(x^2y-xy^2)

A: A partial derivative of a function of several variables is its derivative with respect to one of…

Q: Assume that ƒ is continuous on [0,1] and that f(0) = 2 and f(1) = 1. Show that the graph of y = f(x)…

A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

Q: Show that f(z) =xy+iy is everywhere continuous but is not analytic

A: Answer

Q: Find a value of k making h(x) continuous on [0, 5]. kx 0 < x < 2 x + 1, 2<x < 5 h(x) =

A: The given function is hx=ekxif 0≤x&lt;2x+1if 2≤x≤5. The function is continuums. That is,…

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Q: Use logarithmic differentiation to find for y = x4* %3D dx 2x Use LaHopital's rule to calculate the…

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that f (x)dx = A, f(x)dx = B, and…

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Q: Let fc Cx)= Vx if osxs 4 c.(x-4)+2 if x>4 denotes a real constant. - where C Show that lim fe (x) =…

A: The function fcx=x,              if 0≤x≤4cx-4+2   if x&gt;4. Where c is a real constant. We have to…

Q: 2. Show that the function f(x) = cos x is uniformly continuous on

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Q: Show that following function xY (x, y) + (0, 0), f(x, y) = x² + y² (x, y) = (0,0) is continuous,…

A: To check the continuity of given function at origin firstly finding the limit of the function at…

Q: Determine if a function f : R2 → R defined by f (x; y) = { x sin(1/x2+y2) , if (x, y) ≠ (0,0) and 0…

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Q: Suppose f(x) is continuous on [0, 1], with f(0) = 2 and f(1) = 0. Then there must be a value x in…

A: Here given that f(x) is continues.

Q: Find the limit analytically: 2-4x lim 40 5+3x 1. x - 2x-15 lim x 5 2. Find dy/dx of: x' (x- y) -x-y…

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Q: Given function f(x, y) = 1+ xy - x - y. df a) using partial differential involving limits to obtain…

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Q: II. Determine if [x+1]. if x < -1 S x + 1 f(x) = 2 tanh (x+¹) if 1<x< 1 1-x In [cosh (x + 1) + sinh…

A: It is solved whether f(x) is continuous or not.

Q: If f is bounded and integrable in the interval [a, b], (x) cos nx dx3D0. show that lim

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Q: 1. If g(x) is a continuous function on [2 ,9] such that , g(x)dx = 25 and ,9(x)dx = Find value of…

A: Using the properties of definite integral, it can be written as follows.

Q: What can you conclude about a real function f(x,y) that has continuous partial derivatives at a…

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that f(x)dx = A, ] f(x)dx =B, and…

A: Since f and g are continous we used the properties and got answer as D

Q: 4. Use the limit definition of partial derivative to find f, (x, y) and f, (x, y). a) f(x,y)=/x+y

A: As per our guidelines we are supposed to answer only one question i.e. the first question. For the…

Q: 1. Estimate the limit of the signum function s(x) as x approaches zero from the right. (-1, x0

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Q: (xy(x² – y²) f(x,y) ={ x² +y² si x² + y² + 0 0 si (x, y) = (0,0)

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Q: Let f(x) = ln x / sin^-1 x    analyze the limit : lim x--&gt; 1- (from the left) f(x)

A: Given information:The given function is

Q: Consider the sequence of functions {fn} defined on the interval (1, 0), where fn (x) 1 Vx E (1, 0),…

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Q: Let f:Rto R be a continuous function that is convex on ( -infinity,0] and [0,infinity) and has a…

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Q: The partial derivatives of a function f : R? → R were defined by considering the rate of change of…

A: (a) The given function is fx,y=x2y.     The formula for the simultaneous derivative at the point…