How can you sometimes decide from examining ƒx and ƒy that a function ƒ(x, y) is differentiable? What is the relation between the differentiability of ƒ and the continuity of ƒ at a point?

Q: Find a function f such that f ' (x ) = x 3 and the line x + y =0 is tangent to the graph of f

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Q: (b) Test the continuity of the function f(x) = sgn(lx|) at x = 0. Is f(x) %3D differentiable at x 0?…

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Q: Discuss the continuity of the composite function f ∘ g.                                     f(t) =…

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Q: hav is that not the function defined by ху f(x, y) √x² + y² if differentiable at (x, y) + (0;0) if…

A: We have to show that f(x,y) is not differentiable at (0,0)

Q: Show that the function f(z)= Im(z) is nowhere differentiable.

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Q: Use the chain rule to show that for any differentiable function f(x), [e/(e) = f(x)e/(=),

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Q: Let f : R → (−1, 1) be given by f(x) = x/(1 + |x|). Show that f is differentiable at x = 0, with…

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Q: (b) Test the continuity of the function f(x) = sgn(]x|) at x = 0. Is f(x) differentiable at x = 0?…

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Q: Discuss the continuity of the function. f(x, y, z) = 1/(√x2 + y2 + z2)

A: Given rational function is

Q: Prove that y = x- 2 is differentiable and calculate y'.

A: Given: The function, y = x-2

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) = 1 + t3 dt

A: To find F'x Fx=∫0xt21+t3dt

Q: The function f is a composite of two continuous functions: y = cos x and y = x 2, so f is continuous…

A: The function fx is a composition of functions y=cosx and y=x2and it is a continuous function.

Q: Let g (x) = f' (x) be a continuous function. If the derivative of the continuous function g (x)…

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Q: Explain why f (x) = sin (x² + 3r + 1) is a continuous function for all c € R. %3D

A: Let us assume that: g(x)=sin(x)and h(x)=(x2+3x+1)So,…

Q: Suppose f is a continuous function where f(7) = 9 and f'(7) = 6 . The tangent line to the graph of y…

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Q: Suppose that f(a) = f(b) = g(a) = g(b) = 0 and the second derivatives of f and g are continuous on…

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Q: Show that the functions f (x) = I x I and g(x) = x 1/3 are not differentiable at x = 0

A: Derivative of x at x=0:   Plotting: x,  xx

Q: What does it mean for f to be differentiable at a ?

A: we have to define that what we mean by the f is differentiable at a.

Q: Show that the function is differentiable by finding values of ε1 and ε2 as designated in the…

A: Given  f(x,y)=x2y

Q: What does it mean for a function ƒ(x, y) to be differentiable? What does the Increment Theorem say…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: (a) Let f (x) = Vx – 1. Use the limit definition of the derivative to find f' (x).

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Q: Suppose that F(t) is a twice-differentiable function of one variable. Define the function f(x,y) =…

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Q: Let f(x) be a continuous differentiable function that f(x)- 4 satisfies lim =D5 %3D x3 x-3 Which…

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Q: Determine the points of continuity of the function f (x) = x – [x] for all x E R.

A: Solution: The objective is to determine the continuity of the function f(x)=x-x for all x∈R

Q: Compute the right-hand and left-hand derivatives as limits to show thatthe functions in Exercise are…

A: Given figure is,

Q: Show that the function x² – y² 2ху f(x,y) = { x² + y² if (x,y)# 0 if (x,y) = 0 is continuous at…

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Q: Use limits to determine if f(x) = 1 is continuous at x = 0. (z+2)?

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Q: Show that the limit of the function f(x,y) = at (0,0) does not exist. (x+y)² x² + y²

A: Limit of a Function: Like a single-variable function, a multi-variable function can also have limit.…

Q: Determine the points of continuity of the function f (x) [x] for all x E R.

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Q: Let f(x) = V1+x. Estimate the change in f from x = 7 to x = 7.5 using linear approximation. %3D

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Q: Determine if f(x) is differentiable at x = 0 if

A: We will find out the required solution.

Q: Let f(x) = 3√(x + 1) − 4 = (x + 1)1/3 − 4. Use the limit definition of the derivative to show that f…

A: Consider the given function: fx=x+13-4=x+113-4 Vertical tangent at point indicates that at that…

Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

Q: (d) A function not continuous at 0 such that (f(x))4 is continuous at 0.

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Q: Find a function f such that f'(x) = x4 and the line 2x - y = 3 is tangent to the graph of f.

A: Given that

Q: Find the limit of the function f(x) = x²esin(1/x) as x approaches to zero.

A: To find limit of function f(x)=x2esin(1/x) as x approaches to 0. (Using Squeeze theorem)

Q: Consider a function f continuous and differentiable on R with f(2. 5) = -3. 1 and f'(2. 5)=-0. 2.…

A: We have to find

Q: Let f be a differertiable function such that f(-5) = -9 and f'(x) = 2x + 8. What is the…

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Q: Is the function given below continuous/differentiable at x = 3? %3D -3x + 2 x 3

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Q: Analyze continuity of the function.

A: Before going to the continuity, we will be finding the domain of the given function. Note that the…

Q: For the function f(x, y) = xy((x2 − y2) /( x2 + y2) ) define f(0, 0) such that f is continuous at…

A: A function f is continuous at the origin, the function value at the origin must be equal to the…

Q: Let [x² + 1 _ if x 1 f(x) - Is f differentiable at 1?:

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Q: Let g(u) be a differentiable function and let f(x, y) = g(x² + y²), find the direction of maximal…

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Q: 1. Explain what it intuitively means for function f(x,y) to be continuous in some domain D.

A: To explain the meaning of the given statement.

Q: Determine the set of points at which the function is continuous. F(x, y) = (xy)/(1+e^x-y)

A: A function is said to be discontinuous if it is not defined at a certain point  For the function…

Q: Show that the function f(z) = z = is differentiable but not analytic at z = 0

A: Givenf(z)=zz

Q: Suppose that f (1) = 2, ƒ (4) = 7, f' (1) = 5, f' (4) = 3, and f" is continuous. Find the value of…

A: Given data: f(1)=2 f(2)=4 f'(1)=5 f'(4)=3 The objective is to find the value of ∫14xf''(x)dx Apply…