The Mean Value Theorem for differentiable functions f : R → R states the following: f(b) – f(a) Given a < b, 3Ğ E (a,b) such that f'(5) b – - a (a) Use the Mean Value Theorem to prove that if f'(5) > 0 for all ğ E R then f is injective. (b) Consider the formulas f(x) = g(x) = x² – 6x + 2. Find a value a e R and a set B CR such that f: (a,00) → B and 8:(-0,a) → B are both bijective. Explain your answer.

Q: Describe the x-values at which f is differentiable

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Q: Find all differentiable functions f: R→R such that f'(x) = F(x + n) – f(x) n for all real numbers x…

A: As f'(x)=f(x+n)-f(x)n is true for all positive integers values of n thus substitute 1 for n in…

Q: Exercise 1. Let f(x) be the function 2х — 1 if x 2. - (a) Find the value of c that makes f(x)…

A: Given f(x)=2x-1if x&lt;-1cxif -1≤x≤2x-2if x&gt;2

Q: Sketch the graph of a function f such that f (x) ≥ 0 on [0, 1] and f (x) ≤ 0 on [1, 2], whose…

A: Consider the function Then and The graph of the function is shown below,

Q: Let f e E(0, ox) and let a be a positive real number. Define the function fa as follows Sal) = {";…

A: Properties of Laplace transformation

Q: (b) Define a function F:F: R XR → R XR as follows: For all (x, y) ERXR, (F(x, y) = (2x + y, 2x - y)…

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Q: 9: Check Euler's Theorem for the Function

A: We checked Euler's theorem for function.

Q: Consider the function below. (x, y) = x²e^xy (a) Evaluate f(3, 0). (b) Find the domain of f. ORX (0,…

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Q: Derive the autocorrelation function of: x₁ = $xt-1 + εt + 0€t-1 assuming that p < 1, and E(?) = 0²,…

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Q: If f:[-5, 5 ] to R is a differentiable function and if f'(x) does not vanish anywhere, then prove…

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Q: Explain why there does not exist differentiable functions f and g for which f(0) = g(0) = 0 and =…

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Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

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Q: Determine L[f]if 0, f(1) = {1- 1, 0 2. 1,

A: Given that, A function f(t), t ≥ 0, its Laplace transform F(s) = L{f(t)} is defined as

Q: Use the table to determine Df(g(x))) at x = 3. |1 2 3 4 f(x) 1 f(x) |3 -5 1 4 4 1 3 4 3 2 g(x) 2…

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Q: Investigate the continuity of the following function f(2)= if z +0 if z =0 (i) Is f(z)…

A: Calculate the limit:

Q: (a) Let f:Q → Q be the function defined as follows: f(x) = 1 + 2x. Is f(x) a surjective function?…

A: Solution

Q: 2x + 7y + 2z - 5w = 3 3x + 8y + 5z + 2w = 2 4x + 5y + 14z+ 10w = 12 5x + 12y + 8z +4w = 4

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Q: Consider the function given by z-y+2/7–2/9 se r+ y, x>0, y> 0 f(r, y) = k(x), se ¤ = Y n the…

A: function is continuous

Q: Prove that for any function f : X → Y and for any subsets Y₁ and Y₂ of Y, ƒ¯¹(Y₁ NY₂) = ƒ¯¹(Y₁) Ñ…

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Q: Suppose f: X→ Y, and A, B CY. Prove that (a) f¹(AUB) = f¹(A) U f¹(B).

A: Union of two sets A and B is denoted as A∪B  It contains all the elements that are either in A or in…

Q: Let f: Z → Z be defined by f(x) = x² + 1, and let C = {1,2,3}.

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Q: A = [a,b), be an (right half-open) interval of R. 6 (c) (d) a,be R+,b+00, Explain whether f(A) is…

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Q: Suppose f: R → R is a function defined by f(x) = x³. (a) Find f([-1,1]) and f-¹([0, 8]). (b) Show…

A: We shall use the intermediate value theorem to answer this question

Q: Let f and g be functions that are continuous on [a, b] and differentiable on (a, b). Show that there…

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Q: Consider the function f: R → R defined by f(x) = x³ – 3x + 2. Then (A) ƒ is increasing on the…

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Q: Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then…

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Q: (6) If f(z) is analytic function show that: [v] + [0] =

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Q: 5) Find functions f and g so that f • g = H -3 H(x) = Vх — 2

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: