Given a function f: ℝ → ℝ^2, for each x ∈ ℝ write f(x) = [f1(x), f2(x)] ∈ ℝ^2. Where that f1 and f2 are real valued functions of ℝ. Prove f is differentiable at a ∈ ℝ, if and only if, f1 and f2 are differentiable at a, and in that case     f'(a)  = [f1'(a)  f2'(a)] (right part of equation should be in column form)     For the following exercises, use the following definition: A function f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n if there exists a linear transformation g: ℝ^n → ℝ^m such that lim [f(a+h)-f(a)-g(h)]/h =0 h→0

Q: Let 1 f(x) = 2 9, - 2 < x < 5 - The domain of ƒ¯'is the interval [A, B| where A = and B =

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Q: If f(a) = 7 and f(b) = 0, then there exists a number c in the interval (a, b) such that f'(c) 7 a –…

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Q: Provide an example of a function which is continuous on all of R and differentiable on all of R…

A: The function is y=x+4 (shown in the figure above)

Q: Consider a function f:C⟶C, defined by f(z)=7z^2. Then which of the following is true? Check all…

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Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: Suppose that f (0) = 2 and f'(x) 0. Show that f(4) 0.

A: f(0)=2   and f'(x)≤3  for all x&gt;0 ⋯(i)integrate (i) w.r.to x ,we getf(x)≤3x+kput…

Q: f 'C) =7x*+7-3 ond fl0) 3D8

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Q: Finel the ralves of a anel b such that the function aV5-x tb-1 it Hax cos(TT Ix) 9 +1+ b(x²-7)…

A: Note: There is an error in question, for the given function a has 2 values but it cannot be possible…

Q: Determine whether each of these functions from Z to Z is one-to-one. a) f(n)=n−1 b)  f(n)…

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Q: for a -1 2x +5 f (x) = If fis the function defined above, then f' (-1) is A -2 B 3. E) nonexistent…

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Q: Suppose we have a function f on an interval [a,b] where f(a)=f(b)=0 and f''(x) <= 0 for all x on the…

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Q: Use the linut definition to Compute f(x) when f)-at. the derivative f(x) when f)-2+L

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Q: Let 11 f(x) = ( – 2a² – 4)(7z² – 4)" - - f'(x) =

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Q: Let f: A →B and g: B →C be functions. Suppose that f and g have inverses. Prove that g ◦ f has an…

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Q: Let f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not…

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.