Suppose that f: R → R and g : R → R are related by the formula f (x) = g(x)+(x-a)". For which values of n > 0 (n an integer) is f tangent to g at a?

Q: Show there does not exist a continuously differentiable function f on [O, 2] such that f(0) = -1,…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: If f is differentiable at a, then f is continuous at a.

A: If a function fx is differentiable at x=a then function must be continuous at x=a. But a function…

Q: consider the function f defined on [0,∞),f(x)=(x^r)sin(1/x),for x≠0 and f(x)=0 and for x=0, where r…

A: Solve the following

Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

A: The solution is given as

Q: Find an example of a function f:[0,1]¬R such that f is not continuous, but f does satisfy the…

A:

Q: Prove that (r) = tan(r) is uever negative in 0 <r< and show that g(x) = xsinr decreases…

A: We have to prove that f(x)=tanx is never negative in the interval 0,π2. We hence have to show that…

Q: (5) Use the limit definition of derivative to prove the following: (a) For any a e R, (f(ax))' =…

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Q: compute f ' (a) using the limit definition and find an equation of the tangent line to the graph of…

A: Given f(x)=x2-x The limit definition for derivative of a function is given as: Using the above…

Q: 1. Let the function f: R R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Differentiablity analysis problem....

Q: Suppose that the function f: [0, 1] in R is continuous, f(O) &gt; 0, and f(l) = 0.Prove that there…

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Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

Q: Suppose f is differentiable on some open interval containing e and f" (c) = 0 then (c, f(c)) is a…

A:

Q: 3.Show, by means of the example f(t) = sin t, that f ∗ f is not necessarily nonnegative.

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Q: Suppose that f (x) = x·g(x) for some function g that is continuous at 0. Prove that f is…

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Q: Let f(x) = x²+x+ 2 = Z₂ (x) prove that f(x) irreduciable Poly

A: The given function is  f(x)=x2+x+2.

Q: Suppose that  ff"' is continuous and f'(c)  =f"(c) =0 , but f"'(c) &gt;0. Does f have a local…

A:

Q: Suppose that f and g are discontinuous at x = c. Does it follow that f + g is discontinuous at x =…

A: Let us define that Then both f and g are discontinuous at x=2

Q: 7. Suppose that f: R R is differentiable at c and that f(c) = 0. Show that g(x):= |f(x)| is…

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Q: Define h(x) = x³ sin(1/x) for x + 0 and h(0) = 0. Prove that h' is not differentiable at x = 0. .

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Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

A:

Q: Suppose that fx) is continuous on [0, 9] and F(x) is an anti-derivative of f. If F(0)=9 and F(9)=7,…

A: Given that:                           f(x) is continuous in [0, 9].…

Q: 19.D. Show that the function defined by g(x) g² sin (1/x), x * 0, %3D = 0, 0, is differentiable for…

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Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

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Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

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Q: 2. Determine if a function f : [0, 2m] →R defined by f(r) = sin 2x is continuous. %3!

A: A metric space is defined when the points in the space follow the following 3 properties:- The…

Q: Suppose f(z) is analytic for |z| &lt;= 1, and |f(z)| &gt;=1. If f(0) =1, prove that f(z) is…

A: To prove that the given conditions on the analytic function force it to be the constant function…

Q: Suppose f is known to be differentiable and contains a local extremum at c, then which of the…

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Q: Let f(x) = Q[x] be a irr of prime degree p. If fr non-real roots then th

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Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

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Q: Consider g(x) = rº/3. Is g(x) differentiable at x=0? Is g(x) continuous at x=0?

A: We find g'(x) using the power rule  Then we find the left and right-hand limits of g'(x) and g'(0)…

Q: suppose f (x) = Vx (x>0),prove that f is differentiable (0,00) and find f'(x) on

A: Function f(x) = √x (x&gt;0)  We know f(x) =Xn

Q: Prove that f(x) = tan(x) is never negative in 0 < x < ; and show that g(x) = x=l sin x decreases…

A: We have to prove that f(x)=tan(x) is never negative in the interval 0,π2. Then we have to show that…

Q: If f is differentiable at a ,then f is continuous at a

A: Every differentiable function is continuous.

Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

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Q: Suppose F(x) = e-" dt. Find a formula for the function F(x³) expressed using an integral. F(x³) = /

A: Given that: F(x)=∫17xe-t2dt

Q: Suppose that f is continuous on [a.b] that f(x) 20 for all xE[a,b] and that f =0 1. Prove that f(x)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 15. If f+g is differentiable at a, are f and g necessarily differentiable at a ? If f g and f are…

A: Let f be any function that is not differentiable  at the point a and define as: gx=-fx for all real…

Q: ) For a, 3 ER and B > 0, let f(x) ={} x° sin(|x|-#), when x # 0 when x = 0 0, ange of a and B (and…

A: Continuity and differentiability

Q: If f is analytic and has a pole at z = a, then f(2)|| → 0 as z → a in any manner.

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Q: Show that the function f:R to R defined by f(x)=5x-12 is continuous using the E-d definition

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Q: Let f(x) = 2 cosh(4x) – e4x %3D Find the absolute maximum of f(x) on the interval [0, In 3].

A: The given function is                                         f(x)=2 cosh(4x)-e-4x We know that…

Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.

Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

A:

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.