1, when x is a rational number. f(x)= 0, when x is not a rational number. Determine if the function f is integrable in [0,2].

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: Give an example of a (discontinuous) function that does not satisfy the conclusion of the IVT on…

A:

Q: 3. Let f(x) = x, given the interval [0,1]

A: fx=x, x∈0,1 f'x=1&gt;0 for all x∈0,1 So, fx is strictly increasing on the interval 0,1.

Q: In approximating one real root of f (x) = on (1, 3), which endpoint of the interval is constantly…

A:

Q: 02: Let f(x)= x' – 7x+ 2, 0sxs1. 1. Show that f(x) has exactly one real root in the indicated…

A: Answer

Q: Use Theorem 2 to show that 3, if x <-10 2, if – 10 sxso F:R → R,F(x) = -2, if 0< x< 10 -3, if x2 10…

A: so we have to show that given function is discontinuous at 2 and -2 (a)   Discontinuous  at -2…

Q: 1. Suppose that f f(x) dx = In x + Vx+1+C. Can f be a rational function? Explain.

A: To explain f is a rational function or not.

Q: The interval on which f(x) = 3 – 2x – 12x² – x³ is | - concave down is: O (-4, 0) O (-∞, 4) O (4,…

A:

Q: 4 Given that f(2) is analytic at z = 0 and that f G) =, n = 1,2, 3..,find f(2). VE

A:

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: Use the Intermediate Value Theorem to show that f(x) = cos^−1 (x) − e^x has a zero in the interval…

A: The Intermediate Value Theorem: If f is a function which is continuous at every point of the…

Q: Suppose f  is twice differentiable on R  and has threeroots. Show that  f " has at least one real…

A: According to Rolle's theorem, there is a zero of f'x in between two zeroes of f(x).

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

A:

Q: Let I be the interval disjoined from {-1,1} .Prove that the function f(x) = x+(1/x) is…

A: Given that I is any interval disjoint from [-1,1]. Let I=(-∞,-1)∪(1,∞) f(x)=x+1x To show f(x) is…

Q: Let f (x) = 2x3 – 0.0064x? – 1.5x + la (a) Indicate that f(x) = 0 has a root in the interval [-3, 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: Assume that f(x) is a function such that f(x)dx = 6x² + 4x +c. Compute the integral — 5х — 3 - f (x)…

A:

Q: The graph of f(x)= x2 - 1 is concave down on interval | (-1,1) ^O (0,1) BO (-1,0) U (1, ∞ )co (- 00…

A: From the given problem:   fx=xx2-1   As we know that for concave downward:   f''x&lt;0   So,…

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: how that: (f)=X*(-f) x(t) is even, then b₁=0, an= To

A:

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: of of and ду For the function f(x,y) = In (7x + 9y), find

A: The given function,     f(x,y)=ln7x+9y

Q: The function f(x) =x + 1/x is concave upward on none O (-1) O (0,) O (0,1) O (1,)

A:

Q: Find the best quadratic approximation to f on the interval [ -1, 1].                     f(x) = lxl

A:

Q: Use the Cauchy-Riemann Equations to show that f(z) = z is differentiable for 16. all z and find…

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Q: t 0, if x is irrational 1, if x is rational ve a rigorous proof that f is not integrable on the…

A:

Q: Prove that f (x) = x2 takes on the value 0.5 in the interval [0, 1].

A: Here given

Q: Let f(x)={tan(2x) ax+b if x>0 if x<0 For which values of a and b is f(x) differentiable for every…

A:

Q: 1. Let f(x) = 4x − 4x 2 . Verify that f(x) satisfies the conditions of Rolle’s Theorem on the…

A: Note:- According to the company policy we can attempt ?️ only one question so please upload the next…

Q: Consider the function: { 0 ;if x is rational ; if x is irrational f(x) = Use a d, e argument to…

A:

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

A:

Q: What is the largest interval containing zero on which f (x) = sin x is one-to-one?

A: A function f: X -&gt; Y is said to be one-to-one, if the images of distinct elements of X under f…

Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

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: Let f be the function defined on [0, 1] by when x is irrational, f (x) = 1 when x is rational.…

A: Please see the below picture for detailed solution.

Q: Use the Intermediate Value Theorem to show that 2х - 3 if – 3 0 has a zero on the interval [-2, In…

A: By using the Intermediate Value Theorem to show that fx=2x-3x+3  if -3&lt;x≤0-3+2 cosh x  if…

Q: If f is differentiable and non-negative on an interval I in R then /3f + 1 is differentiable on I.

A: The given statement is true. Because when f will be non negative then the derivative of the given…

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

A:

Q: 13. Use the Intermediate Value Theorem to show that f(x) = 2x° – 3 has a zero in the interval [1,2].…

A: Given:-f(x)=2x3-3 using the intermediate value theoremf(x) is continuous in the interval 1,…

Q: If f (x) is a real function such thet f(0) = 0 and f'(x) = Vi-x² for -1 <x< 1, then without using…

A:

Q: Consider the function f(x) = x² – 4x + 4 on the interval [0, 4]. Verify that this function satisfies…

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Q: The function f (x) ,x € [-27, 2n] is continuous on: 1-cos x

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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: Find the absolute extrema of f on [-2, 3] if f(x) = x² - 6x - 1 O √5 O 2

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.