3. Consider a function f(x) = sin() : R → R. x²+1 (a) Show that f is continuous on R.

Q: 1- f(x,y) = In(x² + y?) at (1,1)

A: As per our guidelines we are supposed to answer only one question. Kindly repost others question as…

Q: About what point is the local linear approximation l(x): of the function f(x) = tan x valid? 4

A: According to question, Local linear approximation,lx = x2 +π4-12,         .......1Given Function,fx…

Q: Determine the value of a such that f is differentiable at x = 1. for x <1 a sin(x-1)+6 for x21 S(x)…

A: given , f(x)=x+22-3      for x&lt;1asinx-1+6      for x≥1 we have to determine the value of "a" such…

Q: Let xª sin , x + 0; f(x) ={ 0, x = 0, where a > 0 is a parameter. 1. For which values of a is f(x)…

A:

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: Let A E R"Xn, let b e R", and let n n f (x) = xT Ar + 6Tx = Aikr;xk +> b;x;. i,k=1 i=1 Show that f :…

A: Definition: A function f:ℝn→ℝm is differentiable at x0 with derivative M if M:ℝn→ℝm is linear and…

Q: (tan ) 1/x TC + x 4 * f(x)D. k, For what value of k, f (x) is x =0 continuous at x = 0?

A: The

Q: Assuming that z is a differentiable function of x and y, Let F(x.y.2) = In(x³y) – e" -tan-z. Find F,…

A: We can find the answer as below.

Q: Let f(x) = e2sinz 1. Use the local linear approximation of f at x = 0 to approximate e2 sin(0.01

A:

Q: Let F(x) –3 cos(ť²) dt. Use the Fundamental Theorem of Calculus to find F'(x). F'(z) =

A: Given: F(x)=∫x−1(−3cos(t2))dtF'(x)=ddx∫x−1(−3cos(t2))dt=?

Q: Is the function continuous on the interval 0, 27? 1+(sin z)? O yes O no cannot be determined

A:

Q: Determine f(5) if f is a continuous function that satisfies +sin(a) f+sin(a) f(t)dt = 22² for all x.…

A: I am going to solve the problem by using some simple calculus to get the required result of the…

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: Find the average of the function g (x) = 2 cos 2x on the interval [0, r] -T

A: The function f(x)=2 cos 2x on the interval 0,π

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 continuous on [0, 1]. What is lim n→ ∞ 1 S fo 0 f(x) dx?

A:

Q: "For a continuous function f(x), defined on an interval a , b, we have d f(x) dx = f(x), dx a…

A:

Q: Suppose that f (x) is a continuous function, f(-1) = 6, f(2) = and f(x)dx: = -3 = T. Evaluate the…

A:

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

A:

Q: Find the linearization L(x)of the function at a .

A:

Q: Find an antiderivative F(x) of the function f(x)%3D7sinx/12+cosx such that F(0)=1

A:

Q: Show that the function is continuous. f(x) = cos(x 2)

A:

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: 3.Find the linear approximation to f(x) = cos x at x, = ÷ and use the linear approximation to…

A: We will use formula for linear approximation.

Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx&lt;0x20≤x&lt;12−x1≤x&lt;2x−3x≥2

Q: Find ao of the function f(x) 1-Cosx f(x) =

A:

Q: 4. Determine the set of points at which the function is continuous. cos(xy) b) f(x,y) = e? – y? '

A: we have to determine the set of points at which the function is continuousgiven function,…

Q: Suppose that f (x) has a continuous derivative, сп/2 f(-1/2) = 0, f(T/2) = 1/4 and f (x) dx = 3.…

A: The method of substitution is a very significant method in evaluating definite integral. The…

Q: Is the following piecewise defined function f(x) continuous at x = = 0, [cos(x), x≤0 f(x) = x²+x+1,…

A: The given function is: fx=cos x,       x≤0x2+x+1,  x&gt;0 Check whether the function is continuous…

Q: Determine the set of values of the continuous function defined by the formula: T COSX b) g(x)= sin 6

A:

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: Determine if f is continuous at the indicated values. x = 0 f(x) = 1 sin x x + 0 a) x = 0. ? b) x =…

A: Given

Q: Suppose that g(x) is a continuous function, g(0) = 2, g(1/2) = 1 and g(x)dx = 3. Evaluate the…

A:

Q: Determine f(5) if f is a continuous function that satisfies t 4x+sin(rz) f(t)dt = x for all z. %3D

A:

Q: Consider the function x sin x> 0 f(x) 0, a. Show that f is continuous at x = 0. b. Determine f' for…

A: The given function can be written as (a) To check the continuity at x=0 Since, Therefore, f(x) is…

Q: 4. If f is a differentiable function of x and y with w = f(x, y), x = r cos 0 and y = r sin 0, show…

A: Given, w=f(x,y) x=rcosθ and y=rsinθ

Q: Suppose f is continuous on the closed interval [–1,3]. If F is an antiderivative of f, which of the…

A:

Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

A:

Q: Find the antiderivative F for f(x) = | sin z|, -T <x <0 satisfying F(0) = 1. (1) F(x) = - cos x – 1…

A:

Q: Prove that the function f(x)=sinπ|x| is continuous at x = 0 but not differntiable at the same point.

A:

Q: If z=f(x,y), x= r cos θ and y= sin θ, where f is a differentiable function expressing az/ax and…

A:

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: 2. i) Give an example of a nonzero continuous function f on [0,1] such that f(x)dx = 0.

A: We have to give an example of a nonzero continuous function f  on [0,1] such that  ∫01f(x)dx = 0

Q: What is the continuous extension of f(x) = at point x-1. x -1 x2-1' %3D x +x-2 find dy dx

A:

Q: Find an antiderivative F(x) of the function f(x)= 7sinx / 12+cosx such that F(0)=1.

A: Now,

Q: (2) Is the function f(x) : R → R, sin a 1 ifa # πh,k & Z, ifa = Tk, k ε Z f (x) Borel measurable?

A: Consider the given function, fx=1sinxifx≠πk,k∈Z1ifx=πk,k∈Z   Thus, the given function have infinite…

Q: 3. Consider the function g(x) defined by { x² sin(1/x) if x > 0, g(x) = if x <0. (a) Show that g(x)…

A:

Q: B) Show that f(x) sin for x * 0. for x= 0. is not continuous.

A: