Question

Recall the Mean Value Theorem: If f : [a, b] → R is continuous on [a, b], and differentiable on (a, b), then there exists c ∈ (a, b) such that f(b) − f(a) = f 0 (c)(b − a). Show that this is generally not true for vector-valued functions by showing that for r(t) = costi + sin tj + tk, there is no c ∈ (0, 2π) such that r(2π) − r(0) = 2πr 0 (c).

Answer #1

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Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 9, [0, 2]
Yes, f is continuous on [0, 2] and differentiable on
(0, 2) since polynomials are continuous and differentiable on .No,
f is not continuous on [0, 2]. No,
f is continuous on [0, 2] but not differentiable on (0,
2).Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.There is...

Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous
such that f is twice differentiable on (a, b), meaning f is
differentiable on (a, b), and f' is also differentiable on (a, b).
Suppose further that there exists c ∈ (a, b) such that f(a) >
f(c) and f(c) < f(b).
prove that there exists x ∈ (a, b) such that f'(x)=0.
then prove there exists z ∈ (a, b) such...

1. Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = x3 + x − 5, [0, 2]
a) No, f is continuous on [0, 2] but not differentiable
on (0, 2).
b) Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
c) There is not enough information to verify if this function
satisfies the Mean Value Theorem.
d) Yes, f is continuous on [0,...

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

Does the function satisfy the hypotheses of the Mean Value
Theorem on the given interval? f(x) = 3x2 + 3x + 6, [−1, 1]
No, f is continuous on [−1, 1] but not differentiable
on (−1, 1).
There is not enough information to verify if this function
satisfies the Mean Value Theorem.
Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
No, f is not continuous on [−1, 1].Yes, f is...

1aDoes the function satisfy the hypotheses of the Mean Value
Theorem on the given interval?
f(x) = 4x2 + 3x + 1, [−1,
1]
a.No, f is continuous on [−1, 1] but not differentiable
on (−1, 1).
b.Yes, it does not matter if f is continuous or
differentiable; every function satisfies the Mean Value
Theorem.
c.Yes, f is continuous on [−1, 1] and differentiable on
(−1, 1) since polynomials are continuous and differentiable on
.
d.No, f is not continuous on...

Prove the IVT theorem
Prove: If f is continuous on [a,b] and f(a),f(b) have different
signs then there is an r ∈ (a,b) such that f(r) = 0.
Using the claims:
f is continuous on [a,b]
there exists a left sequence (a_n) that is increasing and
bounded and converges to r, and left decreasing sequence and
bounded (b_n)=r.
limf(a_n)= r= limf(b_n), and f(r)=0.

To illustrate the Mean Value Theorem with a specific function,
let's consider f(x) = x^3 − x, a = 0, b = 5. Since f is a
polynomial, it is continuous and differentiable for all x, so it is
certainly continuous on [0, 5] and differentiable on (0, 5).
Therefore, by the Mean Value Theorem, there is a number c in (0, 5)
such that
f(5) − f(0) = f '(c)(5 − 0).
Now f(5) = ______ , f(0) =...

Let C be a closed curve parametrized by r(t) = sin ti+cos tj
with 0 ≤ t ≤ 2π. Let F = yi − xj be a vector field.
(a) Evaluate the line integral xyds. C
(b) Find the circulation of F over C. (c) Find the flux of F
over C.

prove f(z) = sin(7z) is uniformly continuous with the
use of mean value theorem.

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