Question

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) = ______, and f '(x) = ______ , so this equation becomes

______ = f '(c)(5) =(_____) 5 = ______ ,

which gives c^2 = ______ , that is, c = ± _______ . But c must be in (0, 5), so c = ______ .

The figure illustrates this calculation: The tangent line at this value of c is parallel to the secant line.

Answer #1

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 f(x) = x^3 - x
a) Find the equation of the secant line through (0,f(0)) and
(2,f(2))
b) State the Mean-Value Theorem and show that there is only one
number c in the interval that satisfies the conclusion of the
Mean-Value Theorem for the secant line in part a
c) Find the equation of the tangent line to the graph of f at point
(c,f(c)).
d) Graph the secant line in part (a) and the tangent line in part...

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,...

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...

1. Determine all value(s) of x=c guaranteed to exist
by the Mean Value Theorem for the
function f(x)=x3+8x2−6x+ 27
restricted to the closed interval [−2,1]
2. Explain what your answer found in part a means using the
words "secant" and "tangent"

1) Verify that the function satisfies the three hypotheses of
Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem. (Enter your answers as a
comma-separated list.)
f(x) = 1 − 12x + 2x^2, [2, 4]
c =
2) If f(2) = 7 and f '(x) ≥ 1 for 2 ≤
x ≤ 4, how small can f(4) possibly be?
3) Does the function satisfy the hypotheses of the Mean Value
Theorem...

Determine whether the Mean Value Theorem applies to f(x) =
cos(3x) on [− π 2 , π 2 ]. Explain your answer. If it does apply,
find a value x = c in (− π 2 , π 2 ) such that f 0 (c) is equal to
the slope of the secant line between (− π 2 , f(− π 2 )) and ( π 2
, f( π 2 )) .

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...

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).

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 5 minutes ago

asked 9 minutes ago

asked 17 minutes ago

asked 46 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago