Question

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 )) .

Answer #1

Explain why the mean value theorem applies to
f(x)=3x^2 defined on
[-2,1]. Find c that satisfies the conclusion of the theorem.

Find the maximum value of the directional derivative of the
function f(x,y)=cos(3x+2y) at the point (π/6,−π/8). Give an exact
answer.

Determine whether the Mean Value theorem can be applied to f
on the closed interval [a, b]. (Select all that apply.) f (x) = x7,
[0,1] Yes, the Mean Value Theorem can be applied. No, f is not
continuous on [a, b]. No, f is not differentiable on (a, b). None
of the above. If the Mean Value Theorem can be applied, find all
values of c in the open interval (a, b) such that f ‘(c) = f (b)...

Determine whether Rolle's Theorem can be applied to f
on the closed interval [a, b]. (Select all that
apply.)
f(x) = cos x, [π, 3π]
Yes.
No, because f is not continuous on the closed interval
[a, b].
No, because f is not differentiable in the open
interval (a, b).
No, because f(a) ≠ f(b).
If Rolle's Theorem can be applied, find all values of c in
the open interval (a, b) such that f '(c) = 0.
(Enter your answers...

a.) Can the mean value theorem be used on f(x)= 3x^2 - 6x +2 on
[-5, 7]? If so find c such that f ' (c)= [f(b)- f(a)]/ (b-a)

Determine whether the Mean Value theorem can be applied to
f on the closed interval
[a, b].
(Select all that apply.)
f(x) =
8 − x
, [−17, 8]
Yes, the Mean Value Theorem can be applied.
No, because f is not continuous on the closed interval
[a, b].
No, because f is not differentiable in the open
interval (a, b).
None of the above.
If the Mean Value Theorem can be applied, find all values of
c in the open...

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"

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 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...

# 12
In Exercises 9–12, determine whether Rolle’s Theorem
can be applied to f on the closed interval [a, b]. If Rolle’s
Theorem can be applied, find all values of c in the open interval
(a, b) such that f ′(c) = 0. If Rolle’s Theorem cannot be applied,
explain why not.
12. f (x) = sin 2x, [−π, π]

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