Question

Theorem 4 states “If f is differentiable at a, then f is continuous at a.” Is the converse also true? Specifically, is the statement “If f is continuous at a, then f is differentiable at a” also true? Defend your reasoning and/or provide an example or counterexample (Hint: Can you find a graphical depiction in the text that shows a continuous function at a point that is not differentiable at that point?)

Answer #1

Let f(x) be a continuous, everywhere differentiable function.
What kind information does f'(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f'(x)?
Let h(x) be a continuous function such that h(a) = m and h'(a) =
0. Is there enough evidence to conclude the point (a, m) must be a
maximum or a minimum?...

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,...

Given that a function F is differentiable.
a
f(a)
f1(a)
0
0
2
1
2
4
2
0
4
Find 'a' such that limx-->a(f(x)/2(x−a)) = 2.
Provide with hypothesis and any results used.

Suppose that f is a twice differentiable function and that
its second partial derivatives are continuous. Let h(t) =
f (x(t), y(t)) where x = 2e^ t and y = 2t. Suppose that
fx(2, 0) = 1, fy(2, 0) = 3, fxx(2, 0) = 4, fyy(2, 0) = 1, and
fxy(2, 0) = 4. Find d ^2h/ dt ^2 when t = 0.

Consider the function g(x) = |3x + 4|.
(a) Is the function differentiable at x = 10? Find out using
ARCs. If it is not differentiable there, you do not have to do
anything else. If it is differentiable, write down the equation of
the tangent line thru (10, g(10)).
(b) Graph the function. Can you spot a point “a” such that the
tangent line through (a, f(a)) does not exist? If yes, show using
ARCS that g(x) is not...

Find an example of a function f that is (i) continuous on (1,
4), (ii) not uniformly continuous on (1, 4) and (iii) uniformly
continuous (2, 4)

5. Determine whether the following statements are TRUE or FALSE.
If the statement is TRUE, then explain your reasoning. If the
statement is FALSE, then provide a counter-example. a) The
amplitude of f(x)=−2cos(X- π/2) is -2 b) The period of
g(x)=3tan(π/4 – 3x/4) is 4π/3.
. c) If limx→a f (x) does not
exist, and limx→a g(x) does not exist, then limx→a (f (x) + g(x))
does not exist. Hint: Perhaps consider the case where f and g are
piece-wise...

4. Given the function f(x)=x^5+x-1, which of the following is
true?
The Intermediate Value Theorem implies that f'(x)=1 at some
point in the interval (0,1).
The Mean Value Theorem implies that f(x) has a root in the
interval (0,1).
The Mean Value Theorem implies that there is a horizontal
tangent line to the graph of f(x) at some point in the interval
(0,1).
The Intermediate Value Theorem does not apply to f(x) on the
interval [0,1].
The Intermediate Value Theorem...

Determine whether Rolle's Theorem can be applied to the function
f(x) = x^4-2x^2 [-2,2]. if not find c and explain why

(1 point) Find the degree 3 Taylor polynomial T3(x) centered
at a=4 of the function f(x)=(7x−20)4/3.
T3(x)=
? True False Cannot be determined The function f(x)=(7x−20)4/3
equals its third degree Taylor polynomial T3(x) centered at a=4.
Hint: Graph both of them. If it looks like they are equal, then do
the algebra.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 20 minutes ago

asked 31 minutes ago

asked 37 minutes ago

asked 46 minutes ago

asked 49 minutes ago

asked 51 minutes ago

asked 51 minutes ago

asked 51 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago