The tutorial question I am working on is,
(a) Attempt
We can use mean value theorem since
##(c: \mathbb{R} \rightarrow \mathbb{R}~countinity ) \implies (c: [-d, d] \rightarrow [c(-d), c(d)]~countinity)##
Thus ##c: [-d, d] \rightarrow [c(-d), c(d)] ## is differentiable on ##(-d, d)##, then...
$\tiny{3.2.15}$
Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$.
$f(x)=\sqrt{x} \quad [0,4]$
Are the secant line and the tangent line parallel...
Hi there
I'm prepping for a big test tomorrow and I'm really struggling with this question:If f′′(x)≥−1, x belongs to (−15,15), and f′(1)=3, find the interval over which x is definitely increasing.I'm struggling with substitution because I just don't seem to have enough values. Is there a...
Homework Statement
http://prntscr.com/daze68
What I don't understand:
1. "P be a polynomial with degree n"
do these equations satisfy this description?:
$$p(x) = (x^2 + x)^n$$
$$p(x) = (5x^2 + 2x)^n$$
etc.
2. "C1 is a curve defined by y=p(x)"
c1 is essentially just the curve of the...
Homework Statement
Show that The tangent at (c,ec) on the curve y=ex intersects the chord joining the points (c-1,ec-1) and (c+1,ec+1) at the left of x=c
Homework Equations
Legrange's mean value theorem
The Attempt at a Solution
f'(c)=ec
Applying LMVT at c-1, c+1...
Homework Statement
for ##0<\alpha,\beta<2##, prove that ##\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]##
Homework Equations
Mean value theorem: ##f'(c)=\frac{f(b)-f(a)}{b-a}##
The Attempt at a Solution
I got the answer for the question but I have made an assumption but I don't know if...
Homework Statement
Let ###f## be double differentiable function such that ##|f''(x)|\le 1## for all ##x\in [0,1]##. If f(0)=f(1), then,
A)##|f(x)|>1##
B)##|f(x)|<1##
C)##|f'(x)|>1##
D)##|f'(x)|<1##
Homework Equations
MVT: $$f'(c)=\frac{f(b)-f(a)}{b-a}$$
The Attempt at a Solution
I first tried...
Can anyone give me an easy way to find extremas and how to use the mean value theorem. This is the first thing in calculus where I read and reread and have no idea what to do when I get to the problems. It just doesn't make sense to me..
Any help is appreciated. Thank you.
EDIT: Basically my...
1. Homework Statement
Give a graphical argument that if f(a)=g(a) and f'(x)>g'(x) for all x>a, then f(x)>g(x) for all x>a. Use the Mean Value Theorem to prove it.
2. Homework Equations
3. The Attempt at a Solution
I have sketched a graphical argument to show that f(x)>g(x)...
1. Homework Statement
Give a graphical argument that if f(a)=g(a) and f'(x)>g'(x) for all x>a, then f(x)>g(x) for all x>a. Use the Mean Value Theorem to prove it.
2. Homework Equations
3. The Attempt at a Solution
I have sketched a graphical argument to show that f(x)>g(x)...
I'm so bad with the Mean Value Theorem. Can someone help me prove that, if f(x)=sinx, that, for any given a and b, |sinb-sina|<=|b-a|. Explain if you could too, please. Thanks a lot.
Excuse the typing please, as I am posting from my phone.
Let f have domain [0,infty) and range in R. Suppose as x goes to infinity, f'(x) goes to a constant b. I wish to show that f(x)/x goes to b as x goes to infinity.
I have tried numerous applications of the MVT to solve this and cannot...
Homework Statement
Consider the function g: [0,∞) -> R defined by G(x)=x+e-2x
A. Use the mean value theorem to prove that |g(x2)-g(x1)|<|x2-x1| for all x1,x2 E [0,∞) with x1≠x2.
B. Find all fixed points of g on [0,∞).
Homework Equations
MVT: f'(c) = f(b)-f(a)/b-a.
The Attempt...
Hello!
The Mean Value thereom gives F(b)-F(a) = f(c).(b-a) Where f is F' and c is the value of x at which it's derivate is equal to the average rate of change over the interval a to b for F.
The Fundamental thereom of calculus also gives F(b)-F(a) = (\frac{1}{b-a} \left \left \int...
Homework Statement
Prove for all real x and y that
|sinx - siny| <= |x-y|
Homework Equations
It's a question from the Mean Value Theorem/Rolle's Theorem section.
The Attempt at a Solution
Honestly, I've tried. It looks somewhat similar to the triangle inequality (I think?), but...
Homework Statement
The mass, m(t), in grams, of a tumor t weeks after it begins growing is given by m(t) = [te^t] / 80 .
What is the average rate of change, in grams per week, during the fifth week of growth?
a.) 2.730
b.) 3.412
c.) 6.189
d.) 6.546
e.) 11.131
Homework...
suppose f and g are continues on [0,1]
and differentiable on (0,1)
and f'(x)g(x) differs f(x)g'(x)
for every x existing in (0,1)
prove that there is a point c in [0,1] so g(c)=0
??
for what purpose do i need to know that
"f and g are continues on [0,1]
and differentiable on (0,1)...
suppose f is a continues function on point x_0
prove that g(x)=(x-x_0)*f(x) differentiable on x_0??
calculate g'(x_0)
i tried to think like this:
if f(x) is continues on x_0 then lim f(x) as x->x_0 equals f(x_0)
mvt says f'(c)=[f(a)-f(b)]
cauchys mvt says...
Homework Statement
-x<sin(x)<x
Homework Equations
show the inequality using the mean value theorem.
The Attempt at a Solution
i try to find c but i keep getting tan(x) as the solution.
Homework Statement
Give an example of a function (necessarily discontinuous) that does not satisfy the conclusion of MVT for Integrals
Homework Equations
MVT for \int = \frac{1}{b-a}\int ^{b}_{a} f(x) dx
The Attempt at a Solution
So I should need one point of discontinuity on every interval...
Homework Statement
Let f be diff. on (0,infinity) If the limit of f'(x) as x->infinity and limit of f(n) as n->infinity both exist and are finite, prove limit of f'(x) as x->infinity is 0.
Homework Equations
Mean Value Theorem (applied below)
The Attempt at a Solution
Suppose a>0 and b>0...
The following two questions are practice problems that I have been stuck on.
Homework Statement
Use the Mean Value Theorem to show that e^x > 1 + x for all x > 0
Homework Equations
Mean Value Theorem: If f: [a,b] to R is continuous on [a,b] and differentiable on (a,b) then there...
Homework Statement
I am trying to see the geometric interpretation of the generalized MVT. It is not a homework problem, but would like to know how to interpret the equation
Homework Equations
[f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x)
The Attempt at a Solution
On...
Hello everyone, I'm stuck on a MVT question. Can someone please help me out? Its not really a homework question, I'm doing this question to enhance my understanding of various things.
Q. Where a < x_0 < b, suppose that f(x) is differentiable in (a,b) and f'(x_0) = 0. Suppose also that for...