Mean value theorem Definition and 150 Threads

Homework Statement By using mean value theorem , show that ln(1+x)<x whenever x>0 Homework Equations The Attempt at a Solution So there is another example in my book and they just use the formula f ' (c) = f(b)-f(a) / b-a but I am not sure how to work out my...

Homework Statement If f is strictly decreasing and differentiable on R, then f '(x) ≤ 0 for all x. Homework Equations Mean value theorem The Attempt at a Solution if f is strictly decreasing, then for any a,b\inℝ such that a<b, f(b)<f(a) or f(b)-f(a)<0. By the MVT, there exists a...

Homework Statement Use the Mean Value Theorem to prove that if p>1.then ((1+x)^p)>(1-px) for x in (-1,0)and(0,infinite) i have no idea that what's the relationship between the inequality and the theorem? first i define g(X)=((1+x)^p)-(1-px) then for x=0 f(0)=0.i.e. x not equals to 0,which...

Homework Statement The Attempt at a Solution My book is not explaining very well the steps at solving these problems. There is a step that I'm missing step 1. find 1/(b-a) easy step 2. find the antiderivative of 4 - x, easy, x^2/2 step 3. plug in what the result of the...

As usual, I typed up the problem and my attempt in LaTeX: Maybe I'm not applying the MVT correctly, but my result does not seem to help me solve the problem in anyway. What are your thoughts?

Homework Statement For \mu\geq 0, s\geq 1, prove that (1+s)^{\mu}\geq 1 + s^{\mu} Homework Equations The Attempt at a Solution I have written a proof involving the mean value theorem and derivatives, but there must be a simpler way! I think this should be done purely...

Hey guys, I'm having a lot of trouble with this calculus problem for homework. If anyone can help me and provide the necessary steps if possible, I would greatly appreciate it. Here is the problem: Given the function f(x) = x^2 - 1, find the number(s), c, that satisfy the Mean Value Theorem...

Homework Statement For what values of a,m, and b does the function satisfy the hypothesis of the Mean Value Theorem of the interval [0,2]. Homework Equations (f(b) - f(a))/(b-a) = f'(c)The Attempt at a Solution So, I wanted to make a point of continuity for the whole equation. So, I set the...

f is continues in [0,1] and differentiable in (0,1) f(0)=0 and for x\in(0,1) |f'(x)|<=|f(x)| and 0<a<1 prove: (i)the set {|f(x)| : 0<=x<=a} has maximum (ii)for every x\in(0,a] this innequality holds \frac{f(x)}{x}\leq max{|f(x)|:0<=x<=a} (iii)f(x)=0 for x\in[0,a] (iiiן)f(x)=0 for...

Homework Statement let g be a function mapping x to xcosx-sinx. use the mean value theorem to prove that g(x) < 0 for x in (0,pi]Homework Equations well the function is both continuous and differentiable on the interval so that's a start... The Attempt at a Solution basically i thought i'd...

Homework Statement What set(s) are a and b assumed to be elements of? Does the mean Value Theorem make a universal claim (for all a ... for all b) or an existential claim (there exists a ... there exists b) about a and b? Explain how you came to this conclusion. Homework Equations We...

Homework Statement Prove the Mean Value Theorem for Integrals Proof Let f(x) be defined on [a,b] Let M be the max of f(x) and m be the min of f(x) Then m \leq f(x) \leq M \int_{a}^{b}m \;dx\leq \int_{a}^{b} f(x)\;dx \leq \int_{a}^{b} M\;dx m(b-a) \;dx\leq \int_{a}^{b} f(x)\;dx \leq...

Homework Statement let x0, x1,... be the approximations of pi from the Newton's Method. Use Mean Value theorem to show that |pi-xj+1|=|tan2cj||pi-xj| for some cj between xj and pi Homework Equations pi is defined as smallest positive number r when sin r =0 The Attempt at a Solution I have...

Homework Statement Suppose that f is continuous on [a,b], and dy/dx f(a+)< u < (f(b) - f(a))/(b-a) that there exists a point c so that u(c-a) = f(c)-f(a) Homework Equations The Mean Value Theorem, Intermediate value theoremThe Attempt at a Solution I defined (f(b) - f(a))/(b-a) = dy/dx...

Homework Statement Homework Equations The Attempt at a Solution

Homework Statement At 2:00PM a car's speedometer reads 50km/h. At 2:10PM it reads 65km/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 90km/h^2. Homework Equations Mean Value Theorem If f is continuous on [a,b] and f is differentiable on (a,b) then there...

The claim: If f:[a,b]\to\mathbb{R} is integrable, and \phi:[a,b]\to\mathbb{R} is monotonic (hence continuous almost everywhere), then there exists \xi\in ]a,b[ such that \int\limits_a^b f(x)\phi(x)dx \;=\; \big(\lim_{x\to a^+}\phi(x)\big) \int\limits_a^{\xi} f(x)dx \;+\; \big(\lim_{x\to...

1. Suppose that f and g are continuous on [a,b] and differentiable on (a,b). Suppose also that f(a)=g(a) and f '(x)<g '(x) for a<x<b. Prove that f(b)<g(b). [Hint: Apply the Mean Value Theorem to the function h=f-g]. 2. {[f(b)-f(a)]\b-a}=f'(c) 3. I know: If h(x) = f(x) - g(x) then h(a) = f(a) -...

Homework Statement Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. Homework Equations f(x)=3cos(x), [-pi/4, pi/4] The Attempt at a Solution well i do f(b)-f(a)/b-a and get zero but that's not the answer...

Homework Statement piecewise...f(x) = {3, x=0 -x^2+3x+a, 0<x<1 mx+b, 1<= x <= 2 Homework Equations (f(b) - f(a))/(b - a) = fprime(c) The Attempt at a Solution I am just really lost at what this is trying to ask...sorry :( I tried plugging in stuff in the equation but I'm...

Homework Statement hello , if f(x) is a function which satisfies the mean value theorem , where :- f(x) = \left\{ {\begin{array}{*{20}c} {2x^3 - x + 1\quad \quad x \in [0,1]} \\ {3x^2 - x\quad \quad \quad x \in (1,3]} \\ \end{array}} \right. find the value of (c) by using the...

Homework Statement Let a>b be Real numbers and f, g: [a,b] --> R be continuous and differentiable on (a,b) Show g is injective on [a,b] if g'(x) != 0 for all x in (a,b) Homework Equations Rolle's theorem: Continuity and differentiability (in the conditions above) imply that f(a) =...

Homework Statement Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f(t)=g(t)-h(t), where g and h are the position functions of the two runners.] Homework Equations If this is ever...

Hi, I've been watching the MIT lectures on single variable calculus, and whilst proving FTC, he mentions that we since we know that: <$> f'(x) = g'(x) </$>, then by MVT we know that <$> f(x) = g(x) + C </$>. I have tried searching for somewhere where this implication is spelled out for me...

Suppose that the conditions for the Mean Value Theorem hold for the function f : [a, a + h] → R, so that for some θ ∈ (0, 1) we have f (a + h) − f (a) = hf ′ (a + θh). Fix f and a, and for each non-zero h write θ(h) for a corresponding value of θ. Prove that if f ′′ (a) exists and is non-zero...

Homework Statement Assuming Rolle's Theorem, Prove the Mean Value Theorem. Homework Equations - The Attempt at a Solution I know these definitions: Rolle's Theorem: If y=f(x) is continuous on all points [a,b] and differentiable on all interior points (a,b), and if f(a)...

Homework Statement Use the mean value theorem to show that if x ∈ ℝ>0 then 0 < ( x + 1)^1/5 − x^1/5 < (5x^4/5)^-1 Homework Equations MVT: f(b) = f(a) + f ' (c)*(b-a) The Attempt at a Solution I can see that (5x^(4/5))^-1 is the differential of x^1/5, but I'm not sure what to let...

1. Use the Mean Value Theorem f(x) = sqrtX - 2x at [0,4] so a = 0, b = 4 3. So I found the derivative (which is the slope) and then set the derivative equal to the 1 because of f(b)-f(a)/b-a f'(x) = 1/2sqrtX - 2 so 1/2sqrtX - 2 = 0, then 1/2sqrtX = 2, then my solution is x = 1/16 Just would...

Homework Statement Let f(x) be a twice differentiable function on an interval I. Let f''(x)>0 for all x in I and let f'(c)=0 for some c in I. Prove f(x) is greater than or equal to f(c) for all x in I. Homework Equations Mean value theorem? The Attempt at a Solution f''(x)>0...

Homework Statement Let a0=0 and an+1=cos(an) for n≥0. a) prove that a2n≤a2n+2≤a2n+3≤a2n+1 for all n≥0. b) use mean value theorem to find a number r<1 such that |an+2-an+1| ≤ r|an-an+1| for all n≥0. Using this, prove that the sequence {an} is Cauchy. Homework Equations N/A The Attempt...

Homework Statement how to show using MVT that cos(cos x) is a contraction. Homework Equations | d/dx (cos(cos x)) | = | sin(cos x) sin(x) | < sin 1 < 1 The Attempt at a Solution Using that relation, the original problem is easily solved. My question is, how do we know: |...

Homework Statement Let x ϵ R such that f'(x) = 3x^2. Prove that f(x) = x^3 + c for some c ϵ R using the Mean Value Theorem. Homework Equations The Attempt at a Solution I used two functions f(x) and g(x) that have the same derivative namely f'(x). Applying the theorem I am able...

Homework Statement Let us suppose that, 3≤ f '(x) ≤5 for all x values. Show that 18≤ f(8) - f(2) ≤30. The Attempt at a Solution Alright folks... I am unsure where to start, or where to apply the MVT or the Rolle's Theorem. Thanks

1. Verify that the function satisfies the mean value theorem, then find all numbers c that satisfy the the conclusion of the mvt f(x)=e^(-2x) on the interval [0,3] 2. f'(c)=[f(b)-f(a)]/[b-a] 3. 1. f(x) is a composition of continuous functions, so f(x) is continuous on[0,3]. 2. f(x) is a...

Homework Statement Consider the function f(x)=2x^3−12x^2−72x+6 on the interval [−4,7] . Find the average or mean slope of the function on this interval. Homework Equations MEAN VALUE THEOREM f'(c) = \frac{f(b)-f(a)}{b-a} The Attempt at a Solution When I set this problem up in...

Homework Statement http://img14.imageshack.us/img14/6132/proiqc.jpg Homework Equations The Attempt at a Solution the first 3 are from the textbook so they must be right.. the last 2 I am pretty sure i got right too.. because the 4th one, if f'(x)=0 then f(x)= c .. so its false...

Homework Statement Let g:[0,1] \to \mathbb{R} be twice-differentiable (i.e. both g and g' are differentiable functions) with g''(x) > 0 for all x \in [0,1]. If g(0) > 0 and g(1) = 1, show that g(d) = d for some d \in (0,1) if and only if g'(1) > 1. Homework Equations The Attempt...

Homework Statement Let h be a differentiable function defined on the interval [0,3], and assume that h(0) = 1, h(1) = 2 and h(3) = 2. (c) Argue that h'(x) = 1/4 at some point in the domain. Homework Equations (a) Argue that there exists a point d \in [0,3] where h(d) = d. (b) Argue that at...

I'm reading Div, Grad Curl, and All That, and in coming up with a formula for the divergence, H.M. Schey starts with a small cube centered at (x,y,z), labels the face parallel to the yz-plane as S1 and calculates \int\int_{S_1}\mathbf{F}\cdot\hat{\mathbf{n}}dS=\int\int_{S_1}F_x(x,y,z)dS...

http://en.wikipedia.org/wiki/Mean_value_theorem#First_mean_value_theorem_for_integration" Take a look at the Wikipedia proof. Now, wouldn't it be easier to prove it like this: The ordinary mean value theorem says that G(b)-G(a)=(b-a)G'(\xi) And the fundamental theorem of calculus...

The mean value theorem in electrostatics states that for charge free space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any sphere centered at that point. In its derivation I'm getting a kind of strange result that is not...

Homework Statement If h1 and h2 are contractions on a set B with contraction constants δ1 and δ2 prove that the composite function h2 ° h 1 is also a contraction on B and find a contraction constant. Homework Equations |f(a) - f(b)| ≤ δ |a-b| f '(c) = (f(a)-f(b))/(a-b)...

6. The number c satisfying the Mean Value Theorem for f(x) = sinx on the interval [1,1.5]: So if the MVT is f(b) - f(a) / b-a .997 - .841 / 1.5 - 1 so .156 / .5 so .312 But that isn't the correct answer. Any thoughts?

1. If c is the value defined by the mean value theorem, then for f(x) = e^x - x^2 on [0,1], c= I found the two end points as [0,1] and [1,e-1], so the average slope is .71828... is that the answer then?

Homework Statement Suppose that f(x) is a twice-differentiable function defined on the closed interval [a,b]. If f'(c) = 0 for a < c < b, which of the following must be true? I. f(a) = f(b) II. f has a relative extremum at x = c. III. f has a point of inflection at x = c...

1. The problem statement: greek letters used here: xi, eta The Mean Value theorem applied to f(x,y) = sin(x^2 + y^2) implies with a = 0 and b = 0. sin(x^2 + y^2) = 2 xi cos( xi^2 + eta^2)x + 2 eta cos(xi^2 +eta^2) y find xi and eta or an accurate approximation to them as...

Homework Statement A number a is called a fixed point if f(a)=a. Prove that if f is a differentiable function with f'(x)=1 for all x then f has at most one fixed point. Homework Equations In class we have been using Rolle's Theorem and the Mean Value Theorem. The Attempt at a...

Homework Statement A company introduces a new product for which the number of units sold S is S(t)=200(5-(9/(2+t)) where t is the time in months a) Find the average value of S(t) during the first year b)During what month does S'(t) equal the average value during the first year...

Homework Statement Suppose f is continuos on [a,b]. Show that there exists c in (a,b) such that the integral from a to b of f(x)dx equals (b-a)*f(c) Homework Equations The Attempt at a Solution Tried using the mean value theorem to come up with a solution by rearranging different...

Homework Statement Suppose that 3 is < and equal than f'(x) which is also < and equal to 5 for all vales of x. Show that 18< or equal to f(8) -f(2) < or equal to 30. Homework Equations Mean Value theorem The Attempt at a Solution I have no clue where to start or which values...