Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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20 views

AP Calculus BC - Derivative of inverse problem

Let $g(x)$ be the inverse of the function $f(x)$. Given the following values on the table below, at which value $x=a$ will $g'(a)=1/6$? (No calculator allowed) ...
5
votes
3answers
345 views
0
votes
1answer
17 views

Fourier Integral Theorem?

I have this function $x(t)=\left|\frac{t}{T}\right|rect(\frac{t}{2T})$ The book states that given the function x(t) is piecewise linear, we can use the Fourier theorem to calculate X(f). They get the ...
0
votes
1answer
29 views

AP Calculus BC - Polar curve question

A particle moving along the polar curve given by $r = 2 + 2\sin(\theta)$ has position $(x(t),y(t))$ at time $t$, with $\theta = 0$ when $t = 0$. This particle moves along the curve so that ...
0
votes
1answer
20 views

Chain rule for $f(X(t), Y(t))$ where $X, Y : R \to R^2$

I'm having some trouble on understanding how to calculate the derivative of $g(t)$ with regards to $t$, where $g(t) := f(X(t), Y(t))$ and $X(t)$ and $Y(t)$ are $2d$ vectors. That is $X,Y: R \to R^2$. ...
1
vote
2answers
27 views

Equation of a line tangent to $g(x)$ and parallel to line connecting endpoints of $g(x)$

Let $g(x)$ be a differentiable function defined on the interval $0 \le x \le 16$. Some values of $g(x)$ and its derivative $g'(x)$ are given below. Which of the following is the $x-intercept$ of the ...
3
votes
3answers
68 views

Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
1
vote
1answer
78 views

Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
0
votes
3answers
17 views

Alternating sign Nth derivative

Say I have a function $$ f(x) = \dfrac 1x$$ and I'm looking at its $n^{th}$ derivative and trying to come up with a formula. I can easily get it because if forms a very consistent pattern and it ...
0
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0answers
21 views

velocity and acceleration of a disk rotating at a constant speed

A disk of radius 1 is rotating in the counterclockwise direction at a constant angular speed ω. A bug starts at the center of the disk and moves directly toward edge. The position of the bug at time ...
1
vote
4answers
71 views

$\int \frac{1}{\sqrt{x^2+1}} dx$

So I've seen some options on the internet that are fairly good, but I have this substitution: $x^2+1=t-x$, you square both sides and get $x = (t^2-1)/t$ and $x + 1 = (t^2-1)/2t + 1$. If we call that ...
0
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0answers
44 views

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal.

Find the points on the curve $y=x+e^x$ at which the tangent line is horizontal. The answer was $(0,1)$, but I don't get it. I tried to take the derivative of the function and equal it to $0$ ...
2
votes
2answers
46 views

The Cantor staircase function and related things

The Cantor staircase function https://en.wikipedia.org/wiki/Cantor_function has an interesting property: $\{x\colon f'(x)\neq 0\}$ is a nowheredense nullset. But it it differentiable almost ...
2
votes
1answer
31 views

Find the derivative of each of the the following functions

Find the derivative of each of the the following functions. $f(x)=\sqrt{7+\sqrt{x^3}}$ $\frac{d}{du}\left(\sqrt{u}\right)\frac{d}{dx}\left(7+\sqrt{x^3}\right)$ A: ...
2
votes
1answer
39 views

The set where a derivative vanishes is G-delta

If $f:I\to R$ ($I$ - interval) is differentiable, then $\{x\colon f'(x)=0\}$ is a $G_{\delta}$ set. The lecturer didn't prove this fact and I found no proof in my books. How it can be proven?
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2answers
34 views

Derivative of n x n Invertible Matrix

For an invertible $n$ x $n$ matrix $A$, define $f(A):=A^{-2}$. Calculate the derivative $D\space f(A)$. (i.e. give $D\space f(A)B$ for arbitrary $B$.) I'm not super sure how to go about this?
11
votes
3answers
253 views
+50

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
0
votes
0answers
5 views

Accuracy Rebonato Swaption Approximation Formula among Different Strikes

Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes? My foundings: Let $0 ...
-2
votes
1answer
106 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
0
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0answers
13 views

Equality of mixed partial using double limit

As a student, my teacher told me to give a proof for the equality of mixed partial. The Theorem stated that, Supposed $f$ is a real value function of 2 variable $x$ and $y$ and $f(x, y)$ is defined ...
4
votes
3answers
130 views

Prove that $f(ab) = f(a) + f(b)$

Question : Assume only that $f: (0,\infty)\to{\mathbb{R}}$ is differentiable and that $f'(x) = 1/x$, and $f(1)=0$. Prove that for all $a,b \in(0,\infty)$, $f(ab)=f(a)+f(b)$. [Hint: Let $g(x)=f(ax)$] ...
1
vote
0answers
23 views

Logistic model - solution verification

I'm looking at the Logistic model: $$\begin{cases} \dot{X} = X(1-X)\\ X(0) = X_0 \end{cases}$$ where the phase space is $M = \mathbb{R}$. The solution appears to be $X(t) = \dfrac{1}{1 + ...
2
votes
2answers
69 views

How to properly find supremum of a function $f(x,y,z)$ on a cube $[0,1]^3$?

Solving an applied problem I was faced with the need to find supremum of the following function $$f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3}$$ where $f\colon\ [0,1]^3\backslash\{(1,1,1)\} ...
1
vote
0answers
31 views

Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?

As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64) There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
1
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0answers
20 views

How to prove second order differentiation matrix is of the form..

Given that the matrix: $$D2 = \left[\begin{matrix}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{matrix}\right]$$ is a second-order differentiation matrix in the sense, for a ...
2
votes
3answers
31 views

I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$.

I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$. My logic is I want to find whether is $f'(5)>0$ or $f'(5) < 0$. I need to use the ...
1
vote
1answer
8 views

What does it mean for the difference $V(\overline{x}_L -dx) -V(\overline{x}_L)$ to be of the second order in $dx$

What does it mean for the difference $V(\overline{x}_L -dx) -V(\overline{x}_L)$ to be of the second order in $dx$, where $dx$ is some tiny increment of $x$? What we know about $V$: $V(z) = U(z) - ...
2
votes
2answers
1k views

Differentiating with respect to the limit of integration

I'm confused about problems involving differentiation with respect to the limit of an integral, I just want to check that my understanding is correct. For example, are the following statements ...
0
votes
0answers
23 views

Compute $\lim_{t\to0}\frac{f(2+t,3+t)-f(2,3)}{t}$ using the partial derivatives of $f$

How can I solve the question? I know that I need to work with the definition of the partial derivatives with respect to $x$ and $y$. $$f'_y(2,3)=-3,\quad f'_x(2,3)=2\\ ...
4
votes
3answers
43 views

Differentiability of function for $\Bbb{Q}$ and $\Bbb{R}\setminus \Bbb{Q}$

A function $f:\Bbb{R}\to\Bbb{R}$ is defined by $f(x)=x$, if $x$ is rational; $\sin(x)$ if $x$ is irrational. Show that $f$ is differentiable at $0$ and $f'(0)=1$. Here I'm thinking to apply ...
11
votes
5answers
1k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
2
votes
0answers
73 views

Functions $g(x)/h(x),h(x)/f(x)$ are constant [duplicate]

Suppose $f$, $g$, $h$ are functions from the set of positive real numbers into itself satisfying $f(x)g(y)=h(\sqrt{x^2+y^2})$ for all $x$, $y\in (0,\infty)$. Show that the functions $g(x)/h(x)$, ...
1
vote
2answers
66 views

Functions $f(x)/g(x), g(x)/h(x),h(x)/f(x)$ are constant

Suppose $f,g,h$ are functions from the set of positive real numbers into itself satisfying $f(x)g(y)=h(\sqrt{x^2+y^2})$ for all $x,y\in (0,\infty)$. Show that the functions $f(x)/g(x), ...
1
vote
1answer
61 views

How to find $\frac{\mathrm{d}y}{\mathrm{d}x}$ when both number in front and exponent have fractions?

I'm not sure how to solve this: $\frac{5}{9}x^\frac{2}{3}$. I applied the product rule and have $\frac{2}{3}\frac{5}{9}x^{-\frac{1}{3}}$. $\frac{30}{9}x^{-\frac{1}{3}}$, then ...
2
votes
1answer
380 views

Second derivative of a composite function

Say, we have three Banach spaces $X, Y, Z$ and $g:X \to Y, \ \ f:Y \to Z$ are twice (Fréchet) differenciable. The question is: what is $(f \circ g)''$? Since $(f \circ g)'':X \to ...
1
vote
1answer
18 views

Find the maximum value of the function

So I was just messing around with finding the maximum and minimum values of functions, and I came across this: $$ \text{Find the maximum value of} \,\, f(x)=\frac1{x^{2x^2}}.$$ Any ideas?
6
votes
2answers
106 views

Chain Rule and Vector valued functions?

Let $f: R^n \to R$ be given by $f(x) = \frac{||x||^4} {1 + ||x||^2}$ . Use the chain rule to show that $f$ is differentiable at each $x \in R^n$ and compute $Df(x)$. This vector valued stuff just ...
1
vote
2answers
22 views

$f(x, y) = \prod_{i = 1}^n (1 + xy_i)$, what is ${{{\partial f}\over{\partial x}}\over f}$, geometric series?

Let$$f(x, y) = \prod_{i = 1}^n (1 + xy_i).$$What is$${{{\partial f}\over{\partial x}}\over f}?$$What happens when we use the geometric series?
0
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0answers
24 views

Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
2
votes
4answers
62 views

$f:[0,1] \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in [0,1]$,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in [0,1]$

$f:[0,1] \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in [0,1]$,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in [0,1]$ I used the definition of derivative: $f'(x)=|\lim_{h ...
0
votes
1answer
31 views

Why are scale factors not always unity?

A scale factor in curvilinear coordinates is defined as $$h_v \equiv \left|\frac{\partial\vec{r}}{\partial v}\right|$$ where $\vec{r}=(x,y,z)^T$ is a position vector. The partial differential can be ...
0
votes
2answers
28 views

Prove that there exists some real number θ satisfying 0 < θ < 1 for which f '''(θ) = 0

Let f: D → R be a 3-times differentiable function defined over an open interval D, where 0 ∈ D and 1 ∈ D. Suppose that f(0) = f '(0) = 0 and f(1) = f '(1) = 0. Prove that there exists some real ...
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1answer
24 views

Question about applying the Chain Rule with multiple variables

Let $z = u(x,y)$ and $y = y(x)$ and $u(x,y(x))$ = 0. What is the second derivative of the function $y(x)$? I tried to use chain rule but I keep making mistakes
3
votes
1answer
47 views

Function that is second differential continuous

Let $f:[0,1]\rightarrow\mathbb{R}$ be a function whose second derivative $f''(x)$ is continuous on $[0,1]$. Suppose that f(0)=f(1)=0 and that $|f''(x)|<1$ for any $x\in [0,1]$. Then ...
0
votes
2answers
52 views

What is y'' if $\sin y = y + 5x$?

I got $ 5\sin y / (\cos y - 1)^2$ as my answer, but the correct answer was given as $25\sin y / (\cos y - 1)^3$. My thought process: Derive the original equation to get $y'\cos y = y' +5$ $$y'(\cos ...
6
votes
4answers
10k views

An inflection point where the second derivative doesn't exist?

A point $x=c$ is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point. And a list of possible inflection points will be those points ...
2
votes
5answers
3k views

Check my workings: Show that $\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$

Let $f''$ be continuous on $\mathbb{R}$. Show that $$\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$$ My workings ...
2
votes
2answers
28 views

Is there a solution to this differential equation?

I am trying to find a function $y(x)$ that is a solution to $$ \left(a_3 x^3+a_1 x\right) y''(x)-\left(3 a_3 x^2+2 a_1\right) y'(x)+3 a_3\, x \,y(x)=a_0 x^4+a_2 $$ I tried using mathematica but it ...
1
vote
2answers
28 views

Implicitly finding the derivative of $f^{-1}(x)$ given $f(x)$

Can we find the derivative of the inverse of a function implicitly by finding the derivative of the original function? For example lets say I have $f(x) = e^x$ and I want to find the derivative of ...
1
vote
1answer
33 views

Where i am going wrong in finding normal to curve?

The question is Find the perpendicular distance between the normal to the curve $$x=a\cos t+at\sin t, y=a\sin t-at\cos t$$ and the origin. Equation is given in parameterized form. My attempt ...