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

learn more… | top users | synonyms (2)

0
votes
2answers
35 views

Finding local minimum under constraint

How to find the minimum of $f(x) = ||x-\mu||^2$, where $\mu = (1, 1)$ and $< x, \mu > = 0$ (the inner product is $0$)?
1
vote
1answer
63 views

Getting stonewalled on computation of $2\times 2$ Hessian matrix

The question: Let $z \in R^N$, and let $f(z) = \log[1^T z] \in R$. I am told that the Hessian matrix of this function is the following: $$ H = \frac{1}{1^Tz}\Big[ 1^Tz \mathrm{diag}(z) - zz^T \Big] ...
1
vote
2answers
72 views

Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$ f'(x)=\frac{|x|}{x} $$ and $$ f''(x)=2\delta(x). $$ Can you help me?
0
votes
1answer
28 views

How to get good derivatives results in matlab?

I got the set of some points, and I need to get derivatives from it. I try to use the diff() function in matlab but, well, results kinda not ok. Is there any better ...
1
vote
1answer
389 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
7
votes
2answers
330 views

Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}},\, \,n\in\mathbb N$?

Is there a function that is analytic in an open neighbourhood of $z=0$ and satisfies $$f\left(\!\dfrac 1 n\!\right)=\dfrac 1 {\sqrt{n}},$$ for all natural number $n$? I guess this problem requires ...
0
votes
1answer
38 views

Related Rates Differentiation

Consider a ramp modeled by the function $y = \frac{1}{x}, x>0$. A ball slides down the ramp so that the x-coordinate of its position at any time $t$ seconds is increasing by a rate of $f(x)$ ...
3
votes
0answers
76 views

Related Rates - Differentiation

An airplane is flying at an altitude of 8 miles and passes over a radar station. When the airplane is 12 miles from the base of the station, the radar detects that its horizontal distance is ...
0
votes
4answers
55 views

Derivative of a function!

I can't seem to figure this problem out! It asks to find the derivative of $\displaystyle g(x)=\frac{4e^{x/4}}{(x^2+x)(1-x)}$. Can you help me figure this out? Thanks~
0
votes
1answer
36 views

Derivatives with exponentials!

I have a calculus test that I'm studying for but I've run into some problems! For one, the problem asks to compute the derivative of $f(x)= (1-2x)e^{-x}$. I totally get that ...
-1
votes
1answer
62 views

Convergence, Divergence and Summability of this series

If f(x) is an infinitely differentiable function at x=0 and $f^{(n)}(0)$ is the nth derivative of the function f at zero, then does the series below converge or diverge? $\sum_{n=0}^{\infty} ...
0
votes
0answers
41 views

Efficient approximation of derivatives of an integral

Suppose $ \phi(z) $ is the probit function (http://en.wikipedia.org/wiki/Probit). And $$ Z = \int \phi(\mathbf{w}^\top \mathbf{x}) \mathcal{N}(\mathbf{w}; \mathbf{\mu}, \mathbf{\Sigma}) d\mathbf{w} ...
1
vote
1answer
36 views

Why are the uncertainties so different?

Here is my scenario: I am trying to calculate the uncertainty of the function $y=x^2$, that is, I want to find $\Delta y$, and I found that we can get a great difference in the $\Delta y$, depending ...
0
votes
1answer
67 views

Differentiation of bilinear form w.r.t. matrix

I need to do a derivative of bilinear form: b'C a w.r.t to Kx1 vector t where "b" and "a" are Kx1 vectors and "C" is KxK matrix that depends on vector t (and a and b are independent of t). Does anyone ...
0
votes
1answer
189 views

Find acceleration at the first instant when a car has zero velocity.

The position of the front bumper of a test car under microprocessor control is given by: $x(t)=2.17m+\left(4.8\frac{m}{s^2}\right)t^2-\left(.100\frac{m}{s^6}\right)t^6$ Find its acceleration at the ...
1
vote
3answers
52 views

Question regarding multivariable chain rule…

Suppose that $f : \mathbb R^2 \to \mathbb R$ is some function, and $g :\mathbb R^2 \to\mathbb R$ is defined by $g(x, y) = f(f(x, y), x)$. Write $d(g(x, y))/dx$ and $d(g(x, y))/dy$ in terms of partials ...
2
votes
3answers
41 views

Calc Optimization problem with open top

A rectangular box with an open top has a volume of 4500 ft^3. The base is made of slate, and the sides are made of glass. Slate is 3 times the price of glass per sq. ft. What dimensions minimize the ...
0
votes
1answer
20 views

How do I evaluate the integral of this function?

The substitution is u= 5x+11 This is what I have done but it says the answer is incorrect. $$ \frac{1}{5} \cdot \frac{1}{u^3} \ \rightarrow \ -\frac{1}{5} \cdot \frac{1}{(5x+11)^2} \ $$
0
votes
2answers
38 views

How can I find the derivative G'(x) of this function?

I'm not sure how to do this question because there is another variable t inside it. Find G'(x).
0
votes
3answers
288 views

Need a good non-graphing calculus calculator that can derive and integrate

It's my 3rd time going through Calculus II and I want to pass this class already. I'm not trying to cheat I just see people with fancy calculators and I have never even used a calculator on tests ...
0
votes
2answers
63 views

How does expanding by Taylor's theorem work here?

The problem I am trying to figure out a step in the proof of this book (p. 245), where it goes like this: \begin{equation}\tag{a}\label{eq:equal} F_i(x^0; t + dt) = F_i(x; dt)\end{equation} ...
0
votes
1answer
26 views
0
votes
1answer
87 views

How to calculate extreme values for functions with three variables using derivatives?

I'm given the following function: $$f:\mathbb R^3\to \mathbb R\\ f(x,y,z) = x^3 + y^2 + z^2 + 12xy + 2z$$ I know the algorithm for calculating extreme values for functions with two variables. Does ...
0
votes
1answer
38 views

Quotient rules?

Find an equation of the line tangent to the graph of $$f(x)= \frac{2x^2}{3x-1}$$ at the point $x=1$. I feel like this should be relatively easy. I know how to take the derivative of the top portion, ...
0
votes
1answer
39 views

Example: Differentiable, NOT locally lipschitz?

Everybody Good Evening, I'm wondering: Is being differentiable (Gateaux or Frechet) at ONE point enough for being locally lipschitz? If not, can you provide a counterexample? Thx in advance!
1
vote
1answer
26 views

Optimization of cake pan volume from area of pan

It was difficult to accurately word this question, so hopefully a bit of context will clear that up. Context: I have a cake dish that is made by cutting out squares from the corners of a 25cm by 40 ...
2
votes
3answers
48 views

function has partial derivatives but is not differentiable

Can you write me an example of function which has partial derivatives but is not differentiable? How could I create and prove the function like that?
0
votes
1answer
20 views

Question about this Calculus problem

I have a small question about how to approach this problem: $\dfrac{d}{dx}\left((-4x^2-9)^6 * (6x^2+17)^{15}\right)$ Am I correct in thinking that I should do the Product Rule and THEN the Chain ...
1
vote
0answers
25 views

Prove the partial derivative of the summation of $$(y-g_i+a\sum^n_{j=1} g_j)=-1+na >0$$

I have a function: $$\pi_i^1=y-g_i+a\sum^n_{j=1}g_j,$$ where 0 < a<1< na, and I need to prove this: $$\frac{\partial(\sum^n_{i=1}\pi^1_i)}{\partial g_i}=-1+na>0.$$ I am not very ...
0
votes
2answers
34 views

Finding maximum and minimim of function on an interval. Are there multiple ones?

I found myself stuck at such basic problem. If you're to calculate local maximum and minimum on closed interval, $\langle a, b\rangle$, the $a$ and $b$ may as well be the maximum and minimum points of ...
1
vote
1answer
48 views

Derivative of remainder function.

I cannot find a derivative of remainder function (i.e. derivative of a(x) mod b(x) with respect to x, and ...
4
votes
1answer
457 views

If $f$ is a twice differentiable function and $f(2^{-n}) = 0 $, for all $n \in \mathbb N$, then $f^\prime(0) = f^{\prime\prime}(0) = 0$.

Let $f : \mathbb R \to \mathbb R$ be a twice differentiable function, such that $f(2^{-n}) = 0$, for all $n \in \mathbb N$ . Show that $$f^\prime(0) = f^{\prime\prime}(0) = 0.$$ My attempt. First, ...
0
votes
0answers
44 views

Prove that $P_n(1-x)=(-1)^n P_n(x)$ if $n \geq 1$ P is bernoulli polynomials

A sequence of polynomials is defined inductively as follows: $P_0(x)=1; P'_n(x)=nP_{n-1}(x)$ and $ \int^1_0 P_n(x)dx=0$ if $n \geq 1$ (f) Prove that $P_n(1-x)=(-1)^n P_n(x)$ if $n \geq 1$ I ...
2
votes
0answers
87 views

Is it possible to switch limit from inside to outside of integral in this case?

Let $C$ be an open connected subset of $\mathbb{C}$. Let $f:[a,b]\times C \rightarrow \mathbb{C}$ be a function. Assume $f(-,z):[a,b]\rightarrow \mathbb{C}$ is continuous and $f(t,-):C\rightarrow ...
1
vote
2answers
50 views

Find the partial derivative

How to find the partial derivative of this function, with respect for x? $$ h(x,y)=\frac{1}{\ln(e^x+y)}, y>0 $$
0
votes
2answers
34 views

Derivative Question: How to do the following

How should I take the derivative of the following: $$\frac{t^2}{(1+t^4)^{1/2}}$$ I know the answer and I have tried quotient rule and product rule and I can't seem to succeed.
3
votes
2answers
45 views

Derivative of a polynomial

First of all, I would like to say I'm new to Mathematics StackExchange, so pardon me if there're any mistakes (until I read the right formatting rules!). That said, we are currently learning ...
0
votes
1answer
23 views

Can someone help with this question please? (derivatives)

In this question, you will prove the quotient rule for derivatives using the product and chain rules. Let $h$ and $g$ be differentiable functions, with $g(x) > 0$ for all $x$, and let ...
1
vote
1answer
57 views

Starting velocity by distance, time, and friction

I am writing a game in Javascript, and I just got a big math problem, where $\text{friction} = 0.97$. This is what is being looped every $1000$ / $60$ milliseconds, to make the projectile move all ...
1
vote
2answers
229 views

Derivative of exponential function proof

I'm looking for a straight forward proof using the definition of a derivative applied to the exponential function and substitution of one of the limit definitions of $e$, starting with $e = ...
3
votes
4answers
82 views

Differentiability of $\mathrm{max}(x, y)$ at $x=y$

I am trying to figure out differentiability of $\mathrm{max}(x, y)$. Intuitively, it should not be differentiable at $x=y$, since it changes direction "non-smoothly" at those points. I can not, ...
2
votes
0answers
71 views

Can differentiation be done in other fields besides $\mathbb R$ or $\mathbb C$

I was reading some completely unrelated materials where some differentiation was used, but I realized that the discussion was over some field $k$, where $k$ was not specified. So that made me curious: ...
2
votes
1answer
85 views

evaluate $\int_{0}^{1}\frac{x-1}{\ln x}dx$ [duplicate]

evaluate $\int_{0}^{1}\frac{x-1}{\ln x}dx$,where x is real. Approach: The suggestion is to differentiate $H(m)=\int_{0}^{1}\frac{x^{m}-1}{\ln x}dx$. This leads to ...
6
votes
3answers
80 views

$p(x)\geq 0 \forall x\Rightarrow p(x)+p'(x)+p''(x)+…+p^{(n)}(x)\geq 0$ [duplicate]

$p(x)\geq 0 \forall x \in \mathbb{R} \Rightarrow p(x)+p'(x)+p''(x)+...+p^{(n)}(x)\geq 0$, where p(x) is a polynomial of degree n. I showed: $a_{n}+...+a_{0}\geq 0$, ...
2
votes
2answers
80 views

Prove that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$

Hi everyone I have been trying to prove that that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$ . Heres my attempt: LS: $ \int \limits_a^b f(x) dx$ = $ \int \limits f(b) - \int ...
-1
votes
2answers
46 views

Derivative limit at infinity=1 , then the function limit is infinity? [closed]

Derivative limit at infinity = 1 , then the function limit is infinity? how do i prove this?
1
vote
2answers
63 views

Worst case examples of non-differentiability of the Riemannian distance function

Let $g$ be a $C^\infty$ Riemannian metric on the plane, and let $p$ be a point on the plane. Let $X$ be the set of points $x$ at which the Riemannian distance $d(p,x)$ is not differentiable. How bad ...
0
votes
3answers
61 views

Differentiate $5(\sqrt{x})(x^3−5\sqrt{x}+5)$

I don't exactly remember how to approach this problem: $5(\sqrt{x})(x^3−5\sqrt{x}+5)$ However, I know that derivatives of the Xs are: \begin{align*} \sqrt{x} &\to \frac{1}{2\sqrt{x}} \\ ...
1
vote
2answers
62 views

How can the point of inflection change before the vertical asymptote?

I have to draw a graph of a function which seems to have an inflection point AFTER the vertical asymptote. i.e. f(x) = $\tan^{-1}\left({\frac{x-1}{x+1}}\right)$ Using the quotient rule, I get... ...
4
votes
2answers
56 views

Fundamental theorem of calculus with functions of $x$

I have these two homework questions.Using fundamental theorem of calculus and basic (A-level) facts about intergration find $F'(x)$ when: $$F(x)=\int_{a(x)}^{b(x)} f(t) dt$$ I solved this by appling ...