For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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3
votes
2answers
64 views

why is the chain rule used for the area function $A=\frac{1}{2}xy$

To differentiate the area of a triangle function, $A=\dfrac{1}{2}xy$ with respect to time $t$, my text says to use the chain rule and the product rule. So it would be: ...
0
votes
1answer
40 views

Calculus about inflection point, maxima and minima

How many inflection points does the graph of g(x) have? What is the global maxima and minima ? My answer was : There are 0 inflection point because they did not pass the Y axes so it did not have ...
0
votes
3answers
121 views

Question about acceleration

On this homework problem I'm pretty sure I know how to do it, but the answer choice is not there... ...
0
votes
1answer
38 views

For a 2 variable function, prove that the linear approximation is less than the real value for all x and y

Consider $f : \Bbb R^{2} → \Bbb R$ defined by $f(x,y) = x^{2} + 3y^{4}$. Prove that $f(x,y) ≥ L(x,y)$ for all $(x, y)$ in $\Bbb R^{2}.$ I found the linear approximation: $$L = f(x,y) + f'x(a,b)(x-a) ...
5
votes
2answers
227 views

Integral Calculus, right or wrong?

I have two questions, which I decided and my feedback does not match that of my book, but do not seem to be wrong. Calculate the volume of the solid obtained by rotating the region bounded by the ...
1
vote
3answers
622 views

How to calculate a derivative using the “Power Rule” If it includes a negative exponent?

So my understanding of the power rule is that you take your problem with an exponent like this: $x^5 = 5x^4$ or for $x^n$, $f'(x)=nx^{n-1}$ However, it does not seem to be working for me when ...
3
votes
2answers
76 views

What am I doing wrong with this derivative? (Calculus)

I've been doing derivatives with the formula: Definition of a Derivative: for every $x$ plugin $(x+h)$, then subtract original from the equation. This means for $x^2$, I get: $$\frac{(x+h)^2 - ...
1
vote
1answer
37 views

critical points - calculus

$$f(x) = 5x^{1/5} − x $$ $$f^\prime (x)= x^{-4/5} -1 $$ The question is: Find the $x$ values of all of the critical points. Enter your answers as a comma separated list. As far as I know that ...
0
votes
1answer
105 views

how to find the foci, directrix, center of a polar conic section. ($r=\frac{4}{5-4sin\theta} $)

I've been trying to figure this out for a bit and haven't found an answer. the equation is this: $r=\frac{4}{5-4sin\theta} $ I know I need to match this up to a conic graph so I divide top and ...
1
vote
1answer
28 views

Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
1
vote
1answer
55 views

Find $\lim_{x\to-1} f(x)$ for $f(x) = (x^2 - 2x - 3) / (x+ 1)$

I need to find the following limit: $$\lim_{x\to -1}\frac{x^2 - 2x - 3 }{x + 1}$$ The polynomial is simplified to $\dfrac{(x+1)(x-3)}{x+1}$ Hello, I can solve this by plugging in the value $-1$ ...
1
vote
1answer
429 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
1
vote
1answer
53 views

Calc 3 double integral

Compute the double integral of $f(x,y)=3\sin(5x)$ over the domain $D$ bounded by $x=0, x=\frac{\pi}{10}, y=0, y=\cos(5x)$. I am having trouble solving this double integral. I know that I must go ...
0
votes
2answers
98 views

L'Hôpital's Rule and Infinite Limits

I was wondering if anyone could help me with computing a limit using L'Hôpital's Rule. Using L'Hôpital Rule for the following limit, I get the following result: \begin{equation} \lim_{x \to 0} ...
3
votes
1answer
45 views

$f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$

Is there exists $f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$ for all ...
2
votes
2answers
44 views

Find the value of derivative, given that the tangent line passes through a particular point

If the line tangent to the graph of the function $f$ at the point $(2,7)$ passes through the point $(-3,-3)$ then $f'(2)$ is...? A. 5 B.1 C. 2 D.7 E. Undefined I don't understand how to do this. I ...
0
votes
2answers
58 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
3
votes
1answer
65 views

Integrating via partial fractions

My question is this: Why is it that fractions have to be split up in a very specific manner? For example if I have $\frac{5x}{(x+1)^2}$ this fraction HAS to be split up like this:$$\frac ...
1
vote
0answers
45 views

Dense subsets in $L^1(\mathbb{R})$

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$? set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports; set of above-mentioned functions' derivatives ...
2
votes
1answer
122 views

Proving $\displaystyle \int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$ Exists

As the title says, I need to show that $$\int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$$ exists. After performing the substitution $x = 1/u, dx = -1/u^2 du$, the integral becomes ...
0
votes
1answer
29 views

Integral over smooth, closed curve of vector field

Why doesn't vector field $v:\mathbb{R^3}\rightarrow\mathbb{R^3}$ given by $v(x,y,z)=(x,\cos y,e^z)$ does not meet $$\int_{\gamma} \langle {v\frac{\gamma'}{\|\gamma'\|}\rangle}\ d\sigma_1=0$$for every ...
0
votes
1answer
21 views

Surface measure of $A={\{(x,y,z)\in\mathbb{R^3}:z=f(x,y),x^2+y^2<1}\}$

Function $f:\mathbb{R^2}\rightarrow\mathbb{R},\ f\in C^{\infty}$ is Lipschitz continuous with constant $1$ and $$A={\{(x,y,z)\in\mathbb{R^3}:z=f(x,y),x^2+y^2<1}\}.$$ Why does it imply that ...
1
vote
1answer
77 views

How to establish the formula for area of a triangle using the axioms of area?

We have the following definition: AXIOMATIC DEFINITION OF AREA We assume there exists a class of $M$ of measurable sets in the plane (i.e. subsets of the plane whose area can be defined) and a set ...
2
votes
2answers
95 views

Squeeze theorem

Is there any trick when it comes two find two sequences with the same limit to prove that a third sequence that is between them has also a limit? There are a whole bunch of sequences with the same ...
11
votes
1answer
332 views

Integral: $\int_0^{\infty} e^{-ab\cosh x}\cos\left(ac\sinh(x)+\frac{ix}{2}\right)\,dx$

I am trying to solve this: $$\int_0^{\infty} e^{-ab\cosh x}\cos\left(ac\sinh x+\frac{ix}{2}\right)\,dx$$ I don't have much ideas about the problem. I thought of writing $\cos ...
1
vote
1answer
21 views

completeness of $A=\{(x_1,x_2)\in\mathbb R^2| \max\{ |x_1|,|x_2|\}<1\}$

Let $A=\{(x_1,x_2)\in\mathbb R^2| \max\{ |x_1|,|x_2|\}<1\}$ and $g(x_1,x_2)=\frac14(x_1^2x_2,x_1+1)$. I want to show that $g$ has a fixed point in $A$. So ...
2
votes
1answer
61 views

Solving $\int \frac{\sqrt{1-(f'(x))^2}}{f(x)}dx$

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a smooth function satisfying $f(x) >0$ and $|f'(x)| \leq 1 $ for all $x$. Is it possible to solve the indefinite integral \begin{equation} \int ...
3
votes
1answer
44 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
0
votes
2answers
63 views

Writing a formula with the given limits

I'm stuck on a math question regarding creating possible formulas for a graph with limits. The question is as follows: Sketch a graph of a rational function f that satisfies the following conditions: ...
3
votes
1answer
129 views

Dirichlet's function expressed as $\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)$ [duplicate]

How can we see that Dirichlet's function $$D(x):=\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)= \begin{cases} 1 & x\in\mathbb Q\\ 0 & x\notin\mathbb Q\\ ...
0
votes
1answer
78 views

Help needed in solving a differential equation

Please help me in solving: $$a^2z+\frac{\partial^2z}{\partial x^2}-\frac{\partial^2 z}{\partial y^2}=0$$ ($a$ is a constant) I plugged this in Wolfram Alpha and it outputs that this is a second ...
0
votes
0answers
33 views

A function is discontinuous at all rational points and continuous at all irrational points [duplicate]

Define $f(x)$ for $x\in[0,1]$ by $f(\frac pq)=\frac1q$ if $p$ and $q$ are relatively prime, and $f(x)=0$ if $x$ is irrational. How can we see that $f$ is discontinuous at all rational points and ...
4
votes
2answers
410 views

limits using $ \epsilon - \delta $ to prove two variable function

I'm trying to use the $ \epsilon - \delta $ argument to prove $\lim_{(x,y) \rightarrow (1.1)} \frac{2xy}{x^2+y^2} =1$. I know that I need to show that $\forall \epsilon>0, \exists \delta>0$ ...
2
votes
2answers
32 views

how to calculate integrate about Heaviside

everyone,here I have a question about how to calculate $$\int e^t H(t) dt$$ where $H(t)$ is Heaviside step function thank you for your answering!!
2
votes
4answers
491 views

limit as goes to infinity… I thought it is simple

I have to calculate the Asymptotes in the infinity(and minus infinity) for this function: $f(x)=((x-7)(x+4))^{1/2}$ I know that $\lim_{x \to\infty} f(x)/x= 1$ And I get into trouble with: ...
1
vote
0answers
95 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
1
vote
1answer
182 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
1
vote
5answers
143 views

Derivative of a function is the equation of the tangent line?

So what exactly is a derivative? Is that the EQUATION of the line tangent to any point on a curve? So there are 2 equations? One for the actual curve, the other for the line tangent to some point on ...
3
votes
4answers
149 views

Finding Integral of $y = e^{ax}\sin (bx)$

I was solving questions like $ \int e^{2x} \sin x dx. $ I decided to find the general term for $$\int e^{ax} \sin (bx) $$ which can be directly used in questions like (above). I hope it helps others ...
1
vote
2answers
138 views

Indefinite integral of $x^x$

I've seen many many questions on the internet with answer that it cannot be done with elementary functions. Now I did this integration myself and got a pretty nice result. Since I've seen so many ...
0
votes
4answers
74 views

Decide convergence of this series

How to prove the series $$\sum_{n=1}^\infty \frac {e^n\cdot n!}{n^n}$$ diverges? I tried D'Alambert and result is 1 and I'm stuck with Raabe.
0
votes
3answers
54 views

Evaluate this limit

I'm stuck at evaluating this limit $$\lim_{k \to \infty} \left( \frac{2}{a^{1/k}+b^{1/k}} \right)^k, \quad a,\, b>0$$ I tried binomial expansion but didn't seem to work. Can anybody give me a ...
2
votes
1answer
895 views

How to prove that Δy/Δx = f(x+Δx)-f(x) / Δx?

How do I prove that $$\frac{\Delta y}{\Delta x} = \frac{f(x+Δx)-f(x)} {Δx}.$$ I know that this is the slope formula to find the derivative of a function $y=f(x)$ and I know that the formula for a ...
2
votes
1answer
29 views

if $f([a,b])=[c,d]$ and $[c,d] \subset [a,b]$, is there $x \in [c,d]$ such that $f(x)=x$?

I'm trying to prove something that I'm not sure is correct. Let $f$ be a continuous, differentiable and monotonic function $f:[a,b] \to [c,d]$, where $[c,d] \subset [a,b]$. Is there an $x \in [c,d]$ ...
0
votes
4answers
88 views

Can a limit of a function be not an integer?

I'm just taking calc, and all the teacher's examples gave only integer results. Is it possible to have fractions or decimals?
1
vote
3answers
96 views

What will x=3 be in this equation

Assume f is differentiable and has just one critical point, at x = 3. In parts (a) - (d), you are given additional conditions. In each case decide whether x = 3 is a local maximum, a local minimum, or ...
2
votes
2answers
169 views

Find the inflection points in the graph

The question is which of the $x$-values of the given points are inflection points of the function $f(x)$ itself? I chose $C,F$ and $H$ because at this point the $f'(x)$ is zero. But my answer was ...
3
votes
1answer
112 views

I need help with a double sum [closed]

I need help with the following exercise: determining the value of the next sum: $$\sum _{ m=1 }^{ \infty }{ \sum _{ n=1 }^{ \infty }{ \frac { { m }^{ 2 }n }{ { 3 }^{ m }(n{ 3 }^{ m }+m{ 3 }^{ n }) ...
0
votes
1answer
222 views

True/ False to strength my knowledge in calculus

Q1) For a continuous function $f$ whose domain is all real numbers, if $f'(p)$ is undefined, then $x = p$ could be a local maximum or minimum of $f$. I say that at corners it could be maxima or ...
4
votes
4answers
147 views

What's the integral of $\int_0^\infty \frac{dx}{(x^4+1)^5}$

$$\int_0^\infty \frac{dx}{(x^4+1)^5}$$ My answer would be : $\dfrac{\Gamma(\tfrac{1}{4})\Gamma(\tfrac{19}{4})}{4\Gamma(5)}$ Solution: You can use this technique. – Mhenni Benghorbal