Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

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

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

134529 questions
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Calculating $\lim_{n \to \infty}\frac{n^{3}}{(3+\frac{1}{n})^{n}}$

I need help to calculate this limit: $$\lim_{n \to \infty}\frac{n^{3}}{(3+\frac{1}{n})^{n}}$$
Talik
  • 51
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type of discontinuity

$f(x) = \frac{1}{x}\cdot\sin(\frac{1}{x})\cdot\cos(\frac{1}{x})$ $f : \mathbb R \backslash \{0\} \rightarrow \mathbb R$ I need to specify the type of discontinuity at $x_{0} = 0$ (type 1 - jump, type 2 - essential, or removable). Here is what I…
Lisa
  • 41
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Why is $\int \frac 1u du = \ln u$?

Why, in $u$-substitution if $u$ appears to the $-1$ power it becomes equivalent to the $\ln u$? I don't have a specific problem where that is the case, but I do recall this being a rule of thumb. Can someone explain this to me?
Mone Skratt Henry
  • 847
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What is the difference between implicit differentiation and differentiation?

I know that in problems dealing with rates of change that implicit differentiation is used, however how does this compare to just "regular" differentiation? The terminology differentiation just means finding the derivative
Mone Skratt Henry
  • 847
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finding the limit of $\lim_{x\to 0} \frac{e^x-1}{x^2}$

$$\lim_{x\to 0} \frac{e^x-1}{x^2}$$ It is an expression in form of $\left(\frac{0}{0}\right)$. Using l'Hôpital: $$\lim_{x\to 0} \frac{e^x}{2x}$$ The expression in form of $\left(\frac{1}{0}\right)$ so one-sided limits should be checked $$\lim_{x\to…
gbox
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find a function such...

exercise : A function $f$, continuous on the positive real axis, has the property that $$\int_{1}^{xy}f(t)dt =y\int_{1}^{x}f(t)dt +x\int_{1}^{y}f(t)dt$$ for all $x > 0$ and all $y > 0$. If $f (1) = 3$, compute $f (x)$ for each $x > 0$. My…
Elll
  • 369
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Is $|\ln|x||$ differentiable?

Is $|\ln|x||$ differentiable for all $x$ is defined and continuous? I can see that on the graph that it is not differentiable at $-1$ and $1$, but how can I prove it? So I look at $\lim_{h\to 0+} \frac{|\ln(1-h)|-|\ln(1)|}{h}=\lim_{h\to 0+}…
gbox
  • 12,867
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Open/closed intervals and infinity

I'm wondering whether the definition of closed interval given here is correct. Quoting part of it: If one of the endpoints is $+\infty$ or $-\infty$, then the interval still contains all of its limit points (although not all of its …
user216094
  • 945
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What is the difference between the Fundamental Theorem of Calculus 1 and 2?

I am learning this currently in calculus but I don't understand the actual difference logically. I can answer the questions but I don't know why the answer is what it is. Any help is greatly appreciated.
Mone Skratt Henry
  • 847
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Convergence of a improper integral

Test the convergence of the improper integral $\int_1^\infty \frac{1}{x^2(1+ e^{-x})} dx$. I have tried to change the fraction as $\frac{e^x}{x^2 (1+ e^x)}$ then it can be simplified as $\frac1{x^2} - \frac1{x^2 (1+ e^x)}$ . Now I couldn't solve…
Balaji
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Let $b_0, b_1,...,b_n$ be real numbers with the property that

Let $b_0, b_1,...,b_n$ be real numbers with the property that $$ b_0 + \frac{b_1}{2} + \frac{b_2}{3}+...+\frac{b_n}{n+1}=0 $$ Prove that the equation $$ b_0 + b_1x + b_2x^2+...+b_nx^n=0 $$ Has at least one solution in the interval $(0,1)$ How can I…
james42
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Revolving Beacon confusing calculus problem

This is not homework. I have the solutions, I just can't understand why mine are wrong. The problem states: A revolving beacon $3600$ feet off a straight shore makes $2$ revolutions per minute. How fast does its beam sweep along the shore, (a) at…
jeremy radcliff
  • 4,805
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Why do we take $x=\cos t$, $y=\sin t$ for a parametric circle when we can take the opposite?

Both $x = \cos t$, $y = \sin t$ and $x = \sin t$, $y = \cos t$ describe a circle So why is the first parameterization so commonly used in mathematics, and not the second?
Jason
  • 3,563
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Evaluation of $\displaystyle \int\frac{1}{x^{\frac{1}{3}}+x^{\frac{1}{4}}}dx+\int\frac{\ln(1+x^{\frac{1}{6}})}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}dx$

Evaluation of $\displaystyle \int\frac{1}{x^{\frac{1}{3}}+x^{\frac{1}{4}}}dx+\int\frac{\ln(1+x^{\frac{1}{6}})}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}dx$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{1}{x^{\frac{1}{3}}+x^{\frac{1}{4}}}dx\;,$ Now Put…
juantheron
  • 53,015
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$\lim_{x \rightarrow 0^+}$ $(\sin x)^{\ln x}$

My question is, what is $\lim_{x \rightarrow 0^+}(\sin x)^{\ln x}$? This limit is equal to $$\lim_{x \rightarrow 0^+} e^{\ln(\sin x)\ln x} = e^{\lim_{x\to0^+} \ln(\sin x)\ln x}$$ But what is right hand side limit of $\ln(\sin x)\ln x$ when x goes…
corcia candy
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