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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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Is there a tangent of $x\sin(1/x)$ at $x = 0$?

Edit: From the comment below it seems like the question behind is: How can we determine whether of not the function $f(x) = x\sin(x) / x$ has a tangent at $x=0$. My thought is that one would have to find $$ \lim_{x\to 0} \frac{x\sin(1/x) -…
Russ_Student
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Concavity changes

I'm done being confused by Galois theory and am back to being confused by elementary calculus. I have a polynomial of degree $m-1$ that is bounded by the curve $y = x^m$ and intersects it at a nonzero number of points. Does each of these…
badatmath
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Prove that $\int_0^{\infty}\left(\frac{\log (1+x)}{x}\right)^2dx$ converges.

Prove that $\int_0^{\infty}\left(\frac{\log (1+x)}{x}\right)^2dx$ converges. I'm not sure where to begin here, perhaps show that it is bounded by something? This is an exercise I'm doing to review calculus.
lightfish
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If two functions go to infinity at zero, does the difference go to zero?

If $\lim_{ x\to0} f(x) = \infty$ and $\lim_{ x\to0} g(x) = \infty$, then $\lim_{ x\to0} [f(x) − g(x)] = 0$. True or False??
Blurry8312
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Inverse Taylor series

Taylor series expansions assume that we can expand any "good enough" function in terms of its derivatives. My question is, could we make something similar with integrals, the inverse operator of derivation? I mean, could we expand f as…
riemannium
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Second Derivatives Using Implicit Differentiation

According to my textbook, the second derivative of \begin{equation*} y^{2}+xy-x^{2}=9 \end{equation*} is \begin{equation*} \frac{90}{(2y+x)^{3}}. \end{equation*} The problem states "Express $\frac{d^{2}y}{dx^{2}}$ in terms of $x$ and $y$." I've…
G.P. Burdell
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Calculating $\int \frac{\tan(x) + \tan^3(x)} {e^{\sec^2(x)} + e^{-\sec^3(x)}} \, \mathrm{d}x$

$$\int \; \frac{\tan(x) + \tan^3(x)} {e^{\sec^2(x)} + e^{-\sec^3(x)}} \, \mathrm{d}x$$ Analysis: This integral is complex due to the combination of: Rational function: $\tan(x)$ and $\tan^3(x)$ form a rational function where the degree of the…
Nsnansn Jwjwj
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Finite subset of element from a set always gives a sum of 2 or less

There is a similar question to this already on MSE, but I would like to ask about a different approach: The full question is, "Let $B$ be a set of positive real numbers with the property that adding together any finite subset of elements from $B$…
robert lewison
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Prove that, if limit exists, then there are $\delta>0$ and $M>0$ such that $|f(x)| \le M$

Suppose that $\lim\limits_{x \rightarrow p}f(x) = L$. Prove that there are $\delta>0$ and $M>0$ such that, $0<|x-p|<\delta \implies |f(x)| \le M$ That's how I did: Given that $\lim\limits_{x \rightarrow p}f(x) = L$, then for any $\epsilon>0$ there…
Ruan Carlos
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Prove for $f(x)=x+\frac{1}{x}$

Let $f(x)=x+\frac1{x}$ prove that $|f(x)-f(1)| \le (1+\frac1{x})|x-1|$ for $x>0$ I've tried to get it from the left hand side $|f(x)-f(1)|=|x+ \frac1{x}-2|=|\frac{x^2-2x+1}{x}|=|\frac{x-1}{x}||x-1|$ However, I can't see how to proceed from here to…
Ruan Carlos
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What is the intuitive reasoning behind taking the exponent from a variable and moving it to the front when taking a derivative?

For example in the function $x^2$ if you want to find the slope at a point you would take the derivative of the function at that point which would be the slope of the tangent line at that point. I understand intuitively understand that the tangent…
Stef
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How to calculate $ \int \frac{x^5}{\sqrt{x^2 + 1}} \, \mathrm{d}x $

Problem.1) Evaluate $$ \int \frac{x^5}{\sqrt{x^2 + 1}} \, \mathrm{d}x $$ I'm still learning how to integrate, watching the GOAT of calculus, BlackPenRedPen. it would be quiet helpful if anyone could help me understand this. Thanks.
Ian Hendrick Pugoy
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Show that the function $x^4 – 3x^2 + x-10$ cannot have a root inside $(0,2)$.

Show that the function $x^4 – 3x^2 + x-10$ cannot have a root inside $(0,2)$. Please note that roots of $f'(x)$ cannot be found using a calculator. Attempted the question by calculating $f'(x)$ and assuming that at least one root of $f(x)$ exists in…
Completed
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Integral Relation

Grading some exams I noticed that the following is true: $$\int_1^3 (3x+1)(x-1)dx = \int_1^3(3x+1)dx +\int_1^3(x-1)dx.$$ Of course in general $$\int_a^b f(x)g(x)dx \neq \int_a^bf(x)dx+\int_a^bg(x)dx.$$ But is there a general statement of which the…
Aeryk
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