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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How do you find the limit of $\lim\limits_{x\rightarrow 0^-}\frac { \arcsin{ \frac {x^2-1}{x^2+1}}-\arcsin{(-1)}}{x}$?

I'm trying to calculate the following limit, involving $\arcsin$, where $x$ is getting closer to $0$ from the negative side, so far with no success. The limit is: $$\lim_{x\rightarrow 0^-}\frac { \arcsin{ \frac…
rboy
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Show that a set is not bounded

Show that the set $ \left\{\dfrac{1}{x^2-1}\mid x\in(0,1)\right\} $ is not bounded. We should assume that it is bounded, then try to prove the opposite, but I don't know where to start.
Lyndt
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Evaluation of $ \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right)dx$

Evaluation of $\displaystyle \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right)dx$ $\bf{My\; Try::}$ We can write it as $$I = \displaystyle \int_{0}^{1}\sqrt[4]{1-x^7}dx-\int_{0}^{1}\sqrt[7]{1-x^4}dx$$ Now Using $$\displaystyle \bullet…
juantheron
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Finding equations of tangent lines that are parallel

I have no idea how to do this problem at all. Find equation of the tangent lines to the curve $$y = \frac {x-1}{x+1}$$ that are parallel to the line $x-2y = 2$ I found the derivative of $y = \frac {x-1}{x+1}$ to be $\frac 2{x^2 + 2x +1}$ and…
user138246
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An exponential equation

Need to solve the equation $$(x+1)^{x-1}=(x-1)^{x+1}$$ After applying logarithm on each side one obtains the following equation: $$f(x+1)=f(x-1)\text{, where }f(x)=\ln x/x $$ which doesn't seem to have a solution judging from the graph of $f$. What…
user264653
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Find the derivative of the following integral

Find the derivative of $f(x)= \int_x^0 \frac{\cos(xt)}{t} dt$. My first reaction was to apply the FTOC, but I don't believe I can do this because $\frac{\cos(xt)}{t}$ is not defined at $t=0$ and thus it is not continuous in the interval of…
Poko
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what concept am I supposed to apply if second derivative is given.

Given: $f^{\prime\prime}(x)$ is continuous, $f(\pi) = 0$, and $$\int_0^\pi (f(x)+f^{\prime\prime}(x))\sin(x) \, dx = 2.$$ Find: $f(0)$. I know integration by parts etc, but I do not know which particular concept(s) I'm supposed to apply for this…
TPR
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For nonnegative continuous $f$, if $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0$, find the value of $f(1)$.

Let $f(x)$ be a non-negative continuous function such that $f'(x)-f(x)\leq 0, \forall x\geq 0$ and $f(0)=0,$find the value of $f(1)$. $f'(x)-f(x)\leq 0$$\Rightarrow f'(x)\leq f(x)$$\Rightarrow \frac{f'(x)}{f(x)}\leq 1$$\Rightarrow…
Brahmagupta
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Minimum value of $a+b$

If the graph of $f(x)=2x^3+ax^2+bx$ intersects the $x$-axis at three distinct points, then what is minimum value of $a+b$? Here $a$ and $b$ are natural numbers. My attempt: As the graph intersects the $x$-axis at three distinct points, it has $2$…
Vinod Kumar Punia
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Prove or disprove: if $|f(x)|\leq x^2$ then $f$ is differentiable at $0$

I'm trying to prove/disprove the following statement: If $|f(x)|\leq x^2$ for all $x$, then $f$ is differentiable at $x=0$. My initial attempt: $|f(0)|\leq0\Rightarrow f(0)=0$. Then, because $-x^2\leq f(x)\leq x^2$ we get that…
ikj
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If $3x^2$ is the derivative of $x^3$, how can $f'(x)$ be a linear map?

Suppose I have the function $f(x)=x^3$. The derivative is obviously $f'(x) = 3x^2$. But $3x^2$ is nonlinear since $$f'(3x) = 27x^2$$ $$3f'(x) = 9x^2$$ Therefore this isn't a linear map. Rudin defines the following $$f(x+h)-f(x) = f'(x)h +…
Stan Shunpike
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What is the use of the chain rule?

While studying calculus at home, I reached derivatives, and a book mentioned the chain rule. The book didn't go into much detail, and the internet searches gave me little information, so I was hoping that someone could enlighten me on this…
RK01
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What is the $d$ used in calculus?

I know the letter $d$ is commonly used in calculus and represents a derivative. Does this $d$ act as a variable that can be simplified or as a function of another variable?
RK01
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Prove there exists $x$ such that $f'(x)=\sin x$

Let $f$ be a function which is differentiable on $[0, \frac{\pi}{2}]$, such that $0\leq f'(x)\leq1$ for all $x$ in this interval. I'm asked to prove that there exists $x\in [0, \frac{\pi}{2}]$ such that $f'(x)=\sin x$. I believe that I should use…
user249860
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Calculate the surface area of a solid of revolution

I have to calculate the surface area of the solid of revolution which is produced from rotating $f: (-1,1) \rightarrow \mathbb{R}$, $f(x) = 1-x^2$ about the $x$-axis. I do know there is a formula: $$S=2 \pi \int_{a}^b f(x) \sqrt{1+f'(x)^2}\,…
Huy
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