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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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Proving the value of a limit using the $\epsilon$-$\delta$ definition

I'm trying to solve the problem of showing that $$\lim_{x\to6}\left(\frac{x}{4}+3\right) = \frac{9}{2}$$ using the $\epsilon$-$\delta$ definition of a limit.
morcutt
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What are the coordinates of red point?

The function $f (x)=-x^2+4$ "in red" is moving along the line $y=x+4$ " in black " from green point to black point and becomes in the place of blue graph as shown in the following graph What are the coordinates of red point? I got the…
user373141
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Geometric Interpretation of Total Derivative?

Say that: $$z = xy$$ So: $${\partial z \over \partial x} = y$$ and $${\partial z \over \partial y} = x$$ If we plot in 3D space the 2D surface corresponding to eq1, than take a point on that surface, the tangent with respect to the x axis is y, and…
Andrew Tomazos
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Does monotonicity and derivability of a function $f\colon \mathbb{R}\to\mathbb{R}$ imply bijectiveness?

I have to prove that $f \colon x \mapsto e^{4x} + x^5 + 2$ ($f\colon \mathbb{R}\to\mathbb{R}$) is bijective. The argument given in the solution is that since the first two summands of the image is a bijective function of $x$, then so is $f$.…
Abel
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Calculus: why do we define rate of change as $dy/dx$?

I'm just starting to learn Calculus using Morris Klines' awesome book, "Calculus, an intuitive and physical approach." I really like it so far. I'm just at the beginning, and after learning how to differentiate I was wondering why rate of change is…
Arye Segal
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Integrate $x^2 e^{-x^2/2}$

Is it possible to integrate $$\int_0^{\infty} x^2 e^{-x^2/2}\, \mathrm dx$$ by hand? The answer is $\frac{1}{2\sqrt{2}}$ My apologies if this does not meet the standards of this blog. I will delete it if requested.
Wolfy
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Do limits evaluated at infinity exist?

Here is some limit: $$\lim_{x \to b} f(x)$$ We know that for a limit to exist, we must have $$\lim_{x \to b+} f(x) = \lim_{x \to b-} f(x)$$ So I am confused because, when $b=+\infty$ we can only evaluate this limit from the left side and not the…
Jason
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Number of all positive continuous function $f(x)$ in $\left[0,1\right]$

Number of all positive continuous function $f(x)$ in $\left[0,1\right]$ which satisfy $\displaystyle \int^{1}_{0}f(x)dx=1$ and $\displaystyle \int^{1}_{0}xf(x)dx=\alpha$ and $\displaystyle \int^{1}_{0}x^2f(x)dx=\alpha^2$ Where $\alpha$ is a given…
juantheron
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Is $\lim\limits_{x\to x_0}f'(x)=f'(x_0)$?

Let $f$ be a function defined in the open interval $(a,b)$ and let $x_0\in(a,b)$. Suppose in addition that $f'(x)$ exists for all $x_0\neq x\in(a,b)$. Is the following statement true: If $\lim\limits_{x\to x_0}f'(x)$ exists, then $f'(x_0)$ exists…
boaz
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Floor sum of reciprocal of square root of first $50$ numbers

Find Sum of $$\bigg\lfloor 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\cdots+\frac{1}{\sqrt{50}}\bigg\rfloor$$ $\bf{My\; Try::}$ Let $\displaystyle y=f(x) = \frac{1}{\sqrt{x}}\;,$ Then draw that graph in coordinate axis, We…
juantheron
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Characterization of uniform continuity via sequence

For $f(x)$, defined on the interval $X$, $f(x)$ is uniformly continuous in X if and only if for every sequences $x_{n}$, $y_{n}\in X$, when we have $\lim_{n\rightarrow\infty}(x_{n}-y_{n})=0$, then…
Francis King
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Does a limit at infinity exist?

I use Stewart's (Calculus, 8e) terminology. Infinite limits do not exist. For example we can write $$\lim_{x \rightarrow 0} \frac{1}{x^2} = \infty, $$ but at the same time say that $$\lim_{x \rightarrow 0} \frac{1}{x^2}$$ does not exist. Or at least…
user46234
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Solving the "Spider and Fly problem" using calculus

As part of my homework I've the following problem: Find the shortest path from corner $S$ to corner $F$ (the spider should walk on the surface). I have to use calculus to solve the problem (more specifically using the absolute min/max…
Hanan
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Let $a,b,c,d$ are non-zero real numbers such that $6a+4b+3c+3d=0$,then the equation $ax^3+bx^2+cx+d=0$ has

Let $a,b,c,d$ are non-zero real numbers such that $6a+4b+3c+3d=0$. Then the equation $ax^3+bx^2+cx+d=0$ has: (A) At least one root in $[-2,0]$ (B) At least one root in $[0,2]$ (C) At least two roots in $[-2,2]$ (D) No root in $[-2,2]$ Let…
user1557
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Flawed AP Calc question? Inflection points.

The following question was presented to me by a tutoring student in AP Calculus. It's supposedly from a practice test - not sure if it's official. Here's the issue. Below I've reproduced the complete graph of some continuous function $f(x)$. The…
zahbaz
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