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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Need help with Applications of Differentiation Problem

Question My Working Following the hint + some help I got from tutorial below is what I got ... but I believe I did something wrong ... its not the answer below the question yet $0.955\text{ }m\text{ }min$ (I believe its $0.955 m/min$) UPDATE In…
Jiew Meng
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Limits of functions of several variables

Compute the following limit: $\lim_{(x,y)\rightarrow (0,0)}\frac{x^2-y^2}{\sqrt{x^2+y^2}}$
Bittu
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Prove $f(x) = x^2$ is continuous at $x=4$

I want to show that $f(x) = x^2$ is continuous at $x=4$ and here's how the proof goes: $\forall\epsilon>0$, $\exists\delta>0$ s.t $\forall x$, $|x-4|<\delta$ $\implies |f(x)-16|<\epsilon$ So working backwards we get: $$|f(x)-16|<\epsilon ⇔ |x^2 -…
user3412839
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Volume of a general ellipsoid

Can somebody please show me at least one derivation of the volume of a general ellipsoid? I've been trying to derive by considering it a surface of revolution. The answer I keep getting is $4\pi(abc^3)/3$ but I know it's supposed $4\pi(abc)/3$.…
TheBluegrassMathematician
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The derivative of an integral

How would one interpret: $$\frac{\mathrm d}{\mathrm dx}\int_0^x (F(y)-F(x))\,\mathrm dy$$ I don't think I can use the fundamental theorem of calculus here, can I?
Angada
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To show that the function is continuous at $x=2$

Show that $f(x)=3x^2+2x-1$ is continuous at $x=2$. $f(2)=15$ $f(x)-f(2)=3x^2+2x-16 \Rightarrow f(x)-f(2)=(x-7/3)(x+3)$ Let $|f(x)-f(2)| < \varepsilon\Rightarrow |(x-7/3)(x+3)| < \varepsilon$ $|x-7/3|<\varepsilon$ or…
Vikram
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Finding a $δ$ for the limit $\lim_{x\to 2} x^4 = 16$

$$\lim_{x\to a} x^4 = L$$ for some arbitrary a Picking $a$ to be 2, we get: $$\lim_{x\to 2} x^4 = 16$$ To show that is the limit I tried doing the epsilon-delta definition of a limit to show how to find a $δ$ such that $|f(x) - L| < \epsilon $ for…
Sc4r
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Continuous function?

Consider the function $f(x)=[x]$ on the interval $[0,2]$ where $[x]$ denotes the largest integer less than or equal to x. Is this function continuous? I cant find a reason for it not to be, although im not sure.
user117449
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The gradient of a distance function.

In level set a distance function is defined as: $$ d(\vec{x})=\min(\left|\vec{x}-\vec{x}_{I}\right|) $$ where $\vec{x}_{I}$is a point on the interface, for two spatial dimensions it can be a curve. Furthermore, since $d$ is Euclidean distance,…
withparadox2
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Rate of increase in the area of a square

I really do not understand how to do these problems, so many weird math tricks and rules and I am getting caught up on at least a dozen in this problem. Anyways I am supposed to find: Each side of a square is increasing at a rate of $6 \text{…
user138246
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Finding Limit Function that satisfies the conditions.

I am having some trouble figuring out a few math problems from my Calc 1 class. I am not sure where to start, as all the limits are different. find a function that satisfies the given conditions and then sketch it. sketch a graph of the function…
user131785
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Area Between Two Curves

I have been trying to solve a homework problem that reads: "The region between the graphs of $f(x) = x^{2}+2$ and $g(x) = -5x+2$ has what area?" I have been trying to solve this problem and have come up with the answer of $185/6$ square units, but…
user124539
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given positive real numbers x,y how to demonstrate that is defined as the maximum of x and y?

Is it true that given positive real numbers $x,y$, then we have that $$ \sqrt{x^2 + y^2} \geq \max\{ x, y \} $$ I cant find a counter-example although it seems it is true... Any comments?
ILoveMath
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How can the following be calculated?

How can the following series be calculated? $$S=1+(1+2)+(1+2+3)+(1+2+3+4)+\cdots+(1+2+3+4+\cdots+2011)$$
Paul Manta
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Proof that $ f'(x)=0 $ has $n-1$ roots

I am given $f(x)=(x-1)(x-2)(x-3)...(x-n)$. I am obliged to proof that for $n \ge 1$, the equation $ f'(x)=0 $ has $n-1$ roots. I think I have to somehow use Rolle's theorem to proof that. I will be glad for any help and tips.
Marcin Majewski
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