# Questions tagged [calculus]

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

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### Finding constants making function continuous

Determine the values of $h$ and $k$ so that $$F(x)=\begin{cases} \frac{kx^2-kx+x+10}{x^2-4}, &x\neq2,\\ 3x+h, &x=2 \end{cases}$$ is continuous on $[0,4]$. I extracted k as Common factor and ...
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### Integration over an open interval.

Is there any difference between the following integrations: $$\int_a^b f(x) dx$$ where $(a,b]$ And $$\int_a^b f(x) dx$$ where $[a,b]$
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### Problem Help in Studying for Final.

I am reviewing past material and these were two problems that I was stuck on. I want to make sure that I understand them for my final. Problem 1: An isosceles triangle has two sides of length 10. The ...
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### How to parameterize a vector field in terms of $u$ and $v$ given three components.

I am given the following vector field $$F(x,y,z) = xyz \mathbf i + xy \mathbf j + x^2yz \mathbf k.$$ I need to parameterize the vector field $r(u,v)$ to use it with Stoke’s theorem.
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### How do you solve this?

Let C be the intersection of the cylinder $x ^ 2 + y ^ 2 = 1$ with the plane $x + 2y + 2z = 3$ counterclockwise when viewed from the origin. Let F be a vector field with curl $F = i + 2j - \alpha*k$ ...
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### Can $\ln$ be written as a ratio of polynomials?

Is it possible that $\ln(x)=\frac{p(x)}{q(x)}$ for all $x>0,$ where $p$ and $q$ are polynomials with real coefficients? I think the answer is no. Suppose two such polynomials did exist. Take the ...
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### When to include the boundary points in the convexity analysis?

I was wondering about this: suppose I have a function $f: D \to \mathbb{R}$; suppose $(a, b) \subset D$ ($D$ can either be bounded or unbounded), and say $f$ is convex in $(a, b)$. What is the ...
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By the fundamental theorem of algebra a real nonconstant polynomial $Q$ has factorisation into real prime factors $$Q=g_1^{k_1}g_2^{k_2}\cdots g_l^{k_l}$$ The prime factors $g_j$, all distinct, are ...