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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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Range of $g(x)$ in differential equation

If $g(x)$ is real valued differentiable function defined on $[1,\infty)$ with $g(1)=1$ and $g(x)$ satisfy $\displaystyle g'(x)= \frac{1}{x^2+g^2(x)}$. Then range of $g(x)$ is Try: Let $g(x)= y$. Then $\displaystyle \frac{dx}{dy}=x^2+y^2$. Put…
DXT
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How to show $e^x$ and $\ln x$ do not intersect?

I was trying to solve the following problem from a calculus book Show that the graph of $e^x$ and $\ln x$ do not intersect. I tried to solve the problem by showing the function \begin{align}\label{1} f(x)=e^x-\ln x \tag{1} \end{align} does not…
marya
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How do I determine this integral? $\int_{0}^{+\infty}\sin^2(1/x)\frac{dx}{(4+x^2)^2}$

$$\int_{0}^{\infty}\mathrm dx{\sin^2\left({a\over x}\right)\over (4a^2+x^2)^2}$$ $${\sin^2\left({a\over x}\right)\over (4a^2+x^2)^2}={1-\cos^2\left({a\over x}\right)\over (4a^2+x^2)^2}={1\over 2(4a^2+x^2)^2}-{\cos\left({2a\over x}\right)\over…
user550936
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Using complex exponential to show the indefinite integration of sin(x)sinh(x) dx

Use the complex exponential to evaluate the indefinite integral of $\sin x \sinh x$. Express your answer in terms of trigonometric and/or hyperbolic functions The attached photo is what I have tried so far
Jenny
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The intersection of the curves $y=3^{x-1}\ln x$ and $y=x^x-1$

What is the point of intersection of the curves $y=3^{x-1}\ln x$ and $y=x^x-1$? By hit and trial, I got the intersection point as $(1,0)$. Is there any other way I could find the intersection?
Samar Imam Zaidi
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Given $f : \Bbb{R} \to \Bbb{R}$ is a continuous function. To prove that $\int_{0}^{1}f(x)x^2 dx = \frac{f(c)}{3}$ for some $c \in [0,1]$

My approach was to start by integration by parts. $$\int_{0}^{1}f(x)x^2 dx = \frac{1}{3}(f(1) - \int_{0}^{1} f'(x)x^3 dx)$$ Now if I can bound $(f(1) - \int_{0}^{1} f'(x)x^3 dx)$ by $f(0)$ and $f(1)$ then we can use the intermediate value theorem…
Sayan Chattopadhyay
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Squeeze Theorem and infinite oscillation

How do I determine the limit as $x$ approaches zero of $$\frac{(x+1)\cos(\ln x^2)}{\sqrt{x^2+2}}$$ using the Squeeze Theorem. My suspicion is that it oscillates infinitely often and therefore doesn't have a limit but I don't know how to prove or…
Tightrope
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Question involving mean value theorem

I have a question that involves the application of the Mean Value Theorem: Suppose $f$ is a function differentiable on $[a, b]$ with $a
yoshi
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Area of largest inscribed rectangle in an ellipse. Can I take the square of the area to simplify calculations?

So say I have an ellipse defined like this: $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$ I have to find the largest possible area of an inscribed rectangle. So the area ($A$) of a rectangle is $2x2y=4xy$. Also we can redefine $y$ in terms of…
Jwan622
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What is the value of the given limit?

Possible Duplicate: How can I prove Infinitesimal Limit Let $$\lim_{x\to 0}f(x)=0$$ and $$\lim_{x\to 0}\frac{f(2x)-f(x)}{x}=0$$ Then what is the value of $$\lim_{x\to 0}\frac{f(x)}{x}$$
Aliakbar
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The substitution rule and differentials

The following is from Stewart's 'single variable calculus, 6E' (the bold is mine) $$\int f(g(x))g'(x)dx = \int f(u)du$$ "Notice that the Substitution Rule for integration was proved using the chain rule for differentiation. Notice also that if…
Matt Munson
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Difference between Taylor's theorem and Taylor's series?

What does Taylor's theorem say? How do we use Taylor's theorem to get to Taylor's series? I need the basic idea behind these two. What's going on between them?
Shoaib Ashraf
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Prove a statement involving differentiable function

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$, where $0
user122049
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Hyperbolic functions. Why are they named with trig functions?

I don't really get why these hyperbolic functions are named after trig functions. Can someone enlighten me?
Jwan622
  • 5,704
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Minimizing a functional definite integral

I have a definite integral defined by $$T\left(G\left(g\right)\right)=\int_{g_{1}}^{g_{2}}G(g)\mathrm{d}g$$ where $G$ is a continuous function of a variable $g$, and $g_{1}$ and $g_{2}$ are known numbers. I want to minimize…
James White
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