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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Why is $(x+h)^n$ written like this?

I encountered this in my calculus book: $$f\;'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ $$f(x)=x^n$$ $$\begin{align*} (x + h)^n &= (x + h)(x + h)...(x + h)\\ &=x^n + nhx^{n-1}+ \text{stuff involving }h^2\text{ as factor} \end{align*}$$ I don't…
Andrew
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$f'(x)-xf(x)=0$ has more roots than $f(x)=0$

Let $f(x)$ be a polynomial with real coefficients. Show that the equation $f'(x)-xf(x)=0$ has more roots than $f(x)=0$. I saw the hint, nevertheless I can't prove it clearly. The hint is that $f(x)e^{-x^2/2}$ has a derivative…
Gobi
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limit of $f$ and $f''$ exists implies limit of $f'$ is 0

Prove that if $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f''(x)$ exist, then $\lim\limits_{x\to\infty}f'(x)=0$. I can prove that $\lim\limits_{x\to\infty}f''(x)=0$. Otherwise $f'(x)$ goes to infinity and $f(x)$ goes to infinity,…
user19033
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Visualizing Second Derivative

Given a graph of a simple function $f$ (continuous, no oscillation, smooth). We can roughly tell how big $f'(a)$ is for any point $a$, i.e. by looking at the steepness of the graph at that point. Is there any way to estimate $f''(a)$? Determining…
Gotlim
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How to integrate $e^{-x^2-x}$?

We've just started discussing continuous distributions in my probability class and I've come across this interesting distribution that I'm unsure of how to integrate. Let $c$ be a constant and let $X$ be a random variable with distribution…
mathmorales
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quotient rule difficulties

I'm trying to use the quotient rule to differentiate $\frac{r}{\sqrt{r^2+1}}$ but I'm getting the wrong answer. So far I have $$\begin{align*} \frac {d}{dr} \frac{r}{\sqrt{r^2+1}} &= \frac {\sqrt{r^2+1} \frac {d}{dr} r - r \frac {d}{dr}…
Matt Munson
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Help with integral

I seem to be stuck trying to prove the following integral $$ \int\frac{\cos^mx}{\sin^nx}dx = -\frac{\cos^{m+1}x}{(n-1)\sin^{n-1}x}-\frac{m-n+2}{n-1}\int\frac{\cos^mx}{\sin^{n-2}x} dx + C\,\,(n \neq 1) $$ My thinking so far has been that if I…
emjay
  • 545
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Evaluate the integral: integral of $\int 6^{-2x} dx$

$$\displaystyle \int6^{-2x}dx$$ I got $-\dfrac{6^{-2x}}{2\ln6}$ but I'm not confident that its correct.
user142342
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What are the best sites to get caught up on Calculus?

I'm going back to college this summer and will be taking engineering statistics and calculus based physics. I dropped out of college about 4 years ago and took calculus 1-3 before leaving. I'm worried I have forgotten all of my calculus and won't…
will
  • 161
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Continuity and simplification of a function

I have a question to ask about a function. Suppose a function $$f(x) = \frac{x^2 - x}{ x - 1},$$ we can simplify this function to be $f(x) = x$. Yet, we say that this function is discontinuous at $x = 1$ but after the simplification, we say that the…
Chris Myung Hyun Kim
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maximized profit w/ a cost & demand function

I'm having trouble with this problem: If $C(x) = 14000 + 500x − 4.8x^2 + 0.004x^3$ is the cost function and $p(x) = 4100 − 9x$ is the demand function, find the production level that will maximize profit. (Hint: If the profit is maximized, then the…
maribov
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Estimation of integral

Suppose the function $f(x)$ has a Taylor series expansion. Then $$\int_a^bf(x)dx=\int_a^b(f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2+\cdots)dx=\\…
velut luna
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Ignoring constants when finding derivatives of trig functions

Find the derivative of $3sin^2(6x)$. I know I solve this by using chain rule, my question is how come product rule is not used instead on $3$ and $sin^2(6x)$? According to wolfram alpha, the constant is factored out at the beginning of the problem,…
chopper draw lion4
  • 861
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$\dfrac{\mathrm d}{\mathrm dx}x^x=$?

Let $f(x)=x^x$. What is the derivative of $f$? This function can't be treated by chain rule or product rule or $(e^x)'=e^x$
user17399
  • 193
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Mathematics Engineering: How do you prove the power rule?

Consider the real-valued function $f(x)=x^r$ where $r$ is a real number. (1) For what values of $x$ and $r$, is this function differentiable? (2) How do you prove the power rule, when $f(x)$ is differentiable? It would be nice if the proof is…
Sony
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