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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.

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If $I_{n} = \int_{0}^{1}x^n\cdot e^xdx,$ Then $\lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right) =$

(1) If $\displaystyle I_{n} = \int_{0}^{1}x^n\cdot e^xdx$. Then $\displaystyle \lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{I_{k+1}}{k}\right) = $ $(2)$ Value of $\displaystyle…
juantheron
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Prove that $\exists c\in(0,1)$ such that $f(c)=c^2$

Suppose that a function $f:[0,1] \to \mathbb{ R}$ is continuous and $3 \int_0^1 f(x)dx = 1$. Prove there exists $c \in (0,1)$ such that $f(c) = c^2 $
Pat Green
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Is the integral $\int_0^{\infty} \sin(e^x) \, dx$ convergent?

Can anyone help me with this problem? I've tried substitution method but I do not know how to continue. Let $u = e^x$ and $du = u\,dx$, so $dx = du/u$. So we have, $$ \int \sin(u) \frac{1}{u} du = \int \frac{\sin(u)}{u} du.$$
Vicky J
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Evaluate $\int ^1 _0 x^2 \, dx$ without using the Fundamental Theorem of Calculus

Evaluate $\int ^1 _0 x^2 \, dx$ without using the Fundamental Theorem of Calculus. I can find the Riemann sum of some partition and the intermediate points, what should I do next?
Diane Vanderwaif
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Rational bounds for $\tan^{-1} x$

I want help with this question. Show that for all $x>0$, $$ \frac{x}{1+x^2}<\tan^{-1}x
leena adam
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Calculus: Prove function is increasing

This question is from Stewart's Essential Calculus: Suppose $f$ is differentiable on an interval $I$ and $f'(x) > 0$ for all numbers $x$ in $I$ except for a single number $c$. Prove that $f$ is increasing on the entire interval $I$. The main…
MathBeatsCrosswords
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suspended cable problem with slack

Suspending a cable produces a hyperbolic cosine shape, but what happens if we orientate the problem in such a way where the rope begins at $(0,y)$ and ends at $(x,0)$ and we specify a rope of length $m$ such that the rope can hug the boundaries if…
mymathrp
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Ship A and Ship B frustration

At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 18 knots and ship B is sailing north at 19 knots. How fast (in knots) is the distance between the ships changing at 5 PM? Okay, cool. I got this: Let's setup our…
alvonellos
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Calculating the area of an elliptical region?

Let D be the region enclosed by the ellipse $2x^2 + 3y^2 = 1$ and the line $y = 0$, for $y \le 0$. Using Polar coordinates, evaluate the integral $$\int\int[\sinh(4x^2 + 6y^2)]\,dx\,dy$$ By making the change of variables $x = \frac r{\sqrt…
harold
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There's always a smaller rational $x$ with $x^2\gt2$

Kindly help to prove this problem. Prove that for every positive rational number $s$ satisfying the condition $s^2 > 2$ one can always find a smaller rational number $s - k (k > 0)$ for which $(s - k) (s - k) > 2.$ Request! I had very big problem in…
mehar1212
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How find this $\sum_{k=0}^{100}a_{3k}$

let $$(1+x+x^2)^{150}=\sum_{k=0}^{300}a_{k}x^k$$ find the value $$\sum_{k=0}^{100}a_{3k}$$ My idea: I think we let $x=1,w,w^2$?
math110
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What is the most accurate way to project a total running time based on a 95% complete run?

My friends and I are having an argument. I bet my friend that he could not run a mile in under 12:00 minutes, which he did, but it lead to a question. The mile he ran was on a field the size of half of a mile. He ran two laps. During his run, my…
hiiiiiiiiiiiiiiiii
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derivative of function

I have a simple problem from calculus topics. Suppose we have $$x=at^2,\qquad y=2at$$ and want to find $\dfrac{d^2y}{dx^2}$. There is given sample http://www.mathopolis.com/questions/a.php?id=137&ansno=957 I think that the answer is zero but here…
dato datuashvili
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Forget about the $\cos,\sin$ function, show that $\left|1-x^2/2!+x^4/4!-x^6/6!+...\right|\leq1$

Forget about the $\cos,\sin$ function, show that $\left|1-x^2/2!+x^4/4!-x^6/6!+...\right|\leq1$ I tried to use differentiation, but it doesn't seems helpful. Please help. Thanks.
JSCB
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Using implicit differentation to find the tangent to $3xy=y^2$

Use implicit differentiation to find the equation of the line tangent to the graph of $3xy = y^2$ at $(1,3)$ This is what I tried so far: $$\begin{align*} 3\frac{d}{dx}(x)\frac{d}{dx}(y(x)) &=…
OghmaOsiris
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