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.

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Explicit formula for recurrence relation $a_{n+1} = 2a_n + 1$

Possible Duplicate: Solving a Recurrence Relation/Equation, is there more than 1 way to solve this? How do I find an explicit formula for $a_n$ given $a_0 = 3$ and $a_{n+1} = 2a_n + 1$. I'm guessing it's probably related to the formula for first…
Jonathan
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If $\lim_{x \to \infty} (f(x)-g(x)) = 0$ then $\lim_{x \to \infty} (f^2(x)-g^2(x)) = 0\ $: False?

If $\lim_{x \to \infty} (f(x)-g(x)) = 0$, then $\lim_{x \to \infty} (f^2(x)-g^2(x)) = 0$ My teacher said that the above is false, but I can't find any example that shows that it's false! Can someone explain to me why it's false and also give an…
Lisa
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Calculating the area under $2x^2+1$ using the method of exhaustion

I'm currently reading Tom M Apostol Calculus Volume I, and I'm stuck on part b) of the following question, regarding the method of exhaustion in calculating the area under a curve: Modify the region in the figure below by assuming the ordinate at…
Jamie Fearon
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Prove an inequality

I want to show that following: $$\left(\frac{n^2-1}{n^2}\right)^n\sqrt{\frac{n+1}{n-1}}\leq 1; ~~n\geq 2$$ and $n$ is an integer. After some simplifications, I got left hand-side as $$LHS:\left(1-\frac{1}{n}\right)^{n-\frac{1}{2}}…
Frey
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What does it mean to differentiate in calculus?

I am able to complete the action, however, I am not sure what it really means in calculus to differentiate. I'm just curious as to why I have to to do this.
Mone Skratt Henry
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Find the exact length of the curve $y = \ln (1 − x^2)$ , $ 0 ≤ x ≤ \frac{1}{7}$

$y = \ln (1 − x^2)$ , $ 0 ≤ x ≤ \frac{1}{7}$ Okay, find the derivative: $\frac{1}{1-x^2}$ Set the derivative in this rule: $\int_{a}^{b}\sqrt{1+\left(\frac{\text{d}y}{\text{d}x}\right)^2}\space\space\text{d}x$ Using this rule…
BenCarson2016
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Finding equations of two tangent lines

I have a prolate cycloid: $$\begin{align*} x &= 2 - \pi\cos(t)\\ y &= 2t - \pi\sin(t) \end{align*}$$ over the interval $-\pi \leq t \leq \pi$, crossed itself at point $P$ on the $x$-axis a) Find the equations of the 2 tangent lines at $P$ b) find…
Nick
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Finding the limit of sequence

Find the limit of $$ \lim_{n \to \infty} \left( \sin \frac{n}{n^2+1^2} + \sin \frac{n}{n^2+2^2} + \dotsb + \sin \frac{n}{n^2+n^2} \right) $$ I think it is a Riemann Integral but I didn't find the expression of Riemann Integral.
james brown
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Shortest distance between two general curves using matlab

Given two functions: $F(x)$ and $G(x)$ is there is a way to find out the shortest distance between $F(x)$ and $G(x)$ provided we know that they do not intersect. I tried to consider parametric points on the two curves and applied the distance…
Student
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Evaluate the Integral $\int^2_{\sqrt{2}}\frac{1}{t^3\sqrt{t^2-1}}dt$

$\int^2_{\sqrt{2}}\frac{1}{t^3\sqrt{t^2-1}}dt$ I believe I've done everything right; however, my answer does not resemble the answer in the book. I think it has something to do with my Algebra. Please tell me what I am doing wrong.
BenCarson2016
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$\lim\limits_{x\to-\infty}\sqrt{4x^2+3x}+2x$

I want to solve this limit: $$\lim_{x\to-\infty}\sqrt{4x^2+3x}+2x$$ My try was to multiply by the conjugate, which gave me $$\lim_{x\to-\infty}\dfrac{3x}{\sqrt{4x^2+3x}-2x}$$ But then factoring $x$ out of the denominator and cancelling with the $x$…
Xavier
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Derivation of the optimal firing angle of a dragless projectile with an initial height

The range of a dragless projectile fired at angle $\theta$ above the horizontal with an an initial height can be written non-dimensionally as: $$R^* = \cos \theta \left (\sin \theta + \sqrt{\sin^2 \theta + 2 / Fr}\right)$$ where $R^* = R g / v$, $Fr…
Ben Trettel
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Limits and Derivatives

This is a homework question where I don't quite understand what I am being asked to do: A tank contains $5000$ L of pure water. Brine that contains $30$ g of salt per litre of water is pumped into the tank at a rate of $25$ L/min. Show that the…
Kurt
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What is the difference between d/dt and dy/dt?

What is the difference between d/dt and dy/dt? And when should either be used? I have seen both in my calculus class and don't know which to use in what context.
ErinAjello
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Is $f(x) = x^{5/3} − 5x^{2/3}$ defined over $(-\infty, 0]$?

I've encountered this question: Find and describe all local extrema of $$f(x) = x^{5/3} − 5x^{2/3}.$$ Also indicate on which regions the function is increasing and decreasing. I've managed to find the extrema, but I am not sure whether the function…
AlexandreAlvard
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