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Questions tagged [calculus]

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

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Difference between function, derivative and second derivative?

If you are looking at three graphs: one is the original function, one is the derivative and the other is the second derivative, what is the accepted way of determining which is which? For example ...
18k views

Difference between maximum and minimum?

If I have a problem such as this: We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the ...
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Effect of $k$ on turning point?

In the function $$y=(k-x)e^x ,$$ What is the effect of $k$ on the turning point of the function? I can't see any clear pattern when I change the variable. What are some real-life scenarios to which ...
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Is $f'$ continuous at $0$ if $f(x)=x^2\sin(1/x)$

Let $f(x)=x^2\sin(1/x)$ for $x≠ 0$ and $f(0)=0$ for $x=0$. Is $f'$ continuous at $0$? My attempt: $f'(x)=2x\sin(1/x)-\cos(1/x)$. Since when $x$ goes to $0$, the limit of $\cos(1/x)$ does not ...
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integrate square of $\arctan x$. Tricky

$$\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2}dx$$ I ran across an integral I am having a time solving. The solution merely works out to $\displaystyle\frac{1+x\tan^{-1}x}{\tan^{-1}x-x}$, ...
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$\cos x = kx$, finding $k$ that gives two solutions

$\cos x = 0.3x$ has three solutions. $\cos x = 0.4x$ has one solution. How to find $k$ so that $\cos x = kx$ has two solutions?
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How do I remove the polynomial from a fraction?

TLDR: I want to solve function $(-4x^2-6x+4)/(x^2+1)^2$ for $0$. How can I get the polynomial out of the denominator so I can apply the quadratic formula? Long form: I'm trying to find the ...
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From given equality find that $p$ for which equality have at least one positive root

Having $px^3+(p-3)x^2+(2-p)x=0$ how to find p that this equality have at least 1 positive root ? How can we solve this and similiar things? Because i'm stuck... i did it for quaratic, but i can't ...
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Challenging problems in calculus

I'm looking for a textbook (or website etc.) which contains challenging problems in calculus. Problem in Real Analysis by Titu Andreescu is good, but slightly too advanced for me. Spivak's Calculus is ...
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Limits of sets in relation to their infimum and supremum plus monotonic sequences concept

I have tried to solve the following question but I haven't gotten an answer. Show that: $\lim\ [0,1-1/n] = [0,1)$. $\lim\ [0,1-1/n) = [0,1)$. all the limits having $n\to\infty$.
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Proving that integration is continuous

Define $C([a,b], \mathbb R)$ to be the space of continuous functions $f : [a,b] \to \mathbb R$ with the norm $\| \cdot \|_{\infty}$. Let $H : C([a, b], \mathbb{R}) \rightarrow \mathbb{R}$ be the map ...
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Example of a function that is not twice differentiable

Give an example of a function f that is defined in a neighborhood of a s.t. $\lim_{h\to 0}(f(a+h)+f(a-h)-2f(a))/h^2$ exists, but is not twice differentiable. Note: this follows a problem where I ...
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Prove $\lim\limits_{x→∞} f''(x) = 0$

If $\lim\limits_{x→∞} f(x)$ and $\lim\limits_{x→∞} f''(x)$ both exist, then $\lim\limits_{x→∞} f''(x) = 0.$ You may use the fact that $\lim\limits_{x→∞} f(x)$and $\lim\limits_{x→∞} f'(x)$ both exist, ...
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Evaluating integrals

I am having trouble figuring out an algebraic trick to make this work Evaluate the integral $$\int_1^9\frac{x-1}{\sqrt{x}}dx$$ I know I can turn the integrand into $(x-1) (1/\sqrt{x})$ but I ...
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Polynomial Function and $n$-times differentiable

If a function $f$ is $n$-times differentiable on $\mathbb R$ and $f^{(n)}=0$, prove $f$ is a polynomial of degree $\leq n-1$. A hint would suffice.
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Tangent Lines and Implicit Function

I am asked to find the equations of the tangent lines at three different points for the following function: $$y^{5}-y-x^{2}=-1$$ I am provided with a graph/sketch of the function and asked to find ...
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Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize?

why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i'm missing, or is it just ...
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Nice way of showing the following equality?

Is there a good way of showing that ${d^n\over dx^n}(x^2-1)^n|_{x=1}=2^nn!$? I have tried  binomial expanding the thing then differntiate term-by-term, which seems a bit clumsy. Perhaps there's a ...
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Does the Wronskian have anything to do with the product rule in calculus

Does the Wronskian have anything to do with the product rule in calculus. I ask this because i noticed the form looking similar to the product rule. $$W=g(x)f'(x)-g'(x)f(x)$$ where as the ...
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Speed of a runner

I am trying to do this homework problem for calculus. It is an intro to integrals and I have no idea what I am doing wrong. The speed of a runner is increase steadily during the first $3$ seconds ...
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Behavior of $\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/n^{0.5-\epsilon})}}{2a+b/n^{0.5-\epsilon}}\right)^{\frac{n}{2}}$

I am having trouble expressing the behavior of the following limit: $$\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/n^{0.5-\epsilon})}}{2a+b/n^{0.5-\epsilon}}\right)^{\frac{n}{2}}$$ After some ...
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How to prove that $f(x) = \sum_{k=1}^\infty \frac{\sin((k + 1)!\;x )}{k!}$ is nowhere differentiable

This function is continuous, it follows by M-Weierstrass Test. But proving non-differentiability, I think it's too hard. Does someone know how can I prove this? Or at least have a paper with the proof?...
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Are there times when convergence tests contradict each other?

In my Calc II class, we're just starting convergence tests and all the examples are very convinient and they work perfectly (obviously, since they are examples), but my professor couldn't really ...
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Explanation: Volume of a trapezium

A trough is ﬁlled with water at a rate of 1 cubic meter per second. The trough has a trapezoidal cross section with the lower base of length half a meter and one meter sides opening outwards at an ...
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Plotting $\frac{1}{\ln x}$

I need assistance in plotting the graph of $\frac{1}{\ln x}$. wolframalpha gives this. How to plot this function (both real and imaginary part) using calculus?
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If $\int_V f \;dV = 0$ when can we say that $f=0$ everywhere

If $\int\limits_V f \; \mathrm dV = 0$ can we say that $f=0$ everywhere? Or what conditions are there on concluding this. In particular I want to solve the PDE $\nabla^2 f=f^3$ on the region D=\{(x,...
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Help finding the absolute error with $n$th degree Taylor polynomials

I am trying to estimate the absolute error in approximating $\ln 1.09$ with the $3$rd-order Taylor polynomial centered at $0$. It's been a while since I've taken the Calculus and I'm afraid I need ...
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How to prove that $\lim\limits_{h \to 0} \frac{a^h - 1}{h} = \ln a$

In order to find the derivative of a exponential function, on its general form $a^x$ by the definition, I used limits. \$\begin{align*} \frac{d}{dx} a^x & = \lim_{h \to 0} \left [ \frac{a^{x+h}-a^...