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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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Find a continuous function over reals that takes on every value exactly three times

Find a continuous function over reals that takes on every value exactly three times. I was thinking such a function would have to have two critical points and must go from increasing to decreasing or decreasing to increasing. Something like $x^3$,…
user19405892
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Prove that there does not exist a continuous function over reals for which it takes on every value exactly twice

Prove that there does not exist a continuous function over reals for which it takes on every value exactly twice. Can we replace two by four? an even number? Note: By critical point here I mean one that goes from increasing or decreasing or…
user19405892
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Make $2^8 + 2^{11} + 2^n$ a perfect square

Can someone help me with this exercise? I tried to do it, but it was very hard to solve it. Find the value of $n$ to make $2^8 + 2^{11} + 2^n$ a perfect square. It is the same thing like $4=2^2$.
Mang Sopheak
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Area between two curves, which curve is on top?

Given a question like this: Find the area between ${y = x^2 + 2x - 3}$ and ${y = 2x^2 -5x -3}$. I know how to find the area ${\int y_1 - y_2}$ but how can I tell which one is the top curve? Are there any shortcuts to determining the top curve?
dagda1
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Find the number of rational roots of $f(x)$

There's a polynomial $f(x)$ such that its degree is 3. All the coefficients of $f(x)$ are rational. If $f(x)$ is a tangent to $x$ axis, what can be the possible number of rational roots of $f(x) = 0$ options are : 0, 1, 2, 3, none My approach…
Bazinga
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Slope of curve in $\mathbb{R}^3$

While doing revision, I came across this problem: The surface given by $z=x^2-y^2$ is cut by the plane given by $y=3x$, producing a curve in the plane. Find the slope of this curve at the point $(1,3,-8)$. I tried substituting $y=3x$ into…
yoyostein
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Where is the form $f(g(x))g'(x)$?

I was trying to do the exercises in Apostol's Calculus. I tried to answer an exercise to find the indefinite integral of $\int x\sqrt{1+3x} dx$ via substitution. I've been able to do it via integration by parts, but with substitution, I have no clue…
Red Banana
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Constructing $\exp$ on $\mathbb R$

I am trying to construct the exponential function on $\mathbb R$ by first finding all functions $f$ such that $f = f'$ (which should be all the constant multiples of $\exp$), then characterizing $\exp$ by the initial condition $f(0) = 1$. I intend…
user33661
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Find $\int_{0}^{\frac{\pi}{2}} \dfrac{\sin^n(x)}{\sin^n(x)+\cos^n(x)} dx$.

Find $\displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{\sin^n(x)}{\sin^n(x)+\cos^n(x)} dx$. I was told to switch the limits of integration then add them. How can I do that here?
John Ryan
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integral of continuous function can be very small

Let $f(x)\in\mathbb{R}$, $x\in\mathbb{R}^{+}$, with $f(x)$ continuous. Is the following statement true? $\exists M, \forall y> x\geq M$ such that $\lvert\int_{x}^{y}{f(s)ds}\rvert\leq \varepsilon$ for all $\varepsilon>0$.
81235
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Graph Sketching Understanding

Sketch the graph of the function $y=x(4-x)-83\ln(x)$. Indicate the transition points (local extrema and points of inflection). I understand how to solve this problem mathematically, however, is there any way of analyzing the question to get a better…
Mone Skratt Henry
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Why is $f'(x)= 0$ but $f$ is not a constant function?

Let $f(x) := \arctan(x) + \arctan(1/x)$. Then \begin{align*} f'(x) &= \frac{1}{x^2+1} + \frac{1}{(\frac{1}{x})^2 + 1}\cdot \frac{-1}{x^2} \\ &= \frac{1}{x^2 + 1} + \frac{-1}{x^2 + 1} \\ &= 0 \end{align*} which should mean that $f$ is a constant…
aras
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Proof that the derivative of a constant is zero

I know that the derivative of a constant is zero, but the only proof that I can find is: given that $f(x) = {x}^{0}$, $$ f'(x) = \lim_{h\to 0} \frac {f(x+h) - f(x)}{h} $$ $$ f'(x) = \lim_{h\to 0} \frac {{(x+h)}^{0} - {x}^{0}}{h} $$ and then because…
nworb99
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How can I manipulate $\frac { \sqrt { x+1 } }{ \sqrt { x } +1 } $ to find $M>0$ to prove a limit?

Given the following limit, find such an $M>0$ that for every $x>M$, the expression is $\frac { 1 }{ 3 }$ close to the limit. In other words find $M>0$ that for every $x>M:\left| f(x)-L \right| <\frac { 1 }{ 3 }$ for the following function: $$\lim _{…
Cherry_Developer
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Difference quotient

Where am I going wrong? Find the difference quotient for: $f(x)=2-x-3x^2$ $$\frac{[ 2-(x+h)-3(x+h)^2 ] - [ 2-x-3x^2 ]}{h}$$ $$\frac{2-x-h-3x^2-6hx-3h^2-2+x+3x^2}{h}$$ $$\frac{-3h^2-6hx-h}{h}$$ $$-3h-6x-1$$
AFerrara
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