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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The volume of the water in the hemispherical bowl is given by $V=\frac{\pi}{3}y^2(3R-y)$ when the water is $y$ m deep.

Water is flowing out at the rate of $6m^3$/min from a reservoir shaped like a hemispherical bowl of radius $R=13m.$The volume of the water in the hemispherical bowl is given by $V=\frac{\pi}{3}y^2(3R-y)$ when the water is $y$ m deep.Find $(a)$At…
Brahmagupta
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Derivative of an Integral where the limits of integration are not linear

$y'=\frac{\mathrm{d}}{\mathrm{d}x} \int_{a}^{\sqrt{x}}f(t)dt$ Here, $y$ is the area function given by $y = \int_{a}^{\sqrt{x}}f(t)dt$ Let us say that $y=F(x)$. $$y'=F'(x)=\lim_{h\rightarrow 0}\frac{…
R004
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If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$

I have to do the following exercise: Let $f$ and $g$ two differentiable functions such that $f(0)=g(0)$ and $f'(x)\leq g'(x)$ for all $x$ in $\mathbb{R}$. Prove that $f(x)\leq g(x)$ for any $x\geq0$. Now, I know this is true because the first…
Rage
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Integration and Differentiation

while learning Calculus at College level mathematics classes, we were told that: Differentiation and Integration are opposite or complementary to each other....(1) Differentiation is Tangent to the given curve ....(2) and Integration is Area…
Sanjay Naik
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Integration by Parts with a Jump Discontinuity

I ran into the following problem and its solution: The integration by parts formula $$ \int_{a}^{b}u\frac{dv}{dx}\,dx=uv\bigg|_{a}^{b}-\int_{a}^{b}v\frac{du}{dx}\,dx $$ is known to be valid for functions $u(x)$ and $v(x)$, which are continuous…
wjmolina
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Proving that $f(x)$ is less than or equal to $1+\pi/4$

Suppose $f$ is a real valued differentiable function defined on $[1,\infty)$ with $f(1)=1$. Suppose also that $f$ satisfies $$f'(x)=\frac{1}{x^2+f^2(x)}.$$ The question is to prove that $f(x) \leq 1+\pi/4$ for every $x \geq 1$ I tried to solve the…
Navin
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Find cubic function whose graph has horizontal tangents at two points

Find the cubic function $y = ax^3 + bx^2 + cx + d$ whose graph has horizontal tangents at the points $(-2,7)$ and $(2,1)$. I find the derivative and set equation to zero, but then that only gives me one solution. I can't find all solutions for some…
Itsreason
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Derivative of a split function

We have the function: $$f(x) = \frac{x^2\sqrt[4]{x^3}}{x^3+2}.$$ I rewrote it as $$f(x) = \frac{x^2{x^{3/4}}}{x^3+2}.$$ After a while of differentiating I get the final answer: $$f(x)= \frac{- {\sqrt[4]{\left(\frac{1}{4}\right)^{19}} +…
JohnPhteven
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$\int_0^{\pi/2}sin^p\,\theta\;cos^q\,\theta\;d\theta = \frac{\Gamma{\frac{p+1}{2}}\Gamma{\frac{q+1}{2}}}{2\Gamma{\frac{p+q+2}{2}}},\; p,q > -1$

Show that: $$\int_0^{\pi/2}sin^p\,\theta\;cos^q\,\theta\;d\theta = \frac{\Gamma{\frac{p+1}{2}}\Gamma{\frac{q+1}{2}}}{2\Gamma{\frac{p+q+2}{2}}},\; p,q > -1$$ Here's the question.
Lavios
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Use a linear approximation (or differentials) to estimate the given number.

Use linear approximation (or differentials) to estimate: $$\sqrt {99.2}$$ What am I supposed to do with this? I am not given $x$ or $dx$.
dsta
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How is $dx \over dy$ different from $\partial x \over \partial y$?

Say I have variables $x,y_1,y_2,z_1,z_2$ all $\in \mathbb{R}$ And I have the following equations: $$x = f_1(y_1,y_2)$$ $$y_1 = f_2(z_1,z_2)$$ How does: $$dx \over dz_1$$ differ from: $$\partial x \over \partial z_1$$ or am I confused? Intuitively I…
Andrew Tomazos
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Show such a function has a maximum

Let $f:[0, \infty)$ be a continuous function. $f(0) = 1 $ and $\forall x \in [0, \infty)$ $f(x)\leq \frac{x+2}{x+1}$ Show that $f$ gets a maximal value in $[0, \infty)$. My intuition: if $f(0)$ is the maximum i'm done if not the function you showed…
user21312
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Is $\frac1p(\sin^px+\cos^px)-\frac1q(\sin^qx+\cos^qx)$ constant for some reals $p$ and $q$.

We know for $$f(x)=\dfrac14(\sin^4x+\cos^4x)~~~;~~~g(x)=\dfrac16(\sin^6x+\cos^6x)$$ have $f(x)-g(x)=\dfrac{1}{12}$. My question is Are there other real $p$ and $q$ such that …
Nosrati
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Why are there two Fundamental Theorems of Calculus?

I don't know if this is an appropriate question for this site. If not, I apologize. A colleague of mine maintains that the Second Fundamental Theorem of Calculus shouldn't be taught as such (i.e. that it is "fundamental") but rather be thought of,…
HTG
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Why can't I use the chain rule for the derivative of $x^{x^2}$?

I realize you can do this with implicit differentiation, but I thought you could also take the derivative by using the chain rule. Hence, it should be $2x\cdot x^{x^2-1}$ What's wrong with this?
JobHunter69
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