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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 the area bounded by $ \ x^4+y^4=4(x^2+y^2) \ $ .

Find the area bounded by $ \ x^4+y^4=4(x^2+y^2) \ $ . Answer: The graph is above : Since the region is symmetrical , $ Area =4 \times \int_{0}^{2} \int_{0}^{\sqrt{2+\sqrt{4x^2-x^4+4}}} dxdy $ Am I right ? Is there any help ?
MAS
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A question on Intermediate Value Theorem

Suppose that $f$ is a continuous function and that $f(-1)=f(1)=0$. Show that there is $c\in(-1,1)$ such that $$f(c)=\frac{c}{1-c^2}$$ I am not sure if this is a new question as I set it this morning, after solved a similar question. I wanted to…
pipi
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What does slope of a line mean in 3 dimensional figures?

In 2D geometry where $y=f(x)$ then $f'(a)$ means slope of the tangent line at ($a$, $f(a)$). It means the angle made with the positive $X$ axis Now extending to 3D geometry let's say $z$=$f(x, y)$ so ∂z/∂x at let's say ($A$, $B$, $C$) gives us the…
Nigel Goveas
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Logarithmic differentiation of some functions

Given $y=f(x)$, where $f(x) $ is a positive function, we can write $\ln y = \ln f(x) $. Now let's say that $f$ takes zero values at certain points in an interval. At these points, the natural logarithm of the function is not defined. Take the…
R004
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If $f$ is continuous on $[a,\infty)$ and increasing on $(a,\infty)$, can we say $f(x)>f(a)$.

If $f$ is continuous on $[a,\infty)$ and increasing on $(a,\infty)$ can we say $f(x)>f(a)$ for $x>a$. I think yes, the fact that $f$ is (strictly) increasing on $(a,\infty)$ gives, $$f(x)>f(y)$$ For $x,y \in (a,\infty)$ with $x>y$, in particular…
Ahmed S. Attaalla
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Why is the integral the antiderivative of a function?

What's the demonstration that the antiderivative of a function is the integral?
Lucia
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What is $( \partial u / \partial x )^2$ equal to?

Is it equal to $\partial ^2 u / \partial x ^2$? I am trying to figure that out. I don't think it's the case, but I am just trying to make sure.
question
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Show that if $x \neq 0$ then $e^x-x-1 \neq 0$ without Taylor series.

Show that if $x \neq 0$ then $e^x-x-1 \neq 0$ without Taylor series. If we let $f(x)=e^x-x-1$, the fact that it is monotonically increasing on $[0,\infty)$ and monotonically decreasing on $(-\infty,0]$ allows us to say that $f(x) \geq f(0)=0$. But…
Ahmed S. Attaalla
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Periodic differentiable function

If $f(x)$ is a differentiable function on the real line and periodic with period $T$ does it have at least two stationary points in $(0,T]$?
Ric Simmons
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Finding the tangent line that intersects a quartic at two points.

I'm trying to find a straightforward calc solution to this question. I found an algebraic solution but I don't think it's the quickest way to do it. Thanks in advance! Find the linear function $g(x)=mx+b$ whose graph is tangent to the graph of…
MC Delta T
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Differentiating function with radicals

While working on the problem $g(x)= \sqrt{x^{2}-4x+4}$, I got stuck and plugged the formula into Wolfram Alpha. It simplified alot of things for me, but there is one section I'm having trouble with, since one of my weaknesses is radicals. According…
Jason
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$\lambda$ for which the function $f(x)=2x^3-3(2+\lambda)x^2+12\lambda x$ has exactly one local maxima and exactly one local minima

Let $S$ be the set of real values of parameter $\lambda$ for which the function $f(x)=2x^3-3(2+\lambda)x^2+12\lambda x$ has exactly one local maxima and exactly one local minima. Then the subset of $S$ is $(A)(5,\infty)$…
user1442
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Chapter 9, exercise 10 Michael Spivak's Calculus

The problem is this. "Find $f'(x)$ If $f(x)=g(t+x)$ and $f(t) = g(t+x)$. The answers will not be the same." The function $g(t+x)$ is a constant one, otherwise $f(x)$ would not be a function. Hence $f(x)$ is constant as well. Therefore $f'(x)$ is…
Rogelio
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Why do we seek for the $x$ where $f(x) = 0$?

I'm studying numerical analysis on college, and a question that has been following me (since I've started studying calculus) is why do we seek for the $x$ value where $f(x) = 0$? Can someone exemplify a real situation where this would be important?…
VSMelo
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How to prove the given function is not differentiable analytically?

Well the question presented to me is this. The given function is, $$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{1}{2}x + 2,\,\;\;\;x < 2}\\{\sqrt {2x} ,\;\;\;\;\;\;x \ge 2}\end{array}} \right. $$ Now have to check whether the given…
user449525
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