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

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2answers
138 views

Why is $\lim_{x\to 0} \frac{f(x)}{g(x)} = \lim_{x \to 0} \frac{f(x) - f(0)}{g(x)-g(0)}$ ?

In my lecture notes: Why is $\lim_{x\to 0} \frac{f(x)}{g(x)} = \lim_{x \to 0} \frac{f(x) - f(0)}{g(x)-g(0)}$ and so on. I know its trying to get to "$\frac{\text{change in y}}{\text{change in ...
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1answer
489 views

Prove: $f(x)= \sum_{n=1}^{\infty} \frac{\sin nx + \cos nx }{n^3}$ is a differentiable function, and it's derivative is differentiable continuous.

I need to prove that $$f(x)= \sum_{n=1}^{\infty} \frac{\sin nx + \cos nx }{n^3}$$ is well defined on $\mathbb{R}$, is a differentiable function, and it's derivative is differentiable continuous. I ...
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1answer
150 views

Does the infinite sum of $\frac{n+1}{\ln(n+1)}$ converge?

Does the infinite sum $\displaystyle \sum\frac{(n+1)}{\ln(n+1)}$ converge? I actually know it doesn't since if we use the integral test, and let $\ln(n+1)=u$ and $\displaystyle du=\frac{1}{n+1}dx$, ...
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1answer
89 views

How can I find the volume of a solid of revolution

I am finding a bounded volume. The question says Find the volume of the solid obtained by rotating the region bounded by $y=x^{2}$ and $x=y^{2}$. Rotating about $y=1$ I got an intercept of ...
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2answers
177 views

How does one define cross product

Cross product has a wide application in many field like in physics.Torque and circular motion are its application. But how does one define cross product and why they defined in that way??
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1answer
400 views

Relationship between Kronecker's Approximation Thm and Weyl's Equidistribution Thm?

According to Prof. Wikipedia, the Equidistribution Theorem was proved by Weyl in 1910 and independently by two others around the same time. The theorem states that for $\alpha$ irrational, the ...
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3answers
2k views

Proving the inequality: $[x-(x^2)/2 < \ln(1+x) < x]$ , $x>0$

Prove the inequality: $[x-(x^2)/2 < \ln(1+x) < x]$ , $x>0$ The right side is easy, I used taylor expansion to show that $e^x > 1+x$ since $e^x = 1 + x + x^2/2 + x^3/3! +\cdots $ The ...
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3answers
739 views

what do the brackets mean? $\lim _{n\rightarrow \infty }{n}^{3/2}[\sqrt {{n}^{3}+3}-\sqrt {{n}^{3} -3}] $

Calculate: $\lim _{n\rightarrow \infty }{n}^{3/2}[\sqrt {{n}^{3}+3}-\sqrt {{n}^{3} -3}]$ What do the brackets mean? I know sometimes they are used to denote a function that returns only the integer ...
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1answer
60 views

representing arcsinh as a logarithm

I am trying to understand this equality: $$ \ln{\left|\frac{x}{2}+\sqrt{\frac{x^2}{4}+1}\right|} + C= \ln{|x+\sqrt{x^2+4}|} + C'$$ My teacher didn't really explain it, she just noted that "the ...
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2answers
356 views

Computing taylor series for trigonometric exponential function

How do I compute the taylor series for $\cos(x)^{\sin(x)}$ ? I tried using the $e^x$ rule but I still am not getting to the result: $$\cos(x)^{\sin(x)}=1-\frac{x^3}{2}+\frac{x^6}{8}+o(x^6).$$
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3answers
116 views

Proving a point exists on a differentiable function

I have a homework question which is: If $f : [a,b]->R$ is continuous in $[a,b]$ and differentiable at $(a,b)$ and exists a point $c$ in $(a,b)$ such that $(f(c)-f(a))(f(b)-f(c))<0$ then ...
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2answers
194 views

Finding the limit of a function with a trigonometric exponent

I'm sure i'm just missing the trick here, would appreciate some help. I tried lhopital and it didn't help, perhaps a trig identity?
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2answers
160 views

approximation formula for the integral

Get an approximation formula for the following integral: $$ \sum_{k=1}^n \left( \frac{1}{35} \right)^{k-1}\int_0^{\frac{\pi}{2}}\cos^{2(n-k)+1}(y) \cdot \sin^{2(k-1)}(y) \, dy $$
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3answers
143 views

The Ambiguous Use of Differential in Solving Differential Equations?

Is there any theoretical basis for employing differential methods like separation of variables in solving differential equations? As we all know, differential is formally defined as a linear ...
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1answer
355 views

Sequence of functions: Cauchy or not

The question is $f_n(x) = ax^n + b \cos(x/n)$ is a sequence of functions where $f_n: [0,1] \to \mathbb{R}$. Determine for which $a,b \in \mathbb{R}$ values, $f_n$ is Cauchy w.r.t. the sup-norm in ...
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2answers
74 views

Proving a set of a subset of the range of a function has a maximum

I have a homework question which is: $f(x)$ is continuous on $(0,1)$ , $\underset{x\to 0+}{\mathop{\lim }}\,f(x)=-1$ and $\underset{x\to 1-}{\mathop{\lim }}\,f(x)=1$ Let ...
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2answers
647 views

Use Lagrange Remainder Theorem to Prove Inequality

I'm supposed to use Lagrange Remainder Theorem to prove that $$1 + \frac{x}{2} - \frac{x^2}{8} < \sqrt{1+x} < 1 + \frac{x}{2} \text{ } \text{ if } x>0$$ Obviously, the left and right hand ...
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2answers
94 views

Solve $a, b$ in the improper integral

Find $a, b \in\mathbb{R}$ such that $$\int^\infty_1\left(\dfrac{2x^2+bx+a}{x(2x+a)}-1\right) \mathrm{d}x=1$$
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1answer
255 views

Simplifying function notation

For example, in the process of proving that $$\left({\frac{f}{g}}\right)'\left({a}\right)= ...
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1answer
225 views

Proof of Spivak's statement that if $0<|x-a|<\delta$, then $|g(x)-m|< \min(\frac{|m|}{2},\frac{\epsilon|m|^2}{2})$

Michael Spivak, in his Calculus textbook pp 89, has tried to prove that given $\lim \limits_{x\to a} g(x)=m $, where $m\not=0$, $\lim _{x\rightarrow a} (\frac{1}{g})(x)=\frac{1}{m}$. To prove it, he ...
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2answers
130 views

Is $\sqrt{|xy|}$ equal to $\sqrt[4]{x^2y^2}$?

This is a noob calculus question. (1) $\sqrt{|xy|} = \sqrt[4]{x^2y^2}$, are the 2 expressions equal? if yes to (1), then why $\frac{d}{dx}\sqrt{|xy|} \neq \frac{d}{dx}\sqrt[4]{x^2y^2}$ ? as ...
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2answers
328 views

Derivative of this function (quadratic over quadratic)

$$K = \frac{-(\sigma^2 + 3\sigma + 2)}{\sigma^2 - 8\sigma +15}$$ How is this differentiated with respect to $\sigma$? The answer is $$\dfrac{dK}{d\sigma} = \frac{11\sigma^2 - 26\sigma ...
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3answers
131 views

Simplifying a square root

How could I simplify the following equation (C1) to become the next equation (C2) knowing that $r \gg d$ ($d$ is significantly smaller than $r$) $$\begin{align*} C_1 &= \frac{\epsilon_0 \pi ...
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1answer
86 views

Inverse and Derivative of $g'(x)=(1+x^{3})^{-1/2}$ Question

I am stuck on the following question: Suppose that $g$ is differentiable with derivative $g'(x)=(1+x^{3})^{-1/2}$. Show that the inverse function $h=g^{-1}$ satisfies ...
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1answer
95 views

How do I factor this kind of equations?

I was doing some integration by partial fractions exercises and I found this equation:$$\int_{0}^{1}\frac{x^{3}+1}{x^{4}+4x+3},$$ and I don't know how to factor that in order to compute the partial ...
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1answer
280 views

Optimization price per unit

I have no idea how to do this, I tried a lot of things but they don't make sense and I have too many variables. A manufacturer has been selling lamps at the price of \$6/lamp, and at this price ...
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2answers
304 views

What is exponential decay, exactly?

On the web (ie, Wikipedia, and other sites) it seems that exponential decay is always defined as the situation $f\;'(t) = -kf(t)$. However, is not Newton’s Law of Cooling an example of exponential ...
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2answers
145 views

How to integrate this trigonometry function?

The question is $ \displaystyle \int{ \frac{1-r^{2}}{1-2r\cos(\theta)+r^{2}}} d\theta$. I know it will be used weierstrass substitution to solve but i did not have any idea of it.
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1answer
407 views

How do I understand math? [closed]

I know I have asked how to learn math before, and everyone basically told me that I can't memorize and that I have to understand the concepts. Well I didn't think that was possible, and I still don't. ...
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3answers
3k views

Limit of Exponential Function

I am stuck on a question involving the limit of an exponential function, as follows $$\lim_{z \to \infty} \left ( 1-\frac{4}{z+3} \right )^{z-2}$$ The following hint is given: $$ \text{Assume that} ...
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1answer
87 views

Distribution of integration constant (c) in separable differential equation

This equation describes leaking water form a conical tank. We are interested in finding $t$ when $h(t) = 0$ (time it takes to empty the tank). $$ \frac{dh}{dt} = - \frac{5}{6h^{3/2}}, h(0) = 20 $$ ...
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3answers
1k views

Finding a differential

I am trying to compute $\Delta y$ and dy for a given value of x and dx = $\Delta x$ I am given $y=2x-x^2$ x=2 and $\Delta x = -.04$ Not really sure what to do here, feels like something is missing. ...
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1answer
13k views

any number raised to the power of infinity

1) I saw in a book that "the limit as $x$ approaches positive infinity of $e^x$ equals $0$" I want to ask about this? 2) if the $a$ is a negative number and we take a limit like "the limit as $x$ ...
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2answers
96 views

ODE question: $y'+A(t) y =B(t)$, with $y(0)=0, B>0$ implies $y\ge 0$; another proof?

I am trying to prove that the solution for the different equation $$y'+A(t) y =B(t)$$ with initial condition $y(0)=0$ and the assumption that $B\ge 0$, has non-negative solution for all $t\ge ...
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1answer
68 views

I was trying to prove the multiplication rule for limits but I got stuck trying to factor $a_nb_n-LM$

How was the author able to factor the expression from the left side to the expression on the right? $$a_nb_n-LM=(a_n-L)(b_n-M)+M(a_n-L)+L(b_n-M)$$ Thanks!
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1answer
95 views

implicit differentiation - basic question from lang

I'm reading Lang's First Course in Calculus 5e. I am stuck on p. 104, where he introduces Implicit Differentiation. Here's the text: "Find the derivative $dy/dx$ if $x^2 + y^2=7$, in terms of $x$ ...
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1answer
62 views

derivative of $f(t) = 4.9t^2$, dividing by $h$

I can't figure out where I erred in calculating this derivative: $$f(t) = 4.9t^2$$ $$f'(t) = \frac {f(t+h) - f(t)} {h} = \frac {4.9(t+h)^2 - 4.9t^2} {h}$$ $$= \frac {4.9(t^2 + h^2 +2ht) - ...
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1answer
127 views

Limit of a function with restricted domain

Given a function $f(x)$ whose domain is $\left[-4,6\right]$, is it possible to find $$\lim_{x\rightarrow 6^+} f(x)?$$
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1answer
153 views

Specific steepness of function

How can I find the point of a function where the steepness is exactly 45 degrees? In my specific case, the function is $\dfrac{-1}{\exp(x)}$.
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1answer
909 views

Find the maximum area possible of equilateral triangle that inside the given square

How can I find the maximum area possible of equilateral triangle that inside a square whose sides have length a. And how does that triangle look like? Can we construct it (with compass and ...
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1answer
218 views

Multivariate Maximization

For fixed $g$, I want to find maximum $b$ with $$-2b(3t^2(s+1)+6t(s+1)+3s+2)-2g(6ts+3t+6s+2)-3ts^2+6ts+3t-3s^2+3s+1>0$$ for some nonnegative reals $t,s$. Here $g, b$ are also $\geq 0$. Can it be ...
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1answer
537 views

Finding the direction cosines of a vector

I am having difficulty solving a problem presented to me by a fellow classmate. I was given a diagram, but I will describe the problem without one, as it is not necessary to fully understand the ...
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1answer
119 views

Standardized Integration

Why is it that there is an equation into which you can put a function in order to get that function's derivative, but such a thing doesn't exist for integration?
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1answer
680 views

Limit of a rational function where denominator approaches zero

I'm trying to solve $$ \lim_{x\to\infty}\frac{3x+5}{x-4} $$ Since the numerator and denominator both increase without bound, I try to get something more useful by dividing everything by $ x $. $$ ...
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1answer
150 views

Help with a derivative

I need to take the derivative of the following function w.r.t. $x$. (This is the General Beta of the Second Kind density function.) $$\frac{a}{bB(p,q)}\frac{(x/b)^{ap-1}}{(1+(x/b)^a)^{p+q}}$$ ...
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2answers
160 views

Volume of a solid of revolution

Let $[a,b]$ be an interval, $a\geq 0$ and $f:[a,b]\to \mathbb{R}_+$ continuous. I want to calculate the volume of the solid of revolution obtained by rotating the area below the graph of $f$ around ...
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1answer
76 views

The lower bound of the product between two variables

I wonder how I can determine the minimum of the product between variables $x$ and $y$ (in terms of $\theta$), given that both $x < 1 - \theta$ and $y < 1 - \theta$, and $x + y = 1$? So far I ...
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2answers
201 views

Examine the Maximum and Minimum Value

I have to do the following problem, and I need help. Examine the function $f(x,y) = \dfrac{-3x}{x^2+y^2+1}$ with respect to maximum and minimum.
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2answers
158 views

Does this equality always hold?

Is it true in general that $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x} \int_0^{x} f(u,x) \mathrm{d}u = \int_0^{x} \left( \frac{\mathrm{d}}{\mathrm{d}x} f(u,x) \right)\mathrm{d}u +f(x,x )$ ? Thank ...
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2answers
259 views

Sketching a polar curve

Continued off the question I asked earlier, I also have to sketch the curve. $r^2=−4\sin(2\theta)$ So I have to set up a table of values I'm assuming. How do I know what values to choose for ...