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

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2
votes
2answers
43 views

Prove second derivative of $g$ is proportional to $g^2$

From Apostol's Calculus Vol. 1, chapter 6.26, exercise 30: Let $f(x) = \int_0^x (1+t^3)^{-1/2} dt$. $a)$ Prove $f$ is strictly monotonic. $b)$ Let $g$ be the inverse of $f$. Show that the ...
-2
votes
0answers
49 views

How to solve the following integral?

I think the answer is D. but do is missing the + C? Can I multiply by (-1) to get the to answer number C? or I'm just completely confuse.
-1
votes
4answers
60 views

Please help with derivative question [on hold]

Which is the derivative of $\ln(e^{\ln x})$? a) $\ln x$ b) $x$ c) $e^x$ d) $\ln(\ln x)$ e) Other I'm pretty sure the answer is $1/x$ but I'm not confident enough to mark none of the above...
29
votes
2answers
2k views

When does L' Hopital's rule fail?

This thought jumped out of me during my calculus teaching seminar. It is well known that the classical L'Hospital rule claims that for the $\frac{0}{0}$ indeterminate case, we have: $$ ...
2
votes
1answer
30 views

Find slope of tangent line using m$_{tan}=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$ and the point $P = (5,\frac{2}{5})$

Perhaps I'm missing something simple here, but every time I attempt this problem I get the same answer that does not make sense. The question says, use the definition m$_{tan}=\lim\limits_{x\to ...
4
votes
1answer
62 views

Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$?

The exercise asks me this: Is there an injective function such that $f(x^2)-f^2(x)\ge \frac{1}{4}$? ps: $f: \mathbb{R}\to \mathbb{R}$ I really don't know how to start :c, I appreciate hints.
2
votes
2answers
47 views

What is the order of the PDE $\newcommand\pp\partial\frac{\pp^2u}{\pp x^2}+\frac{\pp^3u}{\pp x^2 \pp y}+\frac{\pp^2u}{\pp^2y}=xy\frac{\pp u}{\pp x}$? [on hold]

The order of the differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^3 u}{\partial x^2 \partial y}+\frac{\partial^2 u}{\partial^2 y}=xy\frac{\partial u}{\partial x}$$ is ...
1
vote
0answers
22 views

Distance histogram within cylinder

Suppose I randomly pick a pair of points $x$ and $y$ from inside a cylinder of radius $R$ and length $L$. What is the probability that they are a distance $d$ apart? In other words, I wish to evaluate ...
-1
votes
1answer
44 views

If $f$ is increasing toward $1$, then $\sup\{f(x)\sin x \}=1$

Suppose $f$ is an increasing monotone function in $(0,\infty)$. If $$\lim_{x \to \infty} f(x)=1$$ then $$\sup\{f(x)\sin x\mid x>0\}=1$$ I am not really sure how to approach this, any help will ...
0
votes
2answers
29 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
2
votes
1answer
30 views

Show that this construction is a parallelogram.

Let $ABC$ be a triangle. The middle of the segment $BC$ is denoted by $M$ and the centroid of $ABC$ is rated $G$. We construct $G'$ on the line $GM$ such that $|GM|=\frac{1}{2}|GG'|$ and ...
5
votes
4answers
108 views

Limit of a sequence of products

How do you prove the following? $$\lim_{n\,\to\,\infty}\,\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\ =\ 0$$
2
votes
1answer
26 views

Proof of absolute convergence [duplicate]

I am independently studying calculus using MIT's publicly available materials on OCW. One of the Final Exam practice question is the following: Suppose the series $\sum_{n=1}^{\infty}a_n$ converges ...
1
vote
0answers
24 views

Distance of the plane relative to the base of a Pyramid.

Consider a pyramid is cut by a plane parallel to its base. Question: What is the distance of the plane relative to the base so that the volume of the truncated pyramid so formed is $\frac{3}{8}$ of ...
6
votes
0answers
89 views

Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
0
votes
0answers
24 views

Calculus (implicit differentiation) [duplicate]

Consider the equation $x^2+3y^2+xy=3$. a) Write an expression for the slope of the curve at any point (x,y). The answer is: $dy/dx = (-2x-y)/(6y+x)$ b) Find the equation of the normal line ...
2
votes
1answer
42 views

$f$ differentiable and $f(0)=f(1)=0$. , prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$

Let $f$ be differentiable on $[0,1]$ and $f(0)=f(1)=0$. Also, we know $|f''(x)| \le A$ on $(0,1)$, prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$ I'm guessing I should use taylor ...
2
votes
2answers
70 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
0
votes
4answers
56 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
4
votes
4answers
97 views

Find $\lim\limits_{x \to \infty} \left(\sqrt{x^2+x+1} - \sqrt{x^2-x} \right)$

I am having a tough time with these TYPE of problems looking forward ideas, All ideas will be appreciated
0
votes
0answers
15 views

Deduction of the equation of continuity in one dimension

I need the deduction of the continuity equation in one dimension usig the result: $$\frac{\partial}{\partial t}\int^{b(t)}_{a(t)} \rho(x,t)dx=\rho(x,t)b'(t)-\rho(x,t)a'(t)+\int^{b(t)}_{a(t)} ...
1
vote
2answers
30 views

Finding the intersection of an xy-plane in a 3D-Coordinate System

I found the equation of a sphere that has a center of $(1,-12,8)$ with a radius of 10 and I got the following equation: $(x-1)^2 + (y+12)^2 + (z-8)^2 = 100$ As for finding an intersection for the ...
2
votes
2answers
92 views

Limit of $S(x) = x − x^2 + x^4 − x^8 + x^{16} − x^{32} + \cdots$ as $x$ approached $1$ from below

I have read the following (http://www.math.harvard.edu/~elkies/Misc/sol8.html) but I dont understand the last part of the solution: For positive $x<1$, consider the alternating sum $$S(x) = x − ...
1
vote
1answer
41 views

Finding minimum point of a function using linear algebra

Given a function $$q(x,y)=2x^2-2xy +2y^2$$. Find the minimum point of the following function by first converting it to a matrix form and using the diagonalisation of the matrix to find its minimum ...
4
votes
1answer
64 views

Prove that there exist $a \in [-1,1]$, such that $f'''(a)=3f(1)-3f(-1)-6f'(0)$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function. Prove that there exists $a \in [-1,1]$ such that $$ f'''(a) = 3f(1)-3f(-1)-6f'(0)$$ Any hint/idea's how to ...
1
vote
0answers
30 views

Proving a trigonometric inequality which is related to calculus

I need to prove the following inequality, without the help of a calculator. This problem was given to me in an exam. It is a trigonometric inequality but is somehow related to calculus (as that was ...
1
vote
2answers
31 views

Inverse Trigonometric functions - Boyce & Diprima 2.2.19

The problem asks for a solution to the initial value problem: \begin{align} &\sin(2x)dx+\cos(3y)dy=0\\ &y\left(\frac{\pi}{2}\right)=\frac{\pi}{3} \end{align} The problem is separable and I ...
1
vote
2answers
31 views

Proving that continuous map from subset of $\mathbb{R}^2$ is closed

Let $\delta>0$ and $f:\mathbb B((0,0),\delta)\to\mathbb R$ is a continuous map, where $\mathbb B((0,0),\delta)=\{(x,y) \in \mathbb R^2 \text{ such that } x^2+y^2\leq\delta^2\}$ Prove that there ...
0
votes
0answers
39 views

Show if $f:\mathbb{R}^2\to\mathbb{R}$, $f(x,y)=x+2y+3xy$ is invertible

I tried solving this, i know that for a function to be invertible it must be bijective thus injective and surjective. So I must show for injection that $x_1=x_2$ and for surjective that $y=f(x)$ but ...
-2
votes
1answer
102 views

Calculus (derivatives/slopes) [on hold]

Consider the equation $x^2+3y^2+xy=3$. a) Write an expression for the slope of the curve at any point (x,y). The answer is: $dy/dx = (-2x-y)/(6y+x)$ b) Find the equation of the normal line ...
4
votes
4answers
189 views

Finding $\frac{\mathrm d}{\mathrm dx} x!$

I'm trying to differentiate $x!$ but I just can't seem to do it right. I define the function as follows: $$x! = \prod_{r = 0}^{x}(x-r) \quad,\quad x \in \mathbb N$$ I've tried attempted to try it by ...
1
vote
2answers
14 views

taking partial derivatives of a modulus

Could anyone help me with this problem . Evaluate $\displaystyle\frac{\partial }{\partial x}$ |xy| $\displaystyle\frac{\partial }{\partial y}$ |xy| What i did was to take a point (a,b), then i ...
6
votes
2answers
105 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...
0
votes
0answers
21 views

How to add shipping cost to unit cost? [on hold]

If I have $\mathcal x$ number of units at different unit costs from supplier, after shipping's costs, how much shipping expenses will each unit carry? Basically I want to know how much each unit will ...
0
votes
1answer
25 views

Query regarding an alegbraic inequality

Do we have any inequality of the type $(a-b)^q \geq C_q(a^q-b^q),$ where $q>0$ and $a,b$ are real numbers?
2
votes
2answers
75 views

How to determine $\int e^{2x} \sqrt{e^x+1}dx$?

Determine $\int e^{2x} \sqrt{e^x+1}dx$ Is there a multiplication rule for integration or something?
0
votes
3answers
35 views

Calculus (what is y when x is?)

Given $y>0$ and $$dy/dx = (3x^2+4x)/y$$ If the point $(1,rad10)$ is on the graph relating x and y, then what is $y$ when $x=0$? I'm not sure whether or not to integrate, or just plug in the ...
2
votes
1answer
49 views

How to find the limit $\lim_{x\to 0} ( 4x/\sin 2x + x\cos2x )$?

Compute $$\lim_{x\to0}\left[-\dfrac{4x}{\sin 2x} + \dfrac{x}{\cos 2x}\right]$$ Obviously I can't plug in 0. I noticed the sin and cos are both 2x. Is there a way to combine them into tan? I don't ...
-2
votes
4answers
60 views

Calculus (limits)

Compute $$\lim_{t\to0}\frac{\tan\left(\dfrac {1}4\pi + t\right) - \tan\left(\dfrac{1}4\pi\right)}t$$ Alright, so I'm taking the derivative first. Is there an easier way to take the derivative of ...
1
vote
1answer
40 views

area and limit of the shaded region

Consider the shaded region outside the sector of a circle of radius 12 meters and inside a right triangle. a) write the area A= f(θ) of the region as a function of θ. I found the area of the ...
1
vote
1answer
31 views

Complex roots (review) (advise)

I have to find the complex roots and want a review of my procedure to see if is correct A. $$\sqrt{3i}$$ $$\left |z \right |=3 $$ $$phase= 90^{\circ}=\displaystyle\frac{\pi}{2}$$ ...
1
vote
2answers
53 views

Simple Derivative paradox

Suppose I define $y(x)=x^3$ $${dy(x) \over dx} = 3x^2$$ $${dy(x) \over dy} = 1 = 3x^2 \frac{dx}{dy} = 0\text{ since }x \neq f(y)$$ $1 \neq 0$ If you take the differential $d()$ where $dy(x)$ then ...
0
votes
1answer
58 views

Calculus (advanced integration)

Compute $\int (5^x+2e^{5 \ln x})dx$ The $5\ln x$ part confuses me. So far I have $5^x/\ln 5\:\:$
0
votes
0answers
18 views

Domain of function for a sector

I seem to have forgotten how to find the domain of a sector.The question asks to find the domain of the function 72(tan $\theta$ - $\theta$). This question was based on a previous question asking to ...
1
vote
1answer
45 views

Evaluate the limit, if it exists

In Exercises 5-34, evaluate the limit, if it exists. If not, determine whether the one side limit exists (finite or infinite). 26. $$\lim_{x\to ...
1
vote
0answers
91 views

Problem with trigonometric substitution proof

I'm sad, I can't get it. I know perfectly how to integrate using the mechanical process described in the books, but I want to understand the proof of it. My book (Stewart) says: In general we can ...
0
votes
3answers
15 views

Disk Method Problem, where Axis of Rotation is Shifted from the Y-Axis

Could somebody please check my work here? The textbook answer is 224pi/15, but I'm getting something different. Is it a set up error? Question: Find the volume of the solid generated by revolving ...
0
votes
2answers
48 views

Calculus (value of $c$)

The slope of the tangent line to the graph of $4x^2+cx-2e^y=-2$ at $x=0$ is $4$. Give the value of $c$. What I'm confused about here is the 3 variables. Is it implicit differentiation or something? ...
0
votes
2answers
24 views

How to find the slope of a secant line of the graph of a function?

The point $P(5,-2)$ lies on the curve $y=\large\frac{2}{4-x}$. (a). If $Q$ is the point $(X,\large\frac{2}{4-x})$, find the slope $M_{pq}$ of the secant line $PQ$ (correct to six decimal places). ...
2
votes
2answers
32 views

How to find local maximum of the function $f(x) = x^3-9x^2+24x+4$?

Give the value of x where the function $f(x) = x^3-9x^2+24x+4$ has a local maximum. a) -4 b) 4 c) 2 d) 3 e) -2 I graphed it and I'm not sure how to find the local max