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

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2
votes
0answers
13 views

Coming up with a function or a single graph, given its characteristics (pre-calculus)

Give an example of a function or a single graph which has the following characteristics: Hole at $(3,-1)$ Domain is all real numbers except $3$ Local minimum at $(-1,-2)$ Local ...
1
vote
2answers
30 views

Finite series identity

How would I prove this statement? I know that it's a finite series. I don't know how to approach this at all. $$\sum_{i=1}^N i^3 = \left(\sum_{i=1}^N i \right)^2$$
0
votes
0answers
15 views

Trig substitution triangle restrictions

I apologize if this is a dumb question, or if I am a little slow, but I've been thinking about this for all of yesterday and today and I just can't figure it out, despite googling it. I am confused ...
3
votes
3answers
38 views

Sum of infinite geometric series

How do I evaluate this (find the sum)? It's been a while since I did this kind of calculus. $$\sum_{i=0}^\infty \frac{i}{4^i}$$
0
votes
1answer
19 views

Area under a parabolic trajectory

I have this problem: "prove that the area under the trajectory described by a parabolic shot that has: $f(x)=\tan(\theta)x - (\frac{g}{2v^2\cos^2(\theta)})x^2$ and $x=v\cos(\theta)t$ is defined ...
1
vote
1answer
20 views

Identifying the formula for a quartic graphic

I am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't ...
1
vote
2answers
45 views

Solve L'Hopitals problem

$$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x}{{\sec^2 3x}} $$ I used LH: $$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x \tan x}{6\sec 3x \sec 3x \tan 3x}$$ then: $$\lim_{x\rightarrow ...
0
votes
0answers
14 views

Fiding the most general antiderivative of a function bounded by two x's.

At first I thought this problem would simply become a definite integral since it appears two be bounded by two x's. However, I feel as though I may be wrong and I'm curious as to how I would approach ...
1
vote
1answer
32 views

Use L'Hopital's with this problem?

The problem is: $$\lim_{x\rightarrow 0^+} \left(\frac{1}{x}\right)^{\sin x}$$ I know the answer is $1$ because I checked with my graphing calculator, but how exactly do I get there? I got this far: ...
-3
votes
1answer
8 views

Calculus :Work of an inverted right circular cone

A tank in the shape of an inverted right circular cone has height 6 meters and radius 4 meters. It is filled with 5 meters of hot chocolate. Find the work required to empty the tank by pumping the hot ...
3
votes
2answers
36 views

Trouble solving this differential equation: $x'=3(x-2)$, $x(0)=-1$.

Find the solution of the differential equation x'=3(x-2) given initial value condition of x(0)=-1 Here's my attempt. x'=3(x-2) dx/dt = 3(x-2) dx/x-2 = 3dt int dx/x-2 = int 3dt+c ln|x-2| = 3 + C ...
1
vote
1answer
19 views

Divergence theorem and applying cylindrical coordinates

This time my question is based on this example Divergence theorem I wanted to change the solution proposed by Omnomnomnom to cylindrical coordinates. $$ \iiint_R \nabla \cdot F(x,y,z)\,dz\,dy\,dx = ...
1
vote
2answers
15 views

Parametric Representation for a Square with Side $1$ Centered at the Origin as a Function of the Angle Measured from the Positive $x$-Axis

While playing with some graphics progamming in OpenGL, I've encounterd this problem: Find the Parametric representation for a square with side $1$ centered at the origin as a function of the angle ...
3
votes
2answers
61 views

Since $\lim\limits_{x\to0}\frac{\sin kx}{kx}=1$ for constants $k$, is it also true for general arguments?

To be more specific, is it true that $$\lim_{x\to0}\frac{\sin f(x)}{f(x)}=1~~?$$ I'm tempted to say yes at first glance, so long as $f(x)\to0$ as $x\to0$. The reason I ask is to verify this limit ...
1
vote
2answers
90 views

Prove the limit is $\sqrt{e}$.

How do you show $$\lim\limits_{k \rightarrow \infty} \frac{\left(2+\frac{1}{k}\right)^k}{2^k}=\sqrt{e}$$ I know that $$\lim\limits_{k \to \infty} \left(1+\frac{1}{k}\right)^k=e$$ but I don't ...
0
votes
2answers
22 views

interpreting $(1-t)f(a)+tf(b)$ from $f((1-t)a+tb)\leq (1-t)f(a)+tf(b)$

For a convex function $f((1-t)a+tb)\leq (1-t)f(a)+tf(b)$ holds. I understand how the graph looks like but why is the equation of the secant line $(1-t)f(a)+tf(b)$? Can anyone pleasae give me a ...
0
votes
1answer
24 views

Consider a function $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all real roots of $f(x)$ and $t=|s|$. Then…

the real number $s$ lies in the interval (A)$(-0.75,-0.5)$ (B)$(-0.5,0)$ (C)$(0,1)$ (D)$(-0.25,0)$ and the area of region bounded by $f(x),y=0,x=0$ lies in the interval (A)$(0.75,3)$ ...
1
vote
0answers
20 views

Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.

I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
0
votes
0answers
16 views

HJM Model vs Leibniz integral rule

I state that I'm an electronic engineer (undergraduate), then the my knowledges about advanced mathematics are almost null. A colleague asked to me an help about one point of the proof of the theorem ...
8
votes
1answer
69 views

What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?

This was inspired by similar posts like this one. Define the function, $$F(p) = \lim_{n\to\infty}2^n\sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}_{n \textrm{ square roots}}}$$ We know that, ...
0
votes
1answer
14 views

Calculating the length of a helix

I have a pipe and I want to put a wire through it in a helix form. I need to calculate how long the wire (wl) has to be. I know the internal diameter (id), and therefore the circumference (c). I know ...
-1
votes
1answer
26 views

simple percentage problem [on hold]

a man sold a watch of rs 2400 at a loss of 25%.at what rate should he sold the watch to earn a profit of 25%
0
votes
0answers
8 views

Solving for the poisson rate

Say I have an equation of the form $$ 0 = -a + \sum_{k=0}^\infty f(k, \lambda)R(k)\\ 0= -a + \exp(-\lambda)\sum_{k=0}^\infty \frac{\lambda^{k}}{k!}R(k) $$ where $f()$ is the Poisson mpf, $a$ is a ...
-2
votes
1answer
32 views

Area under the given curve [on hold]

The area under the curve $\displaystyle y = \frac{|x-3| + |x+1|}{|x+3| + |x-1|}$ , $x$-axis and the ordinates at $x = -3$ and $x = 1$
0
votes
1answer
25 views

Finding the gradient of a function.

A function $f=f(x,y)$ has continuous partial derivatives , and assume that maximal directional derivative of $f$ at $(0,0)$ is equal to $100$ and is attained in the direction towards $(3,-4)$ , we ...
0
votes
1answer
84 views

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $?

How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .
1
vote
2answers
31 views

Sketching functions $f(x) = \frac{e^x}{x^2} \quad \text{ and } \quad g(x) = \frac{1}{x}$ - First Derivative test and domain restriction

when working on a "Sketching a function" problem, some textbooks have a step-by-step procedure. The first one is usually stating the Domain of a function. When working with functions like $$ f(x) = ...
-2
votes
0answers
57 views

Rolle's theorem question

Let $f(x)=\sin2x/e^{2x}$. Note that $f$ is continuous on $[0,\pi/2]$, and differentiable on $(0,\pi/2)$, with $f(0)=f(\pi/2)=0$. So by Rolle's theorem, there exists a $c\in(0,\pi/2)$ with $f'(c)=0$. ...
2
votes
0answers
53 views

solving definite integral problems without complex line integral

It is well known that some definite integrals such as $$\int_{0}^{\infty} \frac{dx}{a+\cos{x}}$$ $$\int_{0}^{\infty} \frac{\sin{x}}{x}dx$$ are solved by using complex analysis techniques. (It uses ...
4
votes
4answers
52 views

I'm stuck in this one of trig substitution for fuctions.

I got this: $$\int\frac{dx}{\sqrt{(4x^2-9)^3}}.$$ I know that the answer is: $$\frac{x}{9*\sqrt{4x^2-9}}+c.$$ And with the steps that I know about this type of substitution, I came up here, but.. ...
0
votes
0answers
60 views

Prove √2 exists by Archimedean Axiom [duplicate]

I am trying to prove the existence of the square root of 2. The proof: Let $$S=\{x \in \mathbb{R} ∣x \ge 0, x^2 < 2\}.$$ I understand the proof of LUB, $\alpha$ and so I am at the step where ...
4
votes
1answer
38 views

Integrating $\frac{x^3}{(81-x^2)^2}$

I've been trying to figure out this integral for an hour or so now, but keep failing. I can't figure out where I go wrong: $$I = \int \frac{x^3}{(81-x^2)^2} dx$$ Let $x = 9sin\theta \implies dx = 9 ...
-1
votes
2answers
48 views

What is the error in the following working?

$$\frac{\int_0^1 (1-x^{50})^{100}\mathrm{d}x}{\int_0^1(1-x^{50})^{101}\mathrm{d}x}$$ The question asks us to evaluate 5050 times the above fraction> To solve this i had made the following ...
3
votes
0answers
14 views

Trig substitution using reference triangles

Suppose we are doing a trig substitution and make some substition $x = a \sin \theta \equiv \sin \theta = \frac{x}{a}$ where the domain of x is $|x| \le a$ Then from the reference triangle we can ...
1
vote
1answer
18 views

Class $C^1$ function on a compact set

The problem is: Let g be of class $C^1$ on $\Delta$⊂$ℝ^n$ and K be a compact subset of Δ. Show that there is a number C such that |g(s)-g(t)|≤C|s-t| for every s,t∈K. I have proved that it is true ...
0
votes
3answers
41 views

Integrating trig substitution triangle equivalence

When we integrate certain integrals, such as $$\int \frac{x^2}{\sqrt{16-x^2}} dx$$ We can make a substitution like $x = 4 \sin \theta$ Then we can simplify the above integral to the following: $$8 ...
0
votes
3answers
37 views

Integrate $\frac{x^2}{\sqrt{16-x^2}}$ using trig substitution

During our integration of the following integral, using $x = 4 \sin \theta$ $$\int \frac{x^2}{\sqrt{16-x^2}} dx$$ We eventually come to the following point: $$\int \frac{16 {\sin ^2 \theta} }{4 ...
1
vote
1answer
15 views

Finding the directional derivative.

We need to find the directional derivative of the function , $f(x,y) = x^{2}+y^{2}+xy$ at $P(1,-1)$ in the direction towards origin. The direction towards origin form the point $(1,-1)$ is ...
6
votes
2answers
99 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
0
votes
1answer
23 views

Calculus III Vectors - Projectile problem

A projectile is fired from ground level with an initial speed of $450 m/sec$ and an angle of elevation of 30 degrees. Use that the acceleration due to gravity is $9.8 m/sec^2$. The range of the ...
1
vote
0answers
32 views
2
votes
2answers
46 views

$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$

So the question is: $$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$$ First of all, I know we have to use Lhopital's rule. However, I just don't know how. Second of all, I thought in the end we ...
-6
votes
0answers
45 views

Find the limit: $\lim_{x \to 1}x^2 + 2$ [on hold]

What is the limit as $$\lim_{x\to 1} (x^2+2)?$$ I greatly appreciate your help and I will continue to type to satisfy the minimum length requirement please do not downvote this question as it will ...
0
votes
3answers
22 views

Related Rates Shadow Problem

The question is as follows: A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, (a) at what rate ...
7
votes
2answers
68 views

Summation of the reciprocals of the product of consecutive integers

It is well known that there is a closed formula for: $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{(n)(n + 1)}$$ And likewise for: $$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot ...
-1
votes
3answers
65 views

How to solve a Definite Integral (calculus 2)

I've begun to learn about these types of questions, however I still have a hard time knowing how to solve them and doing the actual computation. These three integrals are examples of the questions I ...
0
votes
2answers
33 views

Domain and range of $f(x,y)=\sqrt{1+x-y^2}$

I need to find the domain and range of $f(x,y)=\sqrt{1+x-y^2}$. Can someone walk me through the proper reasonings in solving this problem? My attempt Domain From looking at the function I get: ...
1
vote
0answers
30 views

Need help analytically solving an integral

For one part of a multiple-part problem, I need to analytically find the value of the integral $$I=\int_{x_L}^{x_R}\frac{dx}{\sqrt{-Ax + B - C/x}}$$ My professor gave us notes that: ...
0
votes
1answer
17 views

which of the following is an equivalence relation of the set S

which of the following is an equivalence relation of the set S I have solved all except d and need your help please
0
votes
1answer
27 views

Parametric curve parametriced by length

Normally you have a parametric curve with a variable t and you increment t to find the point along the curve. Is it possible to have a curve so that given a value it will give you the point on that ...