-3
votes
1answer
40 views

Calculus use of integral [on hold]

Assume that the price of a product is at a constant value of $\$100$ per unit or the marginal function is $MR=f(x)=100,$ where $x$ equals the number of units sold $a)\ $ What is the total revenue ...
-1
votes
2answers
55 views

Maclaurin series of the function $\frac{x^2}{2+3x^2}$

I got this question: Find the Maclaurin series of the function $\frac{x^2}{2+3x^2}$ and find its domain of convergence. I tried using the binomial series $(1+x)^m = 1 + \sum_{k=1}^{\infty}{m \choose ...
1
vote
3answers
110 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
2
votes
3answers
221 views

Summation of Infinite Geometric Series

Determine the sum of the following series: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} $$ My work: $$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} = \sum_{n=1}^{\infty } \frac{-1}{7} ...
2
votes
2answers
96 views

How to I write $\frac{7^{2n}}{4^{3n}}$ as a geometric series?

I am trying to write $$\frac{7^{2n}}{4^{3n}}$$ as a geometric series which has the form:$$\sum\limits_{i=0}^n{ar^n}$$. I'm not sure if I should get in the form $$\left(\frac{7}{4}\right)^{2n}$$ ...
2
votes
0answers
37 views

What is a good technique for evaluating this double integral?

The integral is: $ \int_0^1 \int_0^1 \frac{x^2 - y^2}{(x^2 + y^2)^2} dxdy $. I'm having difficulty finding an appropriate technique for evaluating it. I initially thought that polar coordinates ...
0
votes
2answers
33 views

Could anybody provide a more detailed explanation of a tangent equation in its general form?

In my textbook I'm currently at the topic of a tangent line to an ellipsis and hyperbola. And there I've encountered this statement: If a curve has an equation $$ y = f(x) $$ then an equation of a ...
2
votes
2answers
54 views

Convergence/Divergence of a the series $\sum_{k=1}^{\infty} a_k$, where $a_1=1$ and $\forall 1\leq k\in\mathbb{N},a_{k+1}=\cos(a_k)$

I got this question: Determine wether the series $\sum_{k=1}^{\infty} a_k$ absolutely converges, conditionally converges or diverges, where $a_1=1$ and for each $1\leq k \in\mathbb{N}$, ...
0
votes
3answers
63 views

Can you factor before finding derivative?

Say the function is $y=\frac{x^2-1}{x-1}$ Can you factor functions before finding the derivative or does that not work?
3
votes
2answers
64 views

Finding $f(x)$.

If $$f(x)=1+x+x^2+\displaystyle\int_{0}^{x}e^k f(x-k) dk$$ then how do we find the function $f(x)$? Is there a way to solve it, with or without arriving at a differential equation? This a homework ...
1
vote
1answer
30 views

Is it possible to have a inflection on a vertical asymptote?

I found the derivative of a function to be f'(x)=8/x^3 and thus its second derivative as f''(x)=0/3x^2. After setting the second derivative to zero and doing the substitution into the parent function, ...
2
votes
3answers
87 views

Can an inflection exist if there's no max/min?

Very quick question: if a function doesn't have a maximum nor minimum, can it still have a point of inflection? I believe that these two go hand in hand and without one you can't have the other but ...
0
votes
1answer
66 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
1
vote
6answers
49 views

Using Chain Rule and Product Rule to find derivative

I have to find the derivative of the following function: $$f(x) = (x^3+ 4)(4x^5 + 2x − 5)^{1/2}$$ To start solving this, I've dissected the equation and realize that I must use the product and chain ...
2
votes
1answer
46 views

Simplify: $\ln(x^2 − 4)− \ln(x − 2)− \ln 2$

Simplify: $$\ln(x^2 − 4)− \ln(x − 2)− \ln2$$ $$\ln\dfrac{x^2 − 4}{x − 2}− \ln2$$ $$\ln(x + 2)− \ln2$$ $$\ln(x + 2)/2$$ I got this far, is there any other way to simplify it, or do I stop here?
2
votes
2answers
137 views

Solve for $x$: $\frac1e = e^{2x}$

I tried making it to $e^{-1} = e^{2x}$ and had the exponents equal each other $-1=2x$ and the I solved for $x$, making it $x=-1/2$, but that answer is wrong. please help I don't know why that ...
1
vote
1answer
23 views

How is the power rule applied to whole numbers

For the following function, how does the $+1$ become $0$ when finding its derivative via the power rule? Original function: $f(x) = 6x^2 − 4x^{-1} + 5x^{-2} − 2x + 1$ Derivative: $f '(x) = 12x + ...
0
votes
1answer
32 views

Total derivative proof [closed]

The wikipedia article does not prove it http://en.wikipedia.org/wiki/Total_derivative Neither the top articles in google search. Could somebody help me proving it? I've found this: ...
1
vote
1answer
42 views

Evaluating $\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$

$$\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ \sin^{8}(\pi/6+h))-\sin^{8}(\pi/6) \right ]$$ My Attempt: For $\lim_{h\rightarrow 0}\frac{\sin^{8}(\pi/6+h)-\sin^{8}(\pi/6)}{h}=f'(x)=8\cdot ...
-1
votes
1answer
12 views

Equation of a line with a positive gradient [closed]

Two straight lines passing through the point (0,2) are tangent to the graph of the function y=1-x^2. Find the equation of the line with a positive gradient.
0
votes
1answer
48 views

Help with math steps, chain rule.

I'm trying to to understand the math steps to go from Eqn. (1) to Eqn. (2). $$\tag{1} q(x,t)=\frac{-V_t(1+\delta f(c,g))}{P(x,t)}\cdot \left(\frac{dP_o}{dt}\right)$$ $$\tag{2} \frac{-V_t ...
2
votes
1answer
96 views

Evaluate the limit $\lim \limits_{x \to \infty} \frac{1}{x(x+1)}$ [closed]

How can I evaluate the limit $$\lim_{x \to \infty} \frac{1}{x(x+1)}$$
3
votes
0answers
63 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
0
votes
1answer
16 views

Using differentials, estimate the difference in the deflection between the point midway on the beam and the point 1 10 ft above it

So I've been trying to figure out the problem for about an hour and I cannot figure it out. Question: To study the effect an earthquake has on a structure, engineers look at the way a beam bends when ...
1
vote
3answers
74 views

Applications of calculus

We have the following formula for area $$A = r^2(\sinθ\cosθ-\sqrt{3}\sin(θ)^2)$$ We then need to find what value θ will give maximum area, so we differentiate to get; $$ ...
11
votes
3answers
225 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
12
votes
1answer
148 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
2
votes
1answer
32 views

Check the properties of the eigenfunction corresponding to the distinct eigenvalues of an integral equation

Let $\lambda_1, \lambda_2$ be eigenvalues and $f_1 , f_2$ be the corresponding eigenfunctions for the homogeneous integral equation \begin{align} \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) ...
3
votes
5answers
197 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
4
votes
5answers
389 views

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?

I have the following limit: $$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$ If I was finding the limit of only the terms inside the natural log function, I would have the ...
1
vote
1answer
20 views

Normal line to a curve $C_1$

Find the interval for $a$ so that $(3-a)x+ay+(a^2-1)=0$ is normal to the curve $xy=4$ $(C_1)$. I approached it this way-- $C_1$ is $xy=4$. So, $\dfrac{dy}{dx}$ for $C_1$ is $\dfrac{-4}{x^2}$. ...
0
votes
1answer
24 views

Calculating optimum values of $u$ and $m$ from $\mathbb V(\bar {y_2}\prime)=\frac{S_2^2(n-u\rho^2)}{n^2-u^2\rho^2}$

I have to find optimum sample size in sampling on two occasions. Suppose that the samples are of the same size n on both occasions. In selecting the second sample, $m$ of the units in the first ...
6
votes
3answers
250 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
0
votes
1answer
45 views

Find $\frac{dy}{dx}$ of $y=\sqrt{u}$

Find $\dfrac{dy}{dx}$ of $y=\sqrt{u}$, $u=7-x^2$ This is on my homework and I don't know what to do exactly. Steps would be helpful!
6
votes
5answers
185 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
0
votes
2answers
23 views

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent.

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. $y(x)= x^4-500x+2$ So I know the first thing to do is find the derivative which is: $y'(x) = 4x^3-500$ ...
0
votes
1answer
44 views

How do I go about factoring this polynomial?

I am horrid at factoring and I have to find the inflection points of $ f(x)=x^2(x − 3)^3$. So I to find the inflection points I need to set $f'$ equal to $0$ So I have ...
-4
votes
1answer
38 views

Find the percent increase or decrease

For each function, find the percent increase or decrease that the function models $y=1298 \cdot 1.63^x$ $f(x)=2 \cdot 0.65^x$
1
vote
2answers
50 views

solution of a quadratic equation

If I have an equation of a form: $$x^2+\alpha x + 10 =0$$ my book says that both roots have the same sign because "10" is positive. I'm trying to understand why the book makes this claim. Is there ...
1
vote
3answers
37 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
0
votes
3answers
42 views

Differential equation True/ False

Every continuous function has an antiderivative I thought this statement was false, but it seems that it is true. I thought that it suppose to be every antiderivative is a continuous but the ...
0
votes
0answers
35 views

After removing the parameter from $x=\sec \theta$ and $y=\cos\theta$, why does the domain become $|x|\geq1, |y| \leq1$?

For the parametric equations $x=\sec \theta$ and $y=\cos\theta$ with initial domain $0\leq\theta\lt\frac{\pi}{2}$, $\frac{\pi}{2}\lt\theta\leq\pi$, I understand that you arrive at $y = \frac{1}{x}$ ...
1
vote
0answers
39 views

The volume is to be found

Find the volume of $A=\{(x,y,z) \in \mathbb{R}^3: 2x^2+3y^2 \leq z \leq 4+2x+3y\}$ I know we are to solve it by using triple integral...
4
votes
2answers
308 views

True/ False differential equation

Are the statements in Problems 46-54 true or false? If $F(x)$ is an antiderivative of $f(x)$, then $y=F(x)$ is a solution to the differential equation $\frac{dy}{dx}=f(x)$. If $y=F(x)$ is a solution ...
1
vote
1answer
52 views

How to find the value of $c$ using the mean value theorem?

So I'm doing Mean Value theorem homework which states $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ I have $f(x)=e^{\frac{-x}{2}}$ over the interal [0,12]. Using the mean value theorem I ...
1
vote
1answer
92 views

How Can I figure out when cosine = $\frac{2}{\pi}$?

So I'm doing Mean Value theorem homework which states $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ So I am trying to find $c$ for $f(x)=\sin x$ over the interval $[0,\frac{\pi}{2}]$. So using the Mean Value ...
-1
votes
1answer
27 views

How to find the volume of revolution around a vertical line x [closed]

How can I evaluate the volume of a solid generated by the following lines using the washer method: $y=x$, $y=0$, $y=4$. Rotated about $x=5$. I have tried to find the outer radius of $5-x$ and the ...
1
vote
1answer
72 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
0
votes
1answer
28 views

How do I find the critical values to find the maximum of this function?

The total daily profit in dollars realized by the TKK Corporation in the manufacture and sale of x dozen recordable DVDs is given by the total profit function below. $$P(x) = −0.000001x^3 + 0.001x^2 + ...
3
votes
3answers
141 views

I need help finding the critical values of this function.

So $h(t)=t^{\frac{3}{4}}-7t^{\frac{1}{4}}$. So I need to set $h'(t)=0$. So for $h'(t)$ the fattest I've gotten to simplifying os $h'(t)=\frac{3}{4 \sqrt[4]{t}}-\frac{7}{4\sqrt[4]{t^3}}$ and that is as ...