1
vote
0answers
29 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
69 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
216 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
135 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
29 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
191 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
377 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
17 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
191 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 ...
-3
votes
2answers
45 views

Left and right hand limit [on hold]

I need to find $$\lim_{x\to 1} \{x\}$$ where $\{x\}$ is fractional part of $x$. I know how to find limit for polynomials. Here, I don't know how to find left and right hand limit of $1$. Could you ...
0
votes
1answer
44 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!
5
votes
5answers
175 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
21 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
32 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
33 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
40 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
34 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
305 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
0answers
30 views

Find the center of mass of a region with uniform density [closed]

A region on the graph is bound by the lines $y=x/2$, $y=0$, $x=2$ How can I calculate the center of the mass assuming a uniform density of "p" throughout the region?
-1
votes
1answer
26 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
139 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 ...
1
vote
1answer
35 views

Formula alteration

is there any way to transform the formula$ \frac {1-x}{x-3}$ into something that can be easily sketched, or which will help eliminate $x$ from the denominator?
-5
votes
1answer
31 views

to prove partial derivative of a function f is bounded [duplicate]

Let$$ f(x,y) = \begin{cases} 0 & (x,y)=(0,0) \\ \dfrac{x^3}{x^2+y^2} & (x,y) \neq (0,0) \end{cases}$$ Prove that $D_1 f$ and $D_2 f$ are bounded ...
0
votes
2answers
32 views

Approximating volume using differentials

A closed box with dimension $10$ cm, $8$ cm, $6$ cm, is made of $2$ mm thick plywood. Approximate the volume of material used in making the box. We have $V=xyz$ We can find what the approximate ...
1
vote
1answer
24 views

Estimate the decrease in the period of the satellite to the nearest one-hundredth hour…

According to Kepler's Third Law, the period T (in days) of a satellite moving in a circular orbit x mi above the surface of the earth is given by $T=.0588(1+\frac{x}{3959})^{\frac{3}{2}}$ Suppose that ...
0
votes
1answer
31 views

I need help figuring this error percentage homework problem.

Question: Government economists in a certain country have determined that the demand equation for soybeans is given by $p = f(x) = \frac{55}{2x^2 + 1}$ where the unit price p is expressed in dollars ...
1
vote
2answers
31 views

Finding the power series representation for $\ln(1 -10x)$ via integration.

I'm trying to find the power series representation for $ \ln(1-10x) $ Attempt at solution: $$ \ln(1-10x) = \int {-10\over1-10x} \ dx = -10 \int \sum_{n=0}^\infty (10x)^n dx $$ $$ = -10 ...
4
votes
1answer
58 views

Is it true that $\lim_{x\to a}f(x)=0$ if and only if $\lim_{x\to a}|f(x)|=0$?

Is it true that $\lim_{x\to a}f(x)=0$ if and only if $\lim_{x\to a}|f(x)|=0$? I intuitively think this is true, but really no idea to prove it. Can you give me hints?
0
votes
1answer
40 views

Basic calculus integral estimation problem

You are given the table below. $$\begin{array}{|c|c|c|c|c|c|}\hline x & \color{red}2 & \color{red}4 & \color{red}6 & \color{red}8 & \color{red}{10}\\\hline ...
12
votes
3answers
201 views

How to show that $ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $

How to show that $$ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $$ where $e = \lim \left({1 + \dfrac 1 n}\right)^n$ I'm guessing this can be done using the Squeeze Theorem by applying the AM-GM ...
1
vote
1answer
40 views

Volumes of Revolution Washer Method

I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$. ...
0
votes
1answer
43 views

How to go about solving this question on differentials?

A ring of a planet has an inner radius of approximately 52,000 km (measured from the center of the planet) and a radial width of 19 km. Use differentials to estimate the area of the ring. (Round ...
0
votes
1answer
13 views

Integral, left-hand sum

Could anyone explain why my first answer is wrong? what I did was delta x = 10/5 = 2 $$ 2(2^2+1)+2(4^2+1)+2(6^2+1)+2(8^2+1) = 248 $$ and the second answer was $$ ...
2
votes
0answers
71 views

Given the function $f(x)=(1+x)^n$ Show that $L(x)=1$+nx is the linearization of $f$ at $0$ …

So this is how the question goes. 1. Given the function $f(x)=(1+x)^n$.$$$$ a. Show that $L(x)=1$+nx is the linearization of $f$ at $0$.$$$$ b. A friend claims that the cube root of 1.1 is ...
3
votes
4answers
237 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
0
votes
0answers
40 views

solving the logaritham [duplicate]

I was trying to solve: $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ heres my attempt at it; using logaritham laws and a little algebra we get from $\log_2 x ...
1
vote
3answers
75 views

If $f(3) = 12$ and $f(2.8) = 12.6$. Then approximate $f'(3)$

I am new to this site and old to my problem in calculus. I hope, some one will guide. If $f(3) = 12$ and $f(2.8) = 12.6$. Then approximate $f'(3)$ ? Please let me know
1
vote
2answers
35 views

Is there anything more I can say about a vector function that is parallel to its derivative?

The problem is to find the set of curves in $\mathbb{R}^3$ given by a vector equation $\mathbf{r}(t)$ with the property that the vector $\mathbf{r}'(t)$ is parallel to $\mathbf{r}(t)$ for all $t$ in ...
0
votes
2answers
45 views

Finding the equation of a plane, provided a line and a point?

Question: Given the line $$\begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \\ 2 \\ \end{pmatrix} + t \begin{pmatrix} -2 \\ 4 \\ 7 \\ \end{pmatrix},$$ find a plane which is ...
1
vote
2answers
56 views

Minimizing Question

A closed box constructed from a tin sheet has a square base and a volume of $343 \text{in}^3$. Find the dimensions of the box, assuming the minimum amount of material was used in its construction. ...
1
vote
3answers
377 views

The high power integral

Im trying to solve the indefinite integral $$\int\frac{x}{(x^2+4)^3} \, \mathrm{d}x $$ I tried applying polynimial division and breaking to partial fractions but it didnt help...are there any other ...