0
votes
1answer
20 views

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

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
54 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
24 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
131 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 ...
0
votes
1answer
32 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
23 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
24 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
57 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 ...
11
votes
3answers
189 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
36 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
41 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
65 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
236 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
72 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
34 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
42 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
55 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
368 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 ...
0
votes
2answers
34 views

How to find the equation of a plane given the equation of a line?

Given the line $(x, y, z) = (1, -3, 2) + t(-2, 4, 7)$ , find planes to satisfy the following condition: A plane which is not intersected by the line I'm just starting to learn this and find it a ...
2
votes
3answers
113 views

Evaluating a limit involving trigonometry

I really thank you for your answers to my first question--I could easily solve first problem and a few more ones without another question. But a while later I got another one while studying, then I ...
2
votes
1answer
50 views

find optimized height

Please help me with this word problem: A light is to be placed directly above the center of a circular plot of $r=30\text{ ft}$, at such a height that the edge of the plot will get maximum ...
0
votes
1answer
20 views

How to tell if scalar plane equations are perpendicular to one another?

Just by looking at their equations, how can one tell that the following equations intersect at 90º? In other words, how can you tell that they're perpendicular to each other? Plane 1: 2x − y + 4z + 5 ...
2
votes
3answers
177 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
0
votes
2answers
22 views

A simple optimization problem of reciprocal function

Can someone tell me the answer to this question? I cannot seem to figure it out The function $y=\frac{2}{x}$ is decreasing in?? a.$(0,\infty)$ b.$(-\infty,0)$ c.$(0,2)$ d,$(-\infty,\infty)$ I ...
1
vote
1answer
59 views

Finding functions in a differential equation that satisfies two solutions

"Given a differential equation of the form $a(x)y''+b(x)y'+y=0$, find functions $a(x)$ and $b(x)$ so that $y=x$ and $y=x^2$ are each a solution of this differential equation." I'm really not sure how ...
4
votes
1answer
84 views

Solving for limit of integration

$$\frac{1}{\sqrt{2\pi}} \int^0_{z_a} e^{\frac{-z^2}{2}} \, dz = 0.48 $$ How would I solve for the value of $z_a$ using a calculator?
5
votes
2answers
355 views

Finding a rare case where an incorrect rule works?

"A not uncommon error in calculus is to believe that the product rule for derivatives states that $(fg)' = f'g'$. If $f(x) = e^{3x}$, find a nonzero function g for which $(fg)' = f'g'$." I believe ...
9
votes
2answers
105 views

Divergent of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
0
votes
1answer
24 views

extrema and inflection pt of trig function

Hi please help me solve: $y = \cos^2(x) - \cos(x)$: so far: $y'=-2 \cos(x)\sin(x)+\sin(x)$, $y'' = 2\sin^2(x) - 2\cos^2(x) + \cos(x)$ $y' = 0$ when $x = 0, \pi, \pi/3, 5\pi/3$ i plugged these ...
0
votes
3answers
78 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
-1
votes
1answer
29 views

Don't understand answer from exponential growth question

"A biology student finds a large glass bottle which can be used to grow a bacterial culture. She has a bacterial culture that doubles in size every minute, and with the amount she currently has, she ...
0
votes
1answer
51 views

Why do these acceleration computations contradict each other?

A rocket follows the path $(y-40)^2=160x$ after traveling vertically to the height of 40 feet (making for a very convenient continuity. The problem asks "If the component velocity in the vertical ...
0
votes
1answer
26 views

Using Midpoint Rule to Approximate a Definite Integral

I got this question wrong.. I started by obtaining the following sample points $-.7, -.3, .1, .5, .9$ Next I got my $\Delta x$ with the following computation $\displaystyle{\frac{1.1 - (-.9)}{5} = ...
2
votes
1answer
34 views

Limit of a Rational Trigonometric Function

When solving a trigonometric limit such as: $$\lim_{x \to 0} \frac{\sin(5x)}{\sin(4x)}$$ we rework the equation to an equivalent for to fit the limit of sine "rule": $$\lim_{x \to ...
1
vote
1answer
33 views

a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
0
votes
2answers
68 views

I Don't Understand Error Bounds

I understand they're supposed to give us a limit on how off our approximation of an integral can be, but I don't understand how the formula gives that. What does the second derivative have to do ...
0
votes
1answer
57 views

Calculus Problem general polynomial limit to infinity [duplicate]

I have to solve the following problem for homework for a calculus class. I really have no idea where to start, does anyone have any hints?: let n be a positive integer greater than 0. Let P(x) be a ...
6
votes
4answers
260 views

question on limits and their calculation

In taking each of the limits $$\lim_{x\to -\infty}\frac{x+2}{\sqrt {x^2-x+2}}\quad \text{ and } \quad \lim_{x\to \infty}\frac{x+2}{\sqrt {x^2-x+2}},$$ I find that both give the value $1$, although it ...
0
votes
1answer
28 views

Average Speed using variables

A cyclist travels $m$ km at $v$ km/h and $n$ km at $u$ km/h. What is the average speed of the cyclist? Is there a formula for calculating average speed?
0
votes
1answer
49 views

inverse of a function which contain logarithm

$f(x)=2^{x}(x-1) $is a bijective function . What will be the inverse of $f$ ? For this we want to find another function $g$ with composition of both give identity function. So I want to separate $x$ ...
-1
votes
2answers
70 views

I need help solving this related rates equation.

I need help answering the following question and I'll show you what I have. ! $$x=20,y=\sqrt{2100},z=50, \frac{dy}{dt}=30$$so differentiating $(20)^2+y^2=z^2$ $$2y\frac{dy}{dt}=2z\frac{dz}{dt}$$ And ...
1
vote
0answers
36 views

problem of computing limit

The problem is to prove the following for $n \geq 3$ $$u(0)=\frac{1}{n\alpha (n) r^{n-1}}\int_{\partial B(0,r)} g dS +\frac{1}{n(n-2)\alpha (n)} \int_{B(0,r)} (\frac{1}{|x|^{n-2}} - ...
2
votes
3answers
99 views

How to calculate integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
1
vote
2answers
38 views

Integration related question

how does one integrate $ (x^3-1) /( 4x^3-x) dx$ ? I tried dividing polynomials but it didnt help....