,

but clearly, 1/x diverges so I don't think it was very helpful.

Could someone help me what I need to do please?]]>

I am trying to find the derivative of this problem using the four step process but keep getting stuck when it comes to the third step of f(x+h) - f(x). I do not know what to do once I reach that step. Am I canceling terms out incorrectly? How should I deal with a fraction over a fraction? Any help would be really appreciated.

]]>

I am having trouble with finding a formula of the multiplication 3 formula power series.

\[ \left(\sum_{n=0}^{\infty} a_nx^n \right)\left(\sum_{k=0}^{\infty} b_kx^k \right)\left(\sum_{m=0}^{\infty} c_mx^m \right) \]

Work:

For the constant term:

$a_0b_0c_0$

For The linear term : $(a_1 b_0 c_0 + a_0 b_1 c_0 + a_0 b_0 c_1)x$ + $a_0b_0c_0$

For the quadratic term: $a_2 b_2 c_2 x^6 + a_2 b_2 c_1 x^5 + a_2 b_1 c_2 x^5 + a_1 b_2 c_2 x^5 + a_2 b_2 c_0 x^4 + a_2 b_1 c_1 x^4 +...

Finding a formula for the multiplication of multiple formal power series]]>

I need to do please? I just can't find this algebraic trick. Thank you in advance!

]]>

So

$a(t)=v'(t)

=\dfrac{\dfrac{(50-30)mi}{h}}{\dfrac{h}{6}}

=\dfrac{20 mi}{h}\cdot\dfrac{6}{h}=\dfrac{120 mi}{h^2}$

Hopefully ]]>

I have a problem when there are two variables. The only thing I did was that:

,

but I don't know if it was helpful. Thank you!]]>

If someone could please explain, I would be eternally grateful.]]>

Find all $\alpha \in \mathbb{R}$ such that $(\ln x_1)(x_2^2+x_2)=O(||X||^{\alpha})$ as $||X||\to 0$.

and $|X|| \to \infty$ (note that $x_1>0)$]]>

Since y=-x/2e+1/e+e is on top it is the first function.

A=(the lower boundary is 0 and the top is 2) -x/2e+1/e+e-e^x/2

If you could please help!!]]>

\(\displaystyle f(x)=A\left(((\frac{Q}{n-x}-\frac{R}{x})+(n-x))n\right)+Bnx+C((\frac{R}{x})n)\)

How can I find the value of x? I know that x can be between 1 to n-1. But how do I continue from there? I was thinking there must be...

Minimize a function: Find value of x that result in lowest value of formula]]>

$$y = x^2+1 \textit{and } y = x- x^2$$

Ok I think what question is ... The vertical distance between vertex's]]>

#20 this is the last one of the set

$ty'+(t+1)y=t \quad y(\ln{2}=1,\quad t>0$

rewrite

$y'+\left(\dfrac{t+1}{t}\right)y=1$

$u(t)=e^{\displaystyle\int\dfrac{t+1}{t}dt}=e^{t+\ln |t|}$

well anyway wasn't sure about the (t+1)]]>

$t^3y'+4t^2y=e^{-t},\quad y(-1)=1,\quad t>0$

rewrite

$y'+\dfrac{4}{t}y=\dfrac{e^{-t}}{t^3}$

$u(t)=e^{\int 4/t \, dt}=e^{4\ln \left|t\right|}=t^4$

first bold steps...

typos???]]>

$ ty'+2y=\sin t \quad y(\pi/2)=1 \quad t\ge 0$

$y'+\dfrac{2}{t}y=\dfrac{\sin t}{t}$

so

$u(x) = e^{\int 2/t dt}= e^{t^2/4}$]]>

$\displaystyle y^\prime - \frac{2}{t}y

=\frac{\cos{t}}{t^2};

\quad y{(\pi)}=0, \quad t>0$

$u(t)=e^{2 \ln{t}}$

then

$\displaystyle e^{2\ln{t}}\, y^\prime - \frac{2e^{e^{2\ln{t}}}}{t}y

= \frac{e^{2\ln{t}}\cos{t}}{t^2}$

not sure actually!]]>

Ok all I did was DesmosNot real sure how to take limit

]]>

2)Hyperboloid of one sheet

3) Hyperboloid of two sheet

4)Elliptic Paraboloid

5) Elliptic Cone 6) Hyperbolic Paraboloid

The equation of hyperbolic parabolid is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}\tag{1}$

Now, for example, z=2*x*y is a case of equation of mixed variables of the form $Ax^2+By^2+Cz^2+Dxy+...

How is z=2xy a Hyperbolic Paraboloid in the rotated 45° in the xy-plane?]]>

What is the smallest possible value of the sum of their squares?

$x+y=16\implies y=16-x$

Then

$x^2+(16-x)^2=2 x^2 - 32x + 256$

So far

... Hopefully]]>

$P=\dfrac{100I}{I^2 + I + 4} $

where I is the light intensity (measured in thousands of foot-candles). For what light intensity is $P$ a maximum?

OK I presume first we derive P'

$P'=\dfrac{ -100 (I^2 - 4)}{(I^2 + I + 4)}^2$

$P'(I)=0 $ at $x=\pm 2$

so far !!!]]>

1.1) If y^'=3y. Then y=e^2x .

1.2) If Then y=ce^kx for some constants c≠1 and k≠1.

1.3) If y^'=2y. Then y=2e^2x is .

1.4) If Then y=ce^x for some constant c .]]>

Find two positive numbers whose product is 100 and whose sum is a minimum

$x(100-x)=100x-x^2=100$

So far

Looks like it's 10+10=20

Doing all my lockdown hw here

since I have no access to WiFi and a PC.

and just a tablet where overkeaf does not work]]>

]]>

$$y=e^{x/2} \textit{ and the line } x=2$$

a. 2e-2 b. 2e c. $\dfrac{e}{2}-1$ d. $\dfrac{e-1}{2}$ e. e-1

Integrate

$\displaystyle \int e^{x/2}=2e^{x/2}$

take the limits

$2e^{x/2}\Biggr|_0^2=2e-2$

which is a.

any suggestions ????]]>

For $t \ge 0$ the position of a particle moving along the x-axis is given by $v(t)=\sin t—\cos t$ What is the acceleration of the particle at the point where the velocity is first equal to 0?

$a. \sqrt{2}$

$b. \, —1$

$c. \, 0$

$d. \, 1$

$e. —\sqrt{2}$

Ok well originally it was given as $x(t)$ but I changed it to $v(t)$

So via W|A $v(t)=0$ at

$t = 1/4 (4 π n + π), n \in Z$

So the first 0 would be $\dfrac{5\pi}{4}$

Hopefully so far ]]>

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$.

$f(x)=\sqrt{x} \quad [0,4]$

Are the secant line and the tangent line parallel?

$\dfrac{{f(b)-f(a)}}{{b-a}}$

then

$f(0)=0 \quad f(4)=2 \quad m=\dfrac{1}{2}$

then

$f'(x)=\dfrac{1}{2\sqrt{x}}

=\dfrac{1}{2}\quad\therefore \quad f'(1)=\dfrac{1}{2}$

then...

3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines]]>

Ok Just have trouble getting this without a function..]]>

Find y'

$$x^2-4xy+y^2=4$$

dy/dx

$$2x-4(y+xy')+2yy'=2x-4y-4xy'+2yy'=0$$

factor

$$y'(-4x+2y)=-2x+4y=$$

isolate

$$y'=\dfrac{-2x+4y}{-4x+2y}

=\dfrac{-x+2y}{-2x+y}$$

typo maybe not sure if sure if factoring out 4 helped]]>

$$F(x)=(7x^6+8x^3)^4$$

chain rule

$$4(7x^6+8x^3)^3(42x^5+24x^2)$$

factor

$$4x^3(7x^3+8)^3 6x^2(7x^3+4)$$

ok W|A returned this but dont see where the 11 came from

$$24 x^{11} (7 x^3 + 4) (7 x^3 + 8)^3$$]]>

Ok I thot I posted this before but after a major hunt no find

Was ??? With these options since if f(x) Is a curve going below the x axis Is possible and () vs []

If this is a duplicate post

What is the link.. I normally bookmark these

Mahalo ahead]]>

OK, this can only be done by observation so since we have v(t) I chose

here the WIP version of the AP Calculus Exam PDF as created in Overleaf

https://documentcloud.adobe.com/link/track?uri=urn:aaid:scds:US:053a75d8-ca5b-4447-bd65-4e580f0de793

the goal is to have 365 problems that align basically where students are first year calculus

always appreciate comments since it needs to be a group effort.]]>

$$

\int \limits_0^1 e^{-x^2} \leq \int \limits_1^2 e^{x^2} dx

$$

I know that $\int \limits_0^1 e^{-x^2} \leq 1$, but don't see how to take it from there.

Any ideas?]]>

$a. \ln\dfrac{2}{x}$

$b. \ln \dfrac{x}{2}$

$c. \dfrac{1}{2}\ln x$

$d. \sqrt{\ln x}$

$e. \ln(2-x)$

ok, it looks slam dunk but also kinda ???

my initial step was

$y=e^x$ inverse $\displaystyle x=e^y$

isolate

$\ln{x} = y$

the overleaf pdf of this project is here .... lots of placeholders...

https://drive.google.com/open?id=1WyjkfLAzhs4qF3RYOgSJrllP4hoKC5d4]]>

I worked this problem for a quite a while last night. I posted it here.

https://math.stackexchange.com/questions/3547225/help-with-trig-sub-integral/3547229#3547229

The original problem is in the top left. Sorry that the negative sometimes gets cut off in the photo, and yes I know it's not fully simplified there.

My first question is the more involved one: Is the algebra in my original work sound? If it is, why doesn't...

Trouble with trig integral]]>

ok well it isn't just adding the areas of 2 functions but is $xf(x)$ as an integrand

Yahoo had an answer to this but its never in Latex so I couldn't understand how they got $\dfrac{7}{2}$]]>

Let $v_1:\begin{pmatrix}1 \\ 1\\ 1\end{pmatrix}, \ \ v_2:\begin{pmatrix}1 \\ 0\\ 1\end{pmatrix}\in \mathbb{R}^3$.

- Let $w=\begin{pmatrix}1 \\ 0 \\2\end{pmatrix}\in \mathbb{R}^3$. If possible, give a linear map $\phi:\mathbb{R}^3\rightarrow \mathbb{R}^2$ such that $\phi (v_1)=\begin{pmatrix}1 \\ 0\end{pmatrix}, \ \ \phi (v_2)=\begin{pmatrix}0 \\ 1\end{pmatrix}, \ \ \phi (w)=\begin{pmatrix}0 \\ 0\end{pmatrix}$.
- Let $w'=\begin{pmatrix}0 \\ 1 \\0\end{pmatrix}\in...

Give a linear map that satisfies given properties]]>

I want to check the existence of the limit $\lim_{x\to 0}\frac{x}{x} $ using the definition.

For that do we use the epsilon delta definition?

If yes, I have done the following:

Let $\epsilon>0$. We want to show that there is a $\delta>0$ s.t. if $0<|x-0|<\delta$ then $|f(x)-1|<\epsilon$.

We have that $\left |f(x)-1\right |=\left |\frac{x}{x}-1\right |=\left |\frac{x-1}{x}\right |=\frac{|x-1|}{|x|}$.

How can we continue? ]]>

ok all I did was count squares so about 21]]>

$\displaystyle g'=2xe^{kx}+e^{kx}kx^2$

we are given $ x=\dfrac{2}{3}$ then

$\displaystyle g'=\dfrac{4}{3}e^\left(\dfrac{2k}{3}\right)+e^\left({\dfrac{2k}{3}}\right)\dfrac{4k}{9}$

ok something is ??? aren't dx supposed to set this to 0 to find the critical point

did a desmos look like k=-3 but ???

254 AP Calculus find k for critical point]]>

$$\dfrac{dy}{dx}=2\sin x$$

with the initial condition

$$y(\pi)=1$$

a. $y=2\cos{x}+3$

b. $y=2\cos{x}-1$

c. $y=-2\cos{x}+3$

d. $y=-2\cos{x}+1$

e. $y=-2\cos{x}-1$

integrate

$y=\displaystyle\int 2\sin x\, dx =-2\cos(\pi)+C$

then plug in $y(\pi)=1$

$-2\cos(\pi)+C=1

\Rightarrow

-2(-1)+C=1

\Rightarrow

C=-1$

therefore

$y=-2\cos(\pi)-1$...

253 AP Calculus Exam .... solution to the differential equation condition]]>

ok from online computer I got this

$\displaystyle\int_0^x e^{-t^2}=\frac{\sqrt{\pi }}{2}\text{erf}\left(t\right)+C$

not sure what erf(t) means]]>

ok I assume we take the derivative and then set it to zero

then set that into f(x)

I think it is (B)]]>

sorry bad post .... ignore]]>

The generalized meaning of this is if we are given a divergent integral, say \(\displaystyle \int _0^{\infty} f(x) ~ dx\), we can (supposedly) change the problem to \(\displaystyle \lim_{ \Lambda \to \infty } \int_0^{ \Lambda } R( \Lambda ) f(x) ~ dx\) such that the...

Regularization of an Non-conergent Integral]]>

We have the below maps:

- $f_1:\mathbb{R}^2\rightarrow \mathbb{R}^2, \ \ \begin{pmatrix}x \\ y\end{pmatrix}\mapsto \begin{pmatrix}-x \\ -y\end{pmatrix}$

- $f_2:\mathbb{R}^3\rightarrow \mathbb{R}^3, \ \ \begin{pmatrix}x \\ y\\ z\end{pmatrix}\mapsto \begin{pmatrix}x \\ -y\\ z\end{pmatrix}$

- $f_3:\mathbb{R}^3\rightarrow \mathbb{R}^3, \ \ \begin{pmatrix}x \\ y\\ z\end{pmatrix}\mapsto \begin{pmatrix}x+2 \\ y-3 \\ z+1\end{pmatrix}$

- ...

Geometric interpretation of maps]]>

Let $a\in \mathbb{R}$. We define the map $\text{cost}_a:\mathbb{R}\rightarrow \mathbb{R}$, $x\mapsto a$. We define also $-f:=(-1)f$ for a map $f:\mathbb{R}\rightarrow \mathbb{R}$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a map and $\lambda\in \mathbb{R}$.

Show that:

- for $a,b\in \mathbb{R}$ it holds that $\text{cost}_a+\text{cost}_b=\text{cost}_{a+b}$.
- for $a\in \mathbb{R}$ it holds that $\lambda\text{cost}_a=\text{cost}_{\lambda a}$.
- ...

Show the properties of the maps]]>

I want to prove the following properties:

- $\left (e^x\right )^y=e^{xy}$
- $\ln (1)=0$
- $\ln \left (x^y\right )=y\ln (x)$
- $a^x\cdot a^y=a^{x+y}$ and $\frac{a^x}{a^y}=a^{x-y}$
- $a^x\cdot b^x=\left (ab\right )^x$ and $\frac{a^x}{b^x}=\left (\frac{a}{b}\right )^x$
- $\left (a^x\right )^y=a^{xy}$

I have done the following:

- We have that $\displaystyle{\left (e^x\right )^y=e^{\ln \left (e^x\right )^y}=e^{y\cdot \ln e^x}=e^{y\cdot...

Properties of exponential/logarithm]]>

ok I chose

ok I got stuck real soon.....

.a find where the functions meet $$\ln x = 5-x$$

e both sides

$$x=e^{5-x}$$

ok how do you isolate x?

W|A returned $x \approx 3.69344135896065...$

but not sure how they got it

b.?

c.?]]>

If you know of good resources (both theories and problems) please let me know!

a) Calculate fourier and inverse fourier transform of f(t).

b) Calculate the limit.

My question is :

it looks like sometimes you use the definition of the transform with the integration of f(t) * e^-iwt ,

other times you use shortcuts from formula sheets...

fourier and inverse fourier transform]]>

If the independent variable of $W(\theta)=2\theta^2$ is restricted to values in the interval [2,6]

What is the interval of all possible values of the dependents variable?]]>

How to find formulas for these$\displaystyle\int x^n\sin(x)\, dx, \displaystyle\int x^n\cos(x)\, dx,$ indefinite integrals when $n=1,2,3,4$ using differentiation under the integral sign starting with the formulas

$$\displaystyle\int \cos(tx)\,dx = \frac{\sin(tx)}{t}, \displaystyle\int \sin(tx)\,dx= -\frac{\cos(tx)}{t}$$ for $t > 0.$

I don't have any idea to solve these indefinite integrals except to solve them recursively using integration by parts.]]>

View attachment 9488

ok I really have a hard time with these took me 2 hours to do this

looked at some examples but some had 3 variables and 10 steps

confusing to get the ratios set up... ok my take on it is here

see if you can solve it under 5 min

]]>

ok I chose

image due to graph, I tried to duplicate this sin wave on desmos but was not able to.

so with sin and cos it just switches to back and forth for the derivatives so thot a this could be done just by observation but doesn't the graph move by the transformations

well anyway???]]>

or is there some more anointed way to do this.

ok this one baffled me a little

but isn't $g = e^{-t^2}$

and the graph of that has an inflection point at x=1]]>

ok these always baffle me because f(t) is not known. however if $f'(t)>0$ then that means the slope is aways positive which could be just a line. but could not picture this to work in the tables.

Im sure the answer can be found quickly online but I don't learn by copy and paste. d was attractive but where would the slope be?

so any sugest..]]>

What is the total distance traveled by the particle from time $t = 0$ to $t = 3$

ok we are given $v(t)$ so we do not have to derive it from a(t) since the initial $t=0$ we just plug in the $t=3$ into $v(t)$ so

$6(3) - (3)^2=18-9=9$.

hopefully

any sugestions?

I thot I might of posted this earlier but I could not find it]]>

I tried to do this just by observation, but kinda hard with a piece wise function

so would presume

= 2x\biggr|_1^3 + \left(\dfrac{x^2}{2}-x\right)\biggr|_3^5=4+6=10$

i wasn't sure about the notation of limits when you have an inequality in the function

I have \(\displaystyle \int \frac{x^2 - 5x + 16}{(2x + 1)(x - 2)^2} \, dx\)

and I got to this part:

\(\displaystyle x^2 - 5x + 16 = A(x - 2)^2 + B(x - 2)(x + \frac{1}{2}) + c(x + \frac{1}{2})\)

So do i need to distribute all of these and factor out or is there a simpler way? I found a solution where they are just saying oh x = 2 and plugging it in but I am so lost on how they are getting that random value...

Partial Fraction Decomposition Evaluation]]>

Since t>0 we can multiply both sides with heaviside stepfunction (lets call it \theta(t)).

What I am unsure about is what happens with the integral part and how do we inpret the resulting expression?

What will it result in and how will be laplace transform the integral parts? I am also wondering what the laplace transform of y(t) will be.]]>

let $A\neq \emptyset\neq B$ be sets, $C\subseteq A$, $D\subseteq B$ subsets and $f:A\rightarrow B$ a map.

I want to show that the set $\{f^{-1}(\{x\})\mid x\in \text{im} f\}$ is a partition of $A$.

To show that the set $\{f^{-1}(\{x\})\mid x\in \text{im} f\}$ is a partition of $A$, we have to show that the union of $\{f^{-1}(\{x\})\mid x\in \text{im} f\}$ is equal to $A$, the sets $\{f^{-1}(\{x\})\mid x\in \text{im} f\}$ are disjoint and the empty set is not an element of...

Show that the set is a partition of A]]>

Check the below sequences for convergence and determine the limit if they exist. Justify the answer.

- $\displaystyle{f_n:=\left (1-\frac{1}{2n}\right )^{3n+1}}$
- $\displaystyle{g_n:=(-1)^n+\frac{\sin n}{n}}$

I have done the following:

- $\displaystyle{f_n:=\left (1-\frac{1}{2n}\right )^{3n+1}}$

We have that:

\begin{align*}\lim_{n\rightarrow \infty}\left (1-\frac{1}{2n}\right )^{3n+1}&=\lim_{n\rightarrow \infty}\left...

Check convergence of sequences]]>

The function f(x) is defined below:

\[\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.\]

I want to find for which values of a and b the function is differential at x = 1.

The test I was given, is to check the continuity of both f(x) and f'(x). This is fairly easy technically. Checking continuity is only calculating two limits and comparing them.

My question is why this is true. Why the continuity of both f(x) and f'(x) at a point means the function is...

Differentiability and continuity]]>

image due to macros in Overleaf

ok I think (a) could just be done by observation by just adding up obvious areas

but (b) and (c) are a litte ???

sorry had to post this before the lab closes]]>

(x-4) (x²+3)]]>

$\displaystyle I=\int_{0}^{3\pi/2}\int_{0}^{\pi}\int_{0}^{\sin{x}}

\sin{y} \, dz \, dx \, dy$

integrat dz

$\displaystyle I=\int _0^{3\pi/2}\int _0^{\pi }\sin(y)\sin (x)\, dxdy $

integrat dx

$\displaystyle I=\int _0^{3\pi/2}\sin \left(y\right)\cdot \,2dy$

integrat dy

$I=2$

ok I think its correct, took me an hour to do... trig was confusing

typos mabybe]]>

image due to macros in overleaf

well apparently all we can do is solve this by observation

which would be the slope as x moves in the positive direction

e appears to be the only interval where the slope is always increasing]]>

I am trying to solve a limit:

\[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\]

I found many suggestions online, from complex numbers to Taylor approximations.

Finally I found a reasonable solution, but one move there doesn't make sense to me.

I am attaching a picture:

I have marked in blue the move I can't understand. How does the minus from the exponent goes down before the half ?

And one more thing, at the final move, I know that the...

Limit involving a hyperbolic function]]>

$a.4\quad b. 5\quad c. 6\quad d. 7\quad e. 8$

see if you can solve this before see the proposed solution

since

$$f'(x)=7+\dfrac{1}{x}$$

so then

$$f'(x)=7+\dfrac{1}{1}=7+1=8\textit{ is (e)}$$

$$f'(x)=7+\dfrac{1}{x}$$

so then

$$f'(x)=7+\dfrac{1}{1}=7+1=8\textit{ is (e)}$$

ok not real sure what the answer is but I did this (could be easier Im sure}

rewrite as

$y=(2x+1)^3$

exchange x and rename y to g

$x=(2g+1)^3$

Cube root each side

$\sqrt[3]{x}=2g+1$

isolate g

$g=\dfrac{\sqrt[3]{x}-1}{2}$

so

$\left(\dfrac{\sqrt[3]{x}-1}{2}\right)'

=\dfrac{1}{6x^{\dfrac{2}{3}}}$

apply $x=1$...

219 AP calculus Exam .... inverse function]]>

I want to calculate the following limits:

\[\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}\]

using the sandwich rule, where [xa] is the integer part function defined here:

Integer Part -- from Wolfram MathWorld

I am not sure how to approach this. Any assistance will be most appreciated.]]>

Why first terms equal to zero ? ? ?

]]>

A. 0.606

B 2

C 2.242

D 2.961

E 3.747

ok i tried to do a simple graph of y= with tikx but after an hour trying failed

doing this in demos it seens the answer is D

I know you take the differential set it to zero and that should give you the x value of intersection]]>

ok image to avoid typo.... try to solve before looking at suggested solutions

[HR][/HR]

so then we have first $$\displaystyle y=\int 2\sin x \,dx=-2cos x +C$$ if $y(\pi)=1$

then

$$y(\pi)=-2cos(\pi) +C=1 $$

then

$$2cos(\pi)+1=C $$

and

$$2(-1)+1=C=-1$$

and finally

$$y=-2cos{x}-1$$

which is $\textbf{(E)}$

ok I think you could do this by observation if you are careful with signs]]>

Consider the sequence $a_n=\frac{8^n}{n!}$.

It holds that

$$a_{n+1}-a_n=\frac{8^{n+1}}{(n+1)!}-\frac{8^n}{n!}=\frac{8 \cdot 8^n}{(n+1) \cdot n!}-\frac{8^n}{n!}=\frac{8^n}{n!}\left( \frac{8}{n+1}-1\right)=\frac{8^n}{n!} \left( \frac{7-n}{n+1}\right)$$

Since the last term is positive for some $n$ and negative for others, we cannot conclude like that if the sequece is monotonic or not.

I haven't thought of an other criterion which we could use to see if the sequence is...

How to check monotony of sequence]]>

ok this is a snip from stewards v8 15.6 ex

hopefully to do all 3 here

$\displaystyle\int_0^1\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}}\dfrac{z}{y+1} \,dxdzdy$

so going from the center out but there is no x in the integrand

$\displaystyle\int_0^{\sqrt{1 - x^2}} \dfrac{z}{y + 1}dx =\dfrac{ \sqrt{1 - x^2} z}{y + 1}$

and

$\displaystyle\int_0^1 \dfrac{ \sqrt{1 - x^2} z}{y + 1} \, dz=\frac{\sqrt{1-x^2}}{2\left(1+y\right)} $

kinda maybe so far???]]>

(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.

(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.

If you use a major theorem, then cite the theorem, and verify that the conditions of

the theorem are satisfied.

Should I be using Rolle's Theorem/Mean Value Theorem?]]>

$$\int_0^1\int_y^{2y}\int_0^{x+y}

6xy\, dy\, dx\, dz$$

OK this is an even problem # so no book answer

but already ???? by the xy]]>

$$\begin{array}{|c|c|c|c|c|c|c|}

\hline

t\,(minutes)&0&4&9&15&20\\

\hline

W(t)\,(degrees Farrenheit)&55.0&57.1&61.8&67.9&71.0\\

\hline

\end{array}$$

The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W. where W(t) is measured in degrees Fahrenheit and t is measured in minutes.

At time $t=0$. the temperature of the water is $55^o F$.

The water is heated for 30 minutes, beginning at time $t=0$,

Values of...

309 AP Calculus Exam average temperature]]>

ok basically t is 3 hours appart except between 7 and 12 of which I didn't know if we should intemperate.

other wise it is just adding up the 4 $(t)\cdot(R(t))$s.]]>

Question: An area in the first quadrant (x=>0,y=>0) is limited by the axis and the graphs to the functions f(x)=x^2-2 and g(x)=2+x^2/4. When the area rotates around the y-axis a solid is created. Calculate the volume of this solid.

I want to calculate with the disc method. I set h(x)=g(x)-f(x) since g(x)>f(x) in this area. Since x>=0 i check where the lower function crosses the x-axis and get

x^2-2=0 which gives...

Decide volume given two functions]]>

First I list what I am given:

Diameter: 46m

DR: 0.0003m (converted from the 0.03cm that was given)

DV: ???

Volume of a Sphere = 4/3 pi r^3

But volume of a half a Sphere = 2/3 pi r^3

And Radius = Diameter/2

So...dv = 2 pi (23m)^2 (0.0003m)

Using a Calculator I am given 0.3174. The question...

Using differentials to estimate the amount of paint needed for a hemispherical dome...]]>

Question: The limited area in the plane is created when the space between the line y=1 and the graph to the function f(x)=3*x/(x^2+1) rotates around the y-axis. Calculate the volume of the solid.

I want to sum up all the circular discs that make up the body in order to get the volume. One disc has the

base area: pi*radius^2 * the height dy. Since its rotating around the y axis i assume i first need to find the inverse by...

Calculate volume of a solid rotating around the y-axis]]>

ok just posted an image due to macros in the overleaf doc

this of course looks like a sin or cos wave and flips back and forth by taking derivatives

looks like a period of 12 and an amplitude of 3 so....

but to start I was not able to duplicate this on desmos

altho I think by observation alone I think which choices represent the graph.]]>

$\tiny{11.6.(4) } $

$$L=\sum_{n=1}^{\infty}\dfrac{\ln(n+1)}{n+1} $$

using the Ratio Test

$$L=\displaystyle\lim_{n \to \infty}

\left|\dfrac{a_{n+1}}{a_n}\right|

=\lim_{n \to \infty}\dfrac{\ln((n+1)+1)}{((n+1)+1)}

=\lim_{n \to \infty}\dfrac{\ln((n+2)}{n+2}=0$$

thus $L<1$ convergent

Ok I think this is correct but the final limit I did via W|A not sure why it is 0]]>

Assume that g is function such that

(i) g(c)= c+m(x-1)

(ii) f(1) = g(1), and

(iii) \lim_{{x}\to{1}}\frac{f(x)-g(x)}{x-1}

Answer the following questions. Show all of your work, and explain your reasoning.

(a) What are the constants c and m?

(b) How does g compare with the linearization of f at 1?

For a, I have that the constant c=1, but I'm having trouble determining the constant m. I also am not sure what is required to answer part b.]]>

$$S_n= \sum_{n=1}^{\infty} (-1)^{n+1}\frac{\sqrt{n}+6}{n+4}$$

ok by graph the first 10 terms it looks alterations are converging to 0]]>

lim [(x^7)-9(e^x)] / [sqrt(10x-1)+8*ln(x)]

x->infinity

Prompt: Find the limit and the dominant term in the numerator and denominator.]]>

of the series.

$$\sum_{n=1}^{\infty}\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$

(1)

$$a_n=\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$

(2)

$$\left|\dfrac{a_{a+1}}{a_n}\right|

=\left|\dfrac{(-1)^{n+1} x^{n+1}}{\sqrt[3]{n+1}}

\cdot\dfrac{\sqrt[3]{n}}{(-1)^n x^n}\right|

=-\frac{\sqrt[3]{n}x\left(n+1\right)^{\frac{2}{3}}}{n+1}$$

(3) W|A Convergence Interval is

$$-1\le \:x\le \:1$$

ok on (2) I was expecting a different result to take...

11.8.4 Find the radius of convergence and interval of convergence]]>

Ok not sure if I fully understand the steps but presume the first step would be divide both sides deriving

$$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$$

offhand don't know the correct answer

$\tiny{from College Board}$]]>

ok not sure of the next step but

$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$]]>

ok again I used an image since there are macros and image

I know this is a very common problem in calculus but think most still stumble over it

inserted the graph of v(t) and v'(t) and think for v'(t) when the graph is below the x-axis that participle is moving to the left

the integral has a - interval but I think the total is an absolute value...

my take on some of it.

finally this will be my last AP calculus exam question for a while

I was...

314 AP Calculus Exam a particle moves along the x-axis.....]]>

I just posted a image due to overleaf newcommands and graph

ok (a) if we use f(20) then the $B=0$ so their no weight gain.

(b), (c), was a little baffled and not sure how this graph was derived...]]>