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As follows:

Like so:

Find the set of values a for which the range of the function

\(f(x) = \frac{x^2 + a}{x + 1} \)

is the set of all real numbers.

Hello Guest!

\(a\in \mathbb R\ |\ a\neq -1\)

  !

A cylinder and a cone have the same volume. The cylinder has radius x and height y. The cone has a radius of 1/3x. Find the height of the cone in terms of y.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The height of a cylinder     y = V / r²pi

The height of a cone        (n)y = 3V / (r / 3)²pi

                                      (n)y / y = [3V / (r / 3)²pi] / (V / r²pi) = 27

The height of a cone = 27y

Define the sequence of positive integers \(a_n\) recursively by
\(a_1=3\) and \(a_n=3a_{n- 1}\) for all \(n\ge2\).
Determine the last two digits of \(a_{2007}\).

\(\begin{array}{|rcll|} \hline a_1 &=& 3 \\ a_2 &=& 3*a_1 \\ &=& 3*3 \\ &=& 3^2 \\ a_3 &=& 3 * a_2 \\ &=& 3*3^2 \\ &=& 3^3 \\ \dots \\ \mathbf{ a_{2007} } &=& \mathbf{ 3^{2007} } \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline && 3^{2007} \pmod{100} \\ && \boxed { \text{Euler:}\\ 3^{\phi(100)} \equiv 1 \pmod{100} \quad \phi{100}=100*\left(1-\dfrac12 \right)*\left(1-\dfrac15 \right)\\ \phi{100}=40\\ 3^{40} \equiv 1 \pmod{100} }\\ &\equiv& \left(3^{40}\right)^{50}*3^7 \pmod{100} \\ &\equiv& 1^{50}*3^7 \pmod{100} \\ &\equiv& 3^7 \pmod{100} \\ &\equiv& 2187 \pmod{100} \\ \mathbf{ 3^{2007} \pmod{100} } &\equiv& \mathbf{ 87 \pmod{100} } \\ \hline \end{array}\)

Recall that a \(\text{Fibonacci sequence}\) is one in which all entries
from the third onward are the sum of the two entries before it.
If the \(5th\) and \(8th\) entries of a Fibonacci Sequence are \(-19\) and \(79\), respectively,
then what is the first entry of this sequence?

\(\begin{array}{|rcrcrl|} \hline a_1 &=& &a_1& \\ a_2 &=& && &a_2\\ a_3 &=& &a_1& + &a_2 \\ a_4 &=& &a_1& + &2a_2 \\ a_5 &=& &2a_1& + &3a_2 & \qquad\mathbf{ a_5 = 2a_1+3a_2 = -19} \\ a_6 &=& &3a_1& + &5a_2 \\ a_7 &=& &5a_1& + &8a_2 \\ a_8 &=& &8a_1& + &13a_2 & \qquad\mathbf{ a_8 = 8a_1+13a_2 = 79} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline 2a_1+3a_2 &=& -19 \\ 3a_2 &=& -19 -2a_1 \\ \mathbf{ a_2 } &=& \mathbf{ \dfrac{-19-2a_1}{3} } \\ \hline 8a_1+13a_2 &=& 79 \\ 8a_1 + 13 \left( \dfrac{-19-2a_1}{3} \right) &=& 79 \quad | \quad * 3 \\\\ 24a_1 +13(-19-2a_1) &=& 79*3 \\ 24a_1 -13*19-26a_1 &=& 237 \\ -2a_1 -247 &=& 237 \\ -2a_1 &=& 484 \quad | \quad :(-2) \\ \mathbf{a_1} &=& \mathbf{-242} \\ \hline a_2 &=& \dfrac{-19-2a_1}{3} \\\\ a_2 &=& \dfrac{-19-2(-242)}{3} \\\\ a_2 &=& \dfrac{-19+484}{3} \\\\ a_2 &=& \dfrac{465}{3} \\\\ \mathbf{a_2} &=& \mathbf{155} \\ \hline \end{array}\)

\(\begin{array}{|rcr|} \hline a_1 &=& -242 \\ a_2 &=& 155 \\ a_3 &=& -87 \\ a_4 &=& 68 \\ a_5 &=& -19 \\ a_6 &=& 49 \\ a_7 &=& 30 \\ a_8 &=& 79 \\ \hline \end{array}\)

I got it right :)

Oh, it was incorrect for some reason.

[-1.inf) is correct.  Good job!

Height of a cylinder = y

Height of a cone = 27y

Marisol made 300 cookies. S he sold 3/4 of them and 1/3 of the remainder to her neighbor. How many tarts  cookies did she have left. 

Selling 3/4 of them leaves 1/4 *300

   then giving away 1/3 leaves     2/3 (1/4*300) = 50 left

take the terms from each bracket with the highest power and multiply them together.

the degree is the power of the answer.

if you were to just divide, you get 31743 / 281, which is approximately 112.96, which is very close to an integer. This suggests that if you increase 31743 by a few digits it will divide 281. Sure enough, when you increase it by 10, you get 31753, which is divisible by 281

x^2*pi*y is the volume of the cylinder. 

(1/3x)^2*(ay)/3 is the volune of the cone. 

Set them equal to each other and find what a is. 

=^._.^=

3*4*5*x + 1

60x + 1

The largest x can be is 16. 

60*16 + 1

=^._.^=

Factors of 2018   are  1   2    1009   and   2018     and the negatives

(x+2)(x+1009)   would result in b = 1011    the smallest positive

(x-2018)(x-1)      would result in b = - 2019   for the overall smallest integer (but negative)

There are four solutions.

(a) There are 560 ways.

(b) There are 1020 ways.

There 248 ways of arranging the books.

$\frac{1}{(x + 1)(x - 2)} = \frac{-7 - \sqrt{2}}{5}$, so $|A| + |B| + |C| = 14$.

https://web2.0calc.com/questions/answer-it-there-theeblackbadger

Thanks for your post and thoughtful time! 

And how I factor a cubic equation, is I use the rational root theorem to find possible roots. Then I use synthetic division one by one to rule out the false ones. Then it just makes it into (x-n)(x-m)... where n,m,etc. are my roots

I'm back, and I found a reference in What is mathematics, Second edition, pg. 249.

10 000 ( 1 + .046)²    - 10000 = ___________  dollars