hectictar

Mr. Owl, is there supposed to be something more to the sentence

". . . the ratio of participants in debate, music, and community service, respectively, is."

???  

Thank you! And thanks for your help!! 

g(x)  =  \(5-\sqrt{4-x}\)

When  x = 3 , the slope of the curve  \(=\,\lim\limits_{h\to0}\frac{g(3+h)-g(3)}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{(5-\sqrt{4-(3+h)})-(5-\sqrt{4-3})}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{5-\sqrt{4-3-h}-4}{h} \\~\\ =\,\lim\limits_{h\to0}\frac{1-\sqrt{1-h}}{h} \\~\\ =\,\lim\limits_{h\to0}(\,\frac{1-\sqrt{1-h}}{h}\,)\,(\,\frac{1+\sqrt{1-h}}{1+\sqrt{1-h}}\,) \\~\\ =\,\lim\limits_{h\to0}\frac{1-(1-h)}{h+h\sqrt{1-h}} \\~\\ =\,\lim\limits_{h\to0}\frac{h}{h+h\sqrt{1-h}} \\~\\ =\,\lim\limits_{h\to0}\frac{1}{1+\sqrt{1-h}} \\~\\ =\,\frac{1}{1+\sqrt{1-0}} \\~\\ =\,\frac12\)

Look at the graph and see that it does look like that line has a slope of \(\frac12\) .

We want an equation of a line that has a slope of  \(\frac12\)  and passes through the point  ( 3, g(3) ) .

g(3)  =  \(5-\sqrt{4-3}\)  =  5 - 1  =  4

So the equation of our line in point - slope form is

y - 4  =  \(\frac12\)(x - 3)

And in slope - intercept form, this is...     y  =  \(\frac12\)x + \(\frac52\)

P  =  1.14F + 18.31     and     10 ≤ F ≤ 75

The question is.....what value of  F  causes  P  to be  85  ?

Plug in  85  for  P  and solve for  F .

85  =  1.14F + 18.31         Subtract  18.31  from both sides.

66.69  =  1.14F         Divide both sides by  1.14 .

58.5  =  F                  This is between  10  and  75 , so it is a possible value for  F .

The hood must have a flow rate of about  59  liters per second.

Which of the following exhibit inverse variation?

A) The distance traveled as a function of speed.

speed  =  distance / time

A bigger number for the distance causes a bigger number for the speed.     not inverse

B) The total cost as a function of the number of items purchased.

As the number of items purchased goes up, the total cost goes up.     not inverse

C) The area of a circular swimming pool as a function of its radius.

A bigger radius causes a bigger area.     not inverse

D) The number of posts in a 20ft fence as a function of distance between posts.

As the distance between posts gets larger, the number of posts gets smaller.

As the distance between posts gets smaller, the number of posts gets larger.

This is an inverse relationship.

CPhill, how do you "Nultiplt" both sides by 13.4??? 

There are two answers.

5 ± √36     =     5 ± 6     =     5 + 6     or     5 - 6     =     11     or     -1

Remember...

\(\frac{d}{dx}u^3 \,=\, 3u^2\,\cdot\,\frac{d}{dx}u\)

So

 \(f'(x)\,=\,3[x+\frac1x-1]^2\,\cdot\,\frac{d}{dx}[x+\frac1x-1] \\~\\ f'(x)\,=\,3[x+\frac1x-1]^2\,\cdot\,[\frac{d}{dx}x+\frac{d}{dx}\frac1x-\frac{d}{dx}1] \\~\\ f'(x)\,=\,3[x+\frac1x-1]^2\,\cdot\,[1-x^{-2}]\)

To find  f'(2) , plug in  2  for  x .

\(f'(2)\,=\,3[2+\frac12-1]^2\,\cdot\,[1-2^{-2}] \\~\\ f'(2)\,=\,3[\frac32]^2\,\cdot\,[1-\frac1{4}] \\~\\ f'(2)\,=\,3[\frac94]\,\cdot\,[\frac3{4}] \\~\\ f'(2)\,=\,\frac{81}{16}\)

( If   f(x)  is meant to be  \([\frac{x+1}{x-1}]^3\)  then you get a different answer. )

m  =  slope  =  \(\frac{y_2-y_1}{x_2-x_1}\,=\,\frac{-2-1}{-3-0}\,=\,\frac{-3}{-3}\,=\,1\)

Using the point  (0, 1), the equation in point-slope form is

y - 1  =  1(x - 0)      Simplify

y - 1  =  x               Add  1  to both sides.

y  =  x + 1              This is the equation in  y = mx + b  form.