Slope-predictor (4 step process)

[confused_07]

Given f[x]= 4 / sqrt (x+8), use the four-step process to find a slope-predictor function m(x). Then write the equation for the line tangent tot he curve at the point x=8.

Step 1- Write the definetion of m(x)
Step 2- Sub into this definition the formula of the given function f
Step 3- Make algebraic simplifications until Step 4 can be carried out
Step 4- Determine the value of the limit as h -> 0

So, I got:

Step 1- m(x) = f(a+h) - f(a) / h
Step 2- m(x) = lim h ->0 [(4/sqrt x+8 + 4/sqrt h+8) - (4/sqrt x+8)] / h
Step 3- ?????

This where I am stuck..... I was told by someone who has taken calculus that that is how I was supposed to sub in the function. Doesn't look right. Please help.

 

[Count Iblis]

Suggestion: Try to do it for the function 1/sqrt[x] first. You should find
-1/2 x^(-3/2).

 

[tkhunny]

Step 1- m(x) = f(a+h) - f(a) / h

First, fix your notation. What you have written is not correct. It should be: [f(a+h) - f(a)] / h. Order of Operations will help you get it straight.

[(4/sqrt x+8 + 4/sqrt h+8) - (4/sqrt x+8)] / h

Second, fix your notation. You added the square brackets like I suggested in 1). However, no one knows what "sqrt x+8" means. Try "sqrt(x+8)". It's substantially more clear. You had it in the original problem statement. Try to be more consistent.

[(4/sqrt x+8 + 4/sqrt h+8) - (4/sqrt x+8)] / h

?? What are you doing? Have you forgotten how function notation works? Please review it.

You have f(x) = 4/sqrt(x+8). Then:
f(a) = 4/sqrt(a+8)
f(a+h) = 4/sqrt((a+h)+8)

Now set up that quotient again. A little algebra will lead you to the solution.

 

[confused_07]

Please bare with me on this one, as I think I am brain farting simple algebra:

I reset the equation using your help (thank you) and have:

\(\displaystyle [(4/sqrt((x+h)+8)) - (4/sqrt(x+8))]/h\)

Using the common-denominator calculation:

\(\displaystyle [4(sqrt(x+8) - 4(sqrt((x+h)+8)] / [h(sqrt((x+h)+8)(sqrt(x+8))]\)

I am stuck from there. I am sure it is something I am not seeing right. Please help.

 

[galactus]

Yeah, it's always algrebra.

Multiply top and bottom by the conjugate of the numerator.

I am going to skip all the typing. Suffice it to say, when we rationalize the numerator we get:

\(\displaystyle \L\\\frac{\frac{-16h}{(x+8)(x+h+8)}}{h(\frac{4}{\sqrt{x+h+8}}+\frac{4}{\sqrt{x+8}})}\)

The h's cancel and you get:

\(\displaystyle \L\\\frac{\frac{-16}{(x+8)(x+h+8)}}{(\frac{4}{\sqrt{x+h+8}}+\frac{4}{\sqrt{x+8}})}\)

As h tends to 0:

\(\displaystyle \L\\\frac{-16}{(x+8)^{2}}\cdot\frac{\sqrt{x+8}}{8}\)

=\(\displaystyle \L\\\frac{-2}{(x+8)^{\frac{3}{2}}}\)

 

[stapel]

Please bare with me....

I surely hope you mean "please bear with me", 'cause I ain't gettin' nekkid with nobody roundsabout these parts.... :wink:

Eliz.