# calculus graph, please check, thanx

• Jul 13th 2006, 08:48 AM
Qu.

Use graph sketching strategy to sketch the graph of the function f(x) = 4x -15/ x^2 -9

Step 1 The domain of f is all natural numbers except + 3 or - 3 when x^2 - 9 = 0

Step 2 Function f in nether even or odd

step 3 x intercept , 4x -15 = 0, x = 3 3/4
y intercept f (0) = 5/3

step 4

f(x) = 4x -15/ (x-3) (x+3)

f is positive (3 3/4, infinite), and negative on (-infinite, -3), (-3,3), and (3, 3 3/4)

Step 5

the derivative of f(x) = -4x^2 -36 +30x/ (x^2-9)^2

• Jul 13th 2006, 09:28 AM
galactus
Are you wanting the derivative of:

$\displaystyle \frac{-4x^{2}+30x-36}{(x^{2}-9)^{2}}$?.

Rewrite as such and use the product rule:

$\displaystyle (-4x^{2}+30x-36)(x^{2}-9)^{-2}$

$\displaystyle (-4x^{2}+30x-36)(-2)(x^{2}-9)^{-3}(2x)+(x^{2}-9)^{-2}(-8x+30)$

$\displaystyle \frac{16x^{3}-120x^{2}+144x}{(x^{2}-9)^{3}}+\frac{-8x+30}{(x^{2}-9)^{2}}$

Multiply the top and bottom of the right side by $\displaystyle (x^{2}-9)$

and simplify:

$\displaystyle \Large\frac{2(4x^{3}-45x^{2}+108x-135)}{(x^{2}-9)^{3}}$
• Jul 13th 2006, 09:34 AM
ThePerfectHacker
Quote:

Qu.

Use graph sketching strategy to sketch the graph of the function f(x) = 4x -15/ x^2 -9

Step 1 The domain of f is all natural numbers except + 3 or - 3 when x^2 - 9 = 0

Step 2 Function f in nether even or odd

step 3 x intercept , 4x -15 = 0, x = 3 3/4
y intercept f (0) = 5/3

step 4

f(x) = 4x -15/ (x-3) (x+3)

f is positive (3 3/4, infinite), and negative on (-infinite, -3), (-3,3), and (3, 3 3/4)

Step 5

the derivative of f(x) = -4x^2 -36 +30x/ (x^2-9)^2

Sketching curves has always been one of my favorite thing in calculus. I will you show you my way.
-------------------------------------------
Step 1: Find domain
Find the domain of the function,
$\displaystyle f(x)=\frac{4x-15}{x^2-9}$
Note, $\displaystyle x^2-9\not = 0$ thus, $\displaystyle x\not = \pm 3$

Step 2: Symettry
Does $\displaystyle f(-x)=f(x)$
You can see that,
$\displaystyle \frac{4(-x)-15}{(-x)^2-9}=\frac{-4x-15}{x^2-9}$--->No
Does $\displaystyle f(-x)=-f(x)$
You can see that,
$\displaystyle \frac{4(-x)-15}{x^2-9}=\frac{-4x-15}{x^2-9}$ and that,
$\displaystyle -f(x)=\frac{-4x+15}{x^2-9}$---> No

Step 3: Asymptotes
Yes it does at $\displaystyle x=-3,3$ (since numerator is non-zero while denominator is zero).

I countinue later, it takes longgg time.
• Jul 17th 2006, 02:22 AM
ThePerfectHacker- thanks, continued
Horizontal asymptote

y = 0 ?

When m < n, the horizontal asymptote is the line y = 0

where m is the degree of polynomial in the numerator and n is the degree of polynomial in the denominator.
• Jul 24th 2006, 02:29 PM
Quote:

Originally Posted by ThePerfectHacker
Sketching curves has always been one of my favorite thing in calculus. I will you show you my way.
-------------------------------------------
Step 1: Find domain
Find the domain of the function,
$\displaystyle f(x)=\frac{4x-15}{x^2-9}$
Note, $\displaystyle x^2-9\not = 0$ thus, $\displaystyle x\not = \pm 3$

Step 2: Symettry
Does $\displaystyle f(-x)=f(x)$
You can see that,
$\displaystyle \frac{4(-x)-15}{(-x)^2-9}=\frac{-4x-15}{x^2-9}$--->No
Does $\displaystyle f(-x)=-f(x)$
You can see that,
$\displaystyle \frac{4(-x)-15}{x^2-9}=\frac{-4x-15}{x^2-9}$ and that,
$\displaystyle -f(x)=\frac{-4x+15}{x^2-9}$---> No

Step 3: Asymptotes
Yes it does at $\displaystyle x=-3,3$ (since numerator is non-zero while denominator is zero).

I countinue later, it takes longgg time.

• Jul 24th 2006, 02:35 PM
ThePerfectHacker
Quote:

With what?
Tell me extacly you were able to do (post it I will check it) and then procede to tell me what you do not know how to solve.
• Jul 24th 2006, 02:56 PM
ThePerfectHacker - Graph of a function
Found derivative of f

= (-4x +6)(x-6)/(x^2-9)^2

Problems establishing when f is increasing or decreasing and if any stationary points
• Jul 25th 2006, 09:04 AM
macca101
Quote:

Found derivative of f

= (-4x +6)(x-6)/(x^2-9)^2

Problems establishing when f is increasing or decreasing and if any stationary points

• Jul 25th 2006, 09:40 AM
ThePerfectHacker
Quote:

Found derivative of f

= (-4x +6)(x-6)/(x^2-9)^2

Problems establishing when f is increasing or decreasing and if any stationary points

Multiply the numerator (open parantheses),
$\displaystyle \frac{-4x^2+30x-36}{(x^2-9)^2}$
Use quotient rule,
$\displaystyle \left( \frac{u}{v} \right) = \frac{u'v-uv'}{v^2}$
Thus,
$\displaystyle \frac{(-4x^2+30x-36)'[(x^2-9)^2]-(-4x^2+30x-36)[(x^2-9)^2]'}{(x^2-9)^4}$ (1)
Note that,
$\displaystyle (-4x^2+30x-36)'=-8x+30$
And,
$\displaystyle [(x^2-9)^2]'=2(2x)(x^2-9)=4x(x^2-9)$--->Rule of chains.

Now rewrite expression (1) without derivatives,
$\displaystyle \frac{(-8x+30)(x^2-9)^2-4x(-4x^2+30x-36)(x^2-9)}{(x^2-9)^4}$
Factor, that the term $\displaystyle (x^2-9)$ thus,
$\displaystyle \frac{(x^2-9)[(-8x+30)(x^2-9)-4x(-4x^2+30x-36)]}{(x^2-9)^4}$
Cancel,
$\displaystyle \frac{(-8x+30)(x^2-9)-4x(-4x^2+30x-36)}{(x^2-9)^3}$
Open ye olde' parantheses,
$\displaystyle \frac{-8x^3+30x^2+72x-270+16x^3-120x^2+144x}{(x^2-9)^3}$
Thus, combine,
$\displaystyle \frac{8x^3-90x^2+192x-270}{(x^2-9)^3}$
• Jul 25th 2006, 09:29 PM
earboth
Quote:

Originally Posted by ThePerfectHacker
...
Open ye olde' parantheses,
$\displaystyle \frac{-8x^3+30x^2+72x-270+16x^3-120x^2+144x}{(x^2-9)^3}$
Thus, combine,
$\displaystyle \frac{8x^3-90x^2+192x-270}{(x^2-9)^3}$

$\displaystyle \frac{8x^3-90x^2+216x-270}{(x^2-9)^3}$ as galactus has calculated already