Expert: Paul Klarreich - 9/20/2009

Question
For f(x)= 3x^3+4x^2-6x+10, Find the equation of the tangent line  or lines parallel to 3x-4y+10=0.I already determined y= (3/4)x + (10/4) and the derivative of f'(x)= 9x^2+8x-6.

Answer
Questioner: Rick
Country: United States
Category: Calculus
Private: No
Subject: slopes of tangent
Question: For f(x)= 3x^3+4x^2-6x+10,
Find the equation of the tangent line  or lines parallel to 3x-4y+10=0.

>> Did you mean -----------------------OF lines parallel to 3x-4y+10=0??

I already determined  y= (3/4)x + (10/4)

>> Yes, but what conclusion did you draw from it?

and the derivative f'(x)= 9x^2+8x-6.

>> Now how do you use this fact?
................................................
This is all about drawing algebraic-geometric inferences, such as:

"lines parallel to" <--> "having the same slope as"

"f'(x)= 9x^2+8x-6"  <--> "the slope of the tangent line at x is ..."

Now you should have the clue:

"I already determined  y= (3/4)x + (10/4)"  <--> "the slope is  3/4"

Put them together and you will write the equation:

9x^2 + 8x - 6 = 3/4

Solve:

36x^2 + 32x - 24 = 3

36x^2 + 32x - 27 = 0

Alas, this does not factor.  You're on your own now.