Calculus problem help

Question

Determine the equation of the line in slope-intercept form which is both tangent to the curve f(x) and parallel to the line provided:

 

f(x)=2x²+3x+1                  Line:4x+4y=12

 

Answer 1

1.  Find the slope of the given line.

2.  Find the slope of any line parallel to that.

3.  Find the slope of any tangent to f, at (x,f(x)).

4.  Equate the slopes in steps 1 and 2, and solve for x.

5.  Determine the y coordinate of the point of tangency.

6.  Write the equation of the tangent line, using Point Slope form.

Answer 2

Hi Andrew, 

 

The line you have, 4x + 4y = 12, can be rewritten in terms of y:

 

4y = 12 - 4x

y = 3 - x = -x + 3.

 

This means that you want a line with a slope of -1.

 

So let's find when f(x) has a slope of -1.  The derivative (slope) of f(x) for all x is

 

f'(x) = 4x + 3.  We want to set this equal to -1:

 

-1 = 4x + 3

-4 = 4x

x = -1

This means that the slope of -1 occurs when x = -1 as well.  Plugging this into f(x) we get

f(-1) = 2(-1)^2 + 3(-1) +1 = 2 -3 +1 = 0. So since we have a point (-1, 0) and a slope (-1), we can plug into point-slope form and convert into slope-intercept:

 

y - 0 = -1 (x- -1)

y = -1 (x +1)

y = -x -1.

 

Hope that helps!

Let me know if you need any more help!

Dan