Find the Equation of Tangent

Question

Find the equation of each tangent of the function f(x)=x^3−5x^3 +5x−4 which is parallel to the line y = 2x +1

Answer 1

Is your function rule written correctly? There are two x cubed terms in the function rule and I'm wondering if the second one is actually supposed to be x². Can you check on this?

Ok, since you have checked, we will go with the function rule you wrote but we will combine the x³ terms so it simplifies to f(x) = -4x³ + 5x - 4. The tangent(s) line(s) to this function will need to have the same slope as the linear equation, so the slope must equal 2.

Now we take the derivative of f(x) and we get f '(x) = -12x² + 5. We set this equal to 2 and we solve for x. If -12x² + 5 = 2, then -12x² = -3 and, dividing both sides by -12 and then taking square root, we get x = ± 1/2.

To find where the tangent lines will touch the function, we insert both x values into the original function to find the points of tangency. If x = 1/2, y = -2. If x = -1/2, then y = -6. So we will have two tangent lines. We can now use point-slope form to write both equations:

1) using ( .5, -2 ), we have y + 2 = 2( x - .5 ) which, in slope-intercept form, becomes y = 2x - 3.

2) using ( -.5, -6), we have y + 6 = 2( x + .5 ) which converts to y = 2x - 5.