Angle between two tangent lines

[jsmith613]

Homework Statement

http://www.xtremepapers.com/Edexcel/Advanced%20Level/Mathematics/Subject%20Sorted/C3/Elmwood%20Papers/Elmwood%20B.pdf

Question 8(b)

Homework Equations

The Attempt at a Solution

Ok so I found both values of dy/dx for BOTH EQUATIONS

y = x² - 4x → 2x - 4
y = |4x - x²| → |4 - 2x| could someone please claify this, I have never differentiated a modulus before

Thus the two gradients of the lines are 4 and -4 BUT HOW DO I GO ON TO FIND THE angle between the tangents??

 

[HallsofIvy]

$|4x- x^2|= |x(4- x)|$ and is equal to $4x- x^2$ for x between 0 and 4 but equal to $x^2- 4x$ for x< 0 or x> 4. That is, its derivative if 4- 2x for x between 0 and 4 and equal to 2x- 4 for x< 0 or x> 4. At x= 4, there is a "cusp" so technically, there is no derivative. Of course, you are interested in the curve between 0 and 4 so you really want $\lim_{x\to 4^-} 4- 2x= -4$ as you say.

To find the angle between the lines remember that the derivative is the tangent of the angle the curve makes with the horizontal. And that
$$tan(\theta- \phi)= \frac{tan(\theta)- tan(\phi)}{1+ tan(\theta)tan(\phi)}$$

 

[jsmith613]

To find the angle between the lines remember that the derivative is the tangent of the angle the curve makes with the horizontal

I never knew this
is this A-level maths or beyond?
could you please explain why this is true?

 

[SammyS]

To find the angle between the lines remember that the derivative is the tangent of the angle the curve makes with the horizontal.

I never knew this.
is this A-level maths or beyond?
could you please explain why this is true?

It's usually taught in Calculus when you first learn about derivatives representing the slope of the tangent line.

It's often taught in trigonometry that the slope of a line is equal to the tangent of the angle the line makes with the x-axis.

 

[jsmith613]

thanks for this :)
I will note this rule!