Answer:
To find the angles of intersection between the curves f(x) = x^2 and g(x) = |x|, we need to find the points where the two curves intersect.
Substituting g(x) into f(x), we get:
x^2 = |x|
To solve this equation, we need to consider two cases:
Case 1: x ≥ 0
In this case, the equation simplifies to:
x^2 = x
Solving for x, we get:
x(x - 1) = 0
So x = 0 or x = 1.
Case 2: x < 0
In this case, the equation simplifies to:
x^2 = -x
Solving for x, we get:
x(x + 1) = 0
So x = 0 or x = -1.
Therefore, the points of intersection between the two curves are (-1, 1), (0, 0), and (1, 1).
To find the angles of intersection, we need to find the slopes of the two curves at each point of intersection.
At (0, 0), the slopes of both curves are 0.
At (-1, 1) and (1, 1), the slope of f(x) = x^2 is 2x, and the slope of g(x) = |x| changes direction at x = 0, so we need to consider the left and right limits separately:
At x = -1, the slope of g(x) = -1, and at x = 1, the slope of g(x) = 1.
Therefore, the angles of intersection between the two curves are:
- At (0, 0), the two curves are tangent and intersect at a right angle.
- At (-1, 1) and (1, 1), the two curves intersect at acute angles.