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Answer :
Final answer:
The intersection angles between the curves f(x) = x^2 and g(x) = √x are 0 degrees at the point (0,0) and approximately 26.7 degrees at the point (1,1). This is found by first finding the intersection points, then the gradients of the tangents at these points, and finally applying the formula for the angle of intersection.
Explanation:
To solve for the angles of intersection between the curves f(x) = x^2 and g(x) = √x, we first have to find their derivatives which will give us the slopes of the tangents at their intersection points. The derivative of f(x) = x^2 is f'(x) = 2x and the derivative of g(x) = √x is g'(x) = 1/(2√x).
Next, setting f(x) = g(x) gives us the intersection points as (0, 0) and (1, 1). Substituting these into the derivative functions will provide the gradients of the tangents at these points. For the first point, both functions give a gradient of 0, while for the second point, f'(x) gives 2 and g'(x) gives 1/2.
The angle of intersection can then be found using the formula θ = tan^-1|((m2 - m1) / (1 + m1*m2))|, where m1 and m2 are the gradients of the tangents. For the intersection point at (0,0), the gradients are the same and the intersection angle is therefore 0 degrees. For the intersection point at (1,1), the formula gives an angle of approximately 26.7 degrees.
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