Final answer:
To find the angle between two curves at their point of intersection, differentiate the equations of the curves to find the slopes of the tangent lines at the point, and then use the formula for the angle between two lines to calculate the angle.
Step-by-step explanation:
To determine the angle between two curves at their point of intersection, we first need to find the slope of each curve at that specific point. These slopes represent the gradients of the tangent lines to each curve at the point of intersection. The angle between two lines can then be found using the slopes (m1 and m2) of these lines with the formula:
tan(θ) = |(m2 − m1)/(1 + m1 m2)|
Where θ is the angle between the two tangent lines. To find the slopes m1 and m2, we differentiate the equations of the curves with respect to x (if the curves are functions of x). After obtaining the slopes, the angle θ can be calculated by taking the inverse tangent (arctan) of the absolute value of the fraction given by the formula.
This method applies the principle that the slope of a curve at a point is equal to the slope of the tangent line at that point. You need to find the derivative of each curve at the point of intersection to determine these slopes. Once the slope is known, the tangent line equation can be constructed, and the angle between these tangent lines can be found as mentioned.