. Let m be the slope of the curve x , then m = dy = . Let θ and θ be the angles at ( , ) and ( , ) respectively. At ( , ) , we come across the indeterminate form of ×∞ in the denominator of tan θ and so we follow the limiting process.
tan θ = lim x y lim x y = ¥ which gives θ = tan ( ) − ∞= π . At ( , ) , m tan θ = = which gives θ = tan − . Fig. .
. . - - Applications of Differential Calculus Example . Find the angle of intersection of the curve y = sin with the positive x -axis.
When the curve y = sin intersects the positive x -axis, y = which gives, x π , , , , . Now, dy = cos . The slope at x = n p are cos( π = − . Hence, the required angle of intersection is tan q = ) − + − ) ( ) ∀ n Example .
If the curves ax by = and cx = intersect each other orthogonally then, show that d Let the two curves intersect at a point ( . This leads to ( c x d y . Let us now find the slope of the curves at the point of intersection ( . The slopes of the curves are as follows : For the curve ax by = , ax by = − For the curve cx = , cx = − Now, if two curves cut orthogonally, then the product of their slopes, at the point of intersection , is − .
Hence, if the above two curves cut