Calculus Curves

Question

The curves f(x)=x2+5x−5 and g(x)=3x+3 intersect at the point (2,9).

Find the angle of intersection, in radians on the domain 0 < t < Pi

Answer 1

slope of the tangent line to the quadratic at x=2 is 9

slope of the line is 3

the line makes an angle with the x axis of tan^-1(3) = about 71.565 degrees

the tangent line makes an angle with the x axis of tan^-1(9) = 83.660

the angle between the line and the tangent line is 83.660 - 71.565 = about 12.095 degrees

convert to radians by multiplying by pi/180

or about 3.14159/180

pi/180 times 12.095 = about 0.0672 pi radians = about 0.2111 radians

Answer 2

f^'(x) = 2x + 5 so f^'(2) = 9. g^'(x) = g^'(2) = 3.

So on a calculator (in radians mode) we can do tan^-1(9) - tan^-1(3), which will give the angle of intersection.