Question

Find an equation of the tangent line to the curve y= tan[(pix^2)/(4)] at the point (1,1)

Answer 1

let u = pi x^2/4
d/dx tan u = sec^2 u * du/dx
so
slope = sec^2 [(pix^2)/(4)] * pi x/2
if x = 1
slope = m = sec^2 [pi/4] * pi/2
m = (pi/2) / (1/cos pi/4)
but cos pi/4 = 1/sqrt 2
so m = (pi/2)/sqrt2 = (pi/4) sqrt 2
so
y = (pi/4) sqrt 2 x + b
1 = (pi/4) sqrt 2 + b
b = 1 - (pi/4) sqrt 2 = (pi/4)( 4/pi - sqrt2)