How do you determine the y-intercept of the tangent line to the curve ##y=sqrt(x^2+4)## at ##x=3##?
February 20th, 2023
to find the tangent line of the curve, take the derivative, to determine the slope of the tangent line.
y=##sqrt(x^2+4)##
y=##(x^2+4)^(1/2)##
y’=##1/2####(x^2+4)^(-1/2)##(2x)
y’=##x/sqrt(x^2+4)##
because you want the tangent line at x=3, plug it in to y’ to find the slop at that point, since we know that the derivative is the slope at a certain point.
y'(3)=##3/sqrt(3^2+4)##=##3/sqrt(13)##
at x=3, the point on the curve is (3,##sqrt13##)
now, you can create the tangent line
(y-##sqrt13##)= ##3/sqrt13##(x-3)
y-##sqrt13##=##3/sqrt13##x-##9/sqrt13##
y=##3/sqrt13##x-##9/sqrt13##+##sqrt13##
so your y-intercept for the tangent line at x=3 would be (##9/sqrt13##+##sqrt13##)