What is the equation of the line with slope #3# which is tangent to the curve #f(x)=7x-x^2#?

1 Answer
Sep 1, 2016

#y=3x+4#

Explanation:

If #f(x)=7x-x^2#
then the slope (for any general #x# value) is #f'(x)=7-2x#

when the slope is #m=f'(x)=3#
then #7-2x=3#
#color(white)("XX")-2x=-4#
#color(white)("XX")x=2#

If #x=2# then
#color(white)("XXX")f(color(red)(2))=7*(color(red)(2))-color(red)(2)^2=14-4=color(blue)(10)#

and the point on the curve were #f'(x)=3# occurs at #(color(red)(2),color(blue)(10))#

Therefore, for the tangent, we have a slope of #color(green)m=3# and a point #(color(red)(2),color(blue)(10))#

Using the slope-point form of the equation:
#color(white)("XXX")y-color(blue)(10)=color(green)(3)(x-color(red)(2))#

or
#color(white)("XXX")y=3x+4#