Question

How do you find the equation of the line tangent to `y=3x^2-x^3` at point (1,2)?

Answer 1

`y=3x-1`

Explanation:

The equation of the tangent in `color(blue)"point-slope form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y-y_1=m(x-x_1))color(white)(2/2)|)))`
where m represents the slope and `(x_1,y_1)" a point on the tangent"`

`color(orange)"Reminder: "m_("tangent")=dy/dx" at x=a"`

`y=3x^2-x^3`

`rArrdy/dx=6x-3x^2`

`"at "x=1tody/dx=6-3=3`

`"using "m=3" and "(x_1,y_1)=(1,2)" then"`

`y-2=3(x-1)larrcolor(red)"in point-slope form"`

distributing and simplifying gives an alternative version of the equation.

`y-2=3x-3`

`rArry=3x-1larrcolor(red)" in slope-intercept form"`