Homework Help

Find an equation for the tangent line to the graph `y=6xcos10x`   a `x=pi`

user profile pic

user7230927 | (Level 1) Salutatorian

Posted June 27, 2013 at 1:42 AM via web

dislike 1 like

Find an equation for the tangent line to the graph `y=6xcos10x`   a `x=pi`

1 Answer | Add Yours

user profile pic

mjripalda | High School Teacher | (Level 1) Senior Educator

Posted June 27, 2013 at 2:08 AM (Answer #1)

dislike 1 like

To determine the equation of the tangent line, its slope and point of tangency must be determined.

To solve for its slope, take the derivative of the given curve.

`y=6xcos(10x)`

`y'=6x*(cos(10x))' + cos(10x)*(6x)'`

`y'=6x*(-sin (10x)*10)+cos(10x)*6`

`y'=-60xsin(10x)+6cos(10x)`

Since m=y', plug-in x=pi to y'.

`m=y'=-60(pi)sin(10*pi)+6cos(10*pi)`

`m=0+6*1`

`m=6`

So, the slope of the tangent line is 6.

Next, to get the point of tangency, plug-in x=pi to y.

`y=6xcos(10x)=6*pi*cos(10*pi)=6pi*1`

`y=6pi`

Hence, the point of tangency is (pi, 6pi).

Now that the slope of the tangent line and its point of tangency are known, its equation can now be determined by using the point-slope form.

`y-y_1=m(x-x_1)`

`y-6pi=6(x-pi)`

`y-6pi=6x-6pi`

`y-6pi+6pi=6x-6pi+6pi`

`y=6x`

Therefore, the equation of the tangent line is `y=6x` .

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes