Find the equation of the tangent line to the graph of f(x)=2x+sinx+1 on the interval (0,pi) at the point which is guaranteed by the Mean Value Theorem.

Question

precal · Answer

$$\frac{ f(\pi)-f(0) }{ \pi-0 }=2$$

precal · Answer

f ' (c)=2 + cos(c)
2+ cos(c)=2
cos(c)=0
c=pi/2 and (3pi)/2  but pi/2 is the only solution in the interval (0, pi)

precal · Answer

ok  I am stuck at the last part.  writing the tangent line

precal · Answer

ok is it 
$$y-(\pi+2)=2(x-\frac{ \pi }{ 2 })$$

precal · Answer

@ganeshie8

precal · Answer

@agreene

agreene · Answer

it looks alright to me but I always sucked at making tangent lines lol

precal · Answer

do you see anything incorrect?

agreene · Answer

nothing seems wrong.

precal · Answer

tangent lines are not too bad.  you just need the derivative at the given point and you also, need the (x,y) coordinates from that point.
so in this case $$f ' (\frac{ \pi }{ 2 })=2$$

precal · Answer

$$f (\frac{ \pi }{ 2 })=\pi +2$$

agreene · Answer

yeah, i vaguely remember all of this--I don't use MVT or anything to do with tangent lines in my research and I dont teach calculus 1 so it's a bit hazy lol

precal · Answer

so we just do simple sub
y-y1=m(x-x1)
$$y-(\pi +2)=2(x-\frac{ \pi }{ 2 })$$

ipwnbunnies · Answer

This seems correct to me.

precal · Answer

Thanks  :)