The value of y would be 4, since x + 4 = 0 + 4 = 4 when x = 0. Therefore, d = 4.

[JCatt]

Got a bit of a long and nutty question here.

So I got a nutty question from my Calc 1 class and was wondering if anyone could help me out

A section of road, represented by the line y = x + 4 when x ≤0, is to be smoothly connected to another section of road, represented by y = 4 –x when x ≥4, by means of a curved section of road, represented by a cubic curve y = ax3+ bx2+ cx + d. Find a, b, c and d suchthat the function f(x) is everywhere differentiable (and therefore everywhere continuous), where

{x + 4 when x ≤0

{ ax3+ bx2+ cx + d when 0 < x < 4

{ 4 –x when x ≥4

I honestly don't know where to actually start. I know I have to find a, b, c and d at x = 0 and x = 4. Which would mean that d = 0 but from there I'm truthfully stuck.

Any help would be greatly appreciated.

 

[berkeman]

Got a bit of a long and nutty question here.

So I got a nutty question from my Calc 1 class and was wondering if anyone could help me out

A section of road, represented by the line y = x + 4 when x ≤0, is to be smoothly connected to another section of road, represented by y = 4 –x when x ≥4, by means of a curved section of road, represented by a cubic curve y = ax3+ bx2+ cx + d. Find a, b, c and d suchthat the function f(x) is everywhere differentiable (and therefore everywhere continuous), where

{x + 4 when x ≤0

{ ax3+ bx2+ cx + d when 0 < x < 4

{ 4 –x when x ≥4

I honestly don't know where to actually start. I know I have to find a, b, c and d at x = 0 and x = 4. Which would mean that d = 0 but from there I'm truthfully stuck.

Any help would be greatly appreciated.

Welcome to the PF.

Start by making a sketch of the outer two pieces. And as you say, the middle section has to join up with those two connection points. But that only gives you two equations to solve for 4 variables, so you need to figure out what else to use to help you out. One thing is the clue that the function has to be differentiable with respect to x everywhere... That may help you out.

 

[berkeman]

And actually to be a little clearer about my last hint, the function has to be smooth, with no sharp changes of direction...

 

[JCatt]

Welcome to the PF.

Start by making a sketch of the outer two pieces. And as you say, the middle section has to join up with those two connection points. But that only gives you two equations to solve for 4 variables, so you need to figure out what else to use to help you out. One thing is the clue that the function has to be differentiable with respect to x everywhere... That may help you out.

Thanks for the hint, sadly I still couldn't get much outta it. I've already made the graph and know how what shape the polynomial needs to be, but I don't know how to find a b c and d

 

[berkeman]

Thanks for the hint, sadly I still couldn't get much outta it. I've already made the graph and know how what shape the polynomial needs to be, but I don't know how to find a b c and d

I thought I was practically giving the answer away...

You need to solve for 4 constants, right? So you need 4 equations to do that. Joining up the end points gives you 2 of those equations, right? And smoothing the 2 connections gives you the other 2 equations. Now have at it!

 

[SammyS]

I honestly don't know where to actually start. I know I have to find a, b, c and d at x = 0 and x = 4. Which would mean that d = 0 but from there I'm truthfully stuck.

Any help would be greatly appreciated.

You're wrong about d.

If x=0, what is y ?