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

In summary, the problem involves finding a, b, c, and d to create a cubic curve that smoothly connects two sections of road represented by y = x + 4 when x ≤0 and y = 4 –x when x ≥4. The function f(x) must be everywhere differentiable and therefore everywhere continuous. This means that there must be no sharp changes in direction. To find the values of a, b, c, and d, four equations are needed. Two can be found by joining the end points, while the other two can be found by smoothing the connections. Contrary to the student's assumption, d does not equal 0 when x = 0.
  • #1
JCatt
2
0
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.
 
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  • #2
JCatt said:
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. :smile:
 
  • #3
And actually to be a little clearer about my last hint, the function has to be smooth, with no sharp changes of direction...
 
  • #4
berkeman said:
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. :smile:

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
 
  • #5
JCatt said:
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... :smile:

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!
 
  • #6
JCatt said:
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 ?
 

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

What is Calculus 1 Differential Question?

Calculus 1 Differential Question is a type of mathematical problem that involves finding the rate of change of a function at a specific point. It is a fundamental concept in calculus and is often used in physics, engineering, economics, and other fields.

What is the difference between differentiation and integration?

Differentiation is the process of finding the rate of change of a function, while integration is the process of finding the area under a curve. In other words, differentiation deals with instantaneous rates of change, while integration deals with accumulated change over a given interval.

How do I solve a Calculus 1 Differential Question?

To solve a Calculus 1 Differential Question, you need to first understand the problem and identify the given variables. Then, you can use the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule, to find the derivative of the function. Finally, plug in the given values to find the specific rate of change at the given point.

What are some real-life applications of Calculus 1 Differential Question?

Calculus 1 Differential Question has many real-life applications, including calculating the velocity of a moving object, determining the growth rate of a population, optimizing production and profit in business, and analyzing the flow of fluids in engineering.

What are some common mistakes to avoid when solving Calculus 1 Differential Question?

Some common mistakes to avoid when solving Calculus 1 Differential Question include forgetting to use the correct differentiation rules, making errors in algebraic simplification, and not checking the answer for accuracy. It is also important to double-check the given values and units to ensure the final answer is in the expected form.

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