Create One Equation out of two Equations

[Bob52]

Thank you in advance.

I do not know how to combine two continuous definitive equations (using derivatives and definitive integrals?) into one equation. They have identical tangents at their intersection.

Equation One: y=(x-squared) from 0 to 1
Equation Two: y=(x-to the fourth power) from 1 to 2

Once again, I am looking for a single equation to go from 0 to 2
My prior work was to do a derivative of each equation, and then add them together, then integrate over the defined limits and add those together. It doesn’t work.
Gracious

 

[Deleted member 4993]

Thank you in advance.

I do not know how to combine two continuous definitive equations (using derivatives and definitive integrals?) into one equation. They have identical tangents at their intersection.

Equation One: y=(x-squared) from 0 to 1
Equation Two: y=(x-to the fourth power) from 1 to 2

Once again, I am looking for a single equation to go from 0 to 2
My prior work was to do a derivative of each equation, and then add them together, then integrate over the defined limits and add those together. It doesn’t work.
Gracious

What class gave you this assignment? If it is not class-assignment (self-imposed assignment) - why do you want to do this? What do you think you can do with "one" equation that you could not do with "two" equations?

 

[pka]

I do not know how to combine two continuous definitive equations (using derivatives and definitive integrals?) into one equation. They have identical tangents at their intersection.
Equation One: y=(x-squared) from 0 to 1
Equation Two: y=(x-to the fourth power) from 1 to 2
Once again, I am looking for a single equation to go from 0 to 2

Here is a continuous function on \([0,2]\) having the properties 1 & 2.
\(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \). HERE is its graph.

 

[Steven G]

I do not agree with you. If f(x) = x^2, then f'(1) = 2 while if f(x) = x^4, f'(1) = 4. Sorry but the two functions do NOT have the same tangents at x=1

 

[Steven G]

Here is a continuous function on \([0,2]\) having the properties 1 & 2.
\(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \). HERE is its graph.

I need to remember this technique. Very nicely done.

 

[Bob52]

Thanks!

 

[Bob52]

Jomo, how did you get to the answer!
f(x)=x2⌊|x−2|⌋+x4⌊x⌋. Can you show me (teach me) the steps.
PS I now understand how to calculate (differentiate) whether two functions have the same tangent which, in this problem, they do not. I can even see it in your graph. Thanks for that.

 

[Bob52]

Oops, I meant pka, thanks for solving the problem. Can you show me (teach me) the steps?

 

[Deleted member 4993]

I do not agree with you. If f(x) = x^2, then f'(1) = 2 while if f(x) = x^4, f'(1) = 4. Sorry but the two functions do NOT have the same tangents at x=1

Jomo,

Please use the "reply" button while responding to a specific post. From your response (in #4) - it is very difficult to comprehend - you are disagreeing with whom. When you respond using "respond" button the "responsible" post is included in your answer,

 

[Bob52]

Jomo,

Please use the "reply" button while responding to a specific post. From your response (in #4) - it is very difficult to comprehend - you are disagreeing with whom. When you respond using "respond" button the "responsible" post is included in your answer,

Here is a continuous function on \([0,2]\) having the properties 1 & 2.
\(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \). HERE is its graph.

Here is a continuous function on \([0,2]\) having the properties 1 & 2.
\(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \). HERE is its graph.

What am I missing. If I make x=1 the

Here is a continuous function on \([0,2]\) having the properties 1 & 2.
\(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \). HERE is its graph.

What am I missing. Let’s say x=1, then f(x)=x2⌊|x−2|⌋+x4⌊x⌋=2. when the answer is supposed to be 1. The graph is correct but the formula does not seem correct. Plus what are the steps q

Here is a continuous function on \([0,2]\) having the properties 1 & 2.
\(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \). HERE is its graph.

What am I missing?
Restatement of the problem:
You are in a car that starts at (0,0) and begins accelereting at miles/hour. Than at 1 minute, the car speeds up by accelerating at miles/hour. How far have you gone in 2 hour? Write a single equation (showing the steps) to cover 0 to 2 minutes. Therefore:

The solution given was:
(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \)
When I plug 1 into the solution given, I come up with 2 and the correct answer is 1. The graph is correct at 1. I'm confused. Please help

 

[Bob52]

What am I missing. If I make x=1 the


What am I missing. Let’s say x=1, then f(x)=x2⌊|x−2|⌋+x4⌊x⌋=2. when the answer is supposed to be 1. The graph is correct but the formula does not seem correct. Plus what are the steps q


What am I missing?
Restatement of the problem:
You are in a car that starts at (0,0) and begins accelereting at View attachment 21686 miles/hour. Than at 1 minute, the car speeds up by accelerating at View attachment 21688miles/hour. How far have you gone in 2 hour? Write a single equation (showing the steps) to cover 0 to 2 minutes. Therefore:
View attachment 21690
The solution given was:
(f(x) = {x^2}\left\lfloor {\left| {x - 2} \right|} \right\rfloor + {x^4}\left\lfloor x \right\rfloor \)
When I plug 1 into the solution given, I come up with 2 and the correct answer is 1. The graph is correct at 1. I'm confused. Please help

So sorry, I meant to say Write a single equation (showing the steps) to cover 0 to 2 "hours."