How do i find the exact same gradient for 2 different functions at the same X cord?

[Kaiser25th]

Hello, Im trying to connect 2 functions together on a graph but the requirement is they must have smooth transition points, which is by having the same gradient as i was told. I've been trying for a few days but have been unsuccessful and it's quite stressing. I'll be happy as long as i get to know how i can make the transition of 4-5 functions smooth! Thank you.

 

[blamocur]

Please show us your work so far: https://www.freemathhelp.com/forum/threads/read-before-posting.109846/

 

[Kaiser25th]

Please show us your work so far: https://www.freemathhelp.com/forum/threads/read-before-posting.109846/

Thats the problem, I can not do anything unless I have 4-5 functions with the same gradient resulting in a smooth graph without sharp turns. I dont know how to do that because at the same x cord both functions must have the same gradient which is what im stuck at

 

[Deleted member 4993]

Thats the problem, I can not do anything unless I have 4-5 functions with the same gradient resulting in a smooth graph without sharp turns. I dont know how to do that because at the same x cord both functions must have the same gradient which is what im stuck at

Choose any function.

How would you calculate "gradient" of that function?

 

[Kaiser25th]

By finding the derivative at a specific point.

However to find the gradient the exact same for both functions at the same point is what im wondering how to do

 

[blamocur]

Thats the problem, I can not do anything unless I have 4-5 functions with the same gradient resulting in a smooth graph without sharp turns. I dont know how to do that because at the same x cord both functions must have the same gradient which is what im stuck at

Before using 4-5 functions start with 2. For example, can you combine a parabola and a straight line into one smooth function?

 

[Kaiser25th]

Before using 4-5 functions start with 2. For example, can you combine a parabola and a straight line into one smooth function?

Thats where I dont know how to, i've tried puting those 2 in a piecewise function on a calculator but theres always an error

 

[blamocur]

Thats where I dont know how to, i've tried puting those 2 in a piecewise function on a calculator but theres always an error

Try it without a calculator first. Let's say you have [imath]f(x) = x^2[/imath] for [imath]0\leq x\leq 1[/imath], and [imath]f(x) = ax+b[/imath] for [imath]1\leq x\leq 2[/imath]. Can you figure out values [imath]a[/imath] and [imath]b[/imath] which will make [imath]f(x)[/imath] smooth for [imath]0\leq x\leq 2[/imath] ?