Give an example of a family of linear functions all with the same derivative

Question

Give an example of a family of linear functions all with the same derivative:

 

So I'm not sure what it means by family, in terms of how many I need, but I thought of x^2, and 1/x^-2 and I was going to continue but my teacher said it was wrong. How do I find a linear family with all the same derivatives. Please help me, I am so lost.

Answer 1

A family of linear functions would be y = mx + b (rules out quadratic functions and any others with exponents or roots of the variable).

 

The derivative of any of the above is y' = m.  This means the family can be y = mx + b, with all having the same slope, m, but different y-intercept values, b.

 

example: y = 2x    y = 2x+5   y = 2x +17 etc.    That's the family where m = 2.

Answer 2

A function like x² is not linear.  A family of parallel lines should answer the question.  Write down a few equations for lines that have the same slope (i.e. are parallel) and you will have a family of linear functions with the same derivative, because the slope is the derivative of a linear function.

 

y = mx + b

m = slope

b = y intercept

 

Pick some m, say 3 for example, and pick different b values, say for example b =1, 2, 3,.., n

 

Write the equations for the lines:

 

y = 3x + 1

y = 3x + 2

y = 3x + 4

etc.

 

This is a an example of a family of linear functions with the same derivative.  Or in more general terms,

 

y = mx + b₁

y = mx + b₂

y = mx + b₃

etc.