Question

What are some similarities and differences of the rates of change of linear, quadratic, and exponential functions?

Answer 1

Answer:

linear functions change a constant rate. i.e. the slope of the graph is constant.

d/dx (mx+b) = m

quadratic functions change at a linear rate, slower than the function itself.

The slope of the tangent to the graph is proportional to the x coordinate.

d/dx (ax^2) = 2ax

Slope of tangent to ax^2 at point (x,ax^2) is 2ax.

Exponential functions change at the same rate as the function itself. The slope of the tangent to the graph is proportional to the y coordinate.

The slope of tangent at point (x,exp(x)) is exp(x).

Definition of exp(x) is just exp(0) = 1 and d/dx exp(x) = exp(x).

d/dx exp(ax) = a exp(ax) is not part of the definition, but follows from from the chain rule

Step-by-step explanation: