Respuesta :
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:
The similarities and differences of the rates of change of linear, quadratic, and exponential functions are explained below.
What is rates of change of functions ?
Rates of change is a rate that describes how one quantity changes in relation to another quantity.
So,
Linear functions change a constant rate. i.e. the slope of the graph is constant. Rate of change of Linear functions is also known as the slope.
A linear function is one of the form [tex]y = mx + c[/tex].
i.e. Slope [tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1}}[/tex]
The Linear function produces a parabola.
Quadratic functions changes at a linear rate, slower than the function itself and their rate of change changes at a constant rate.
Quadratic function :
[tex]f(x) = ax^2+ bx + c[/tex],
Where [tex]a, b,[/tex] and [tex]c[/tex] are variables.
The quadratic function produces a parabola.
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 exponential functions have a constant ratio.
Exponential function :
[tex]f(x)= ab^x[/tex]
Where [tex]b[/tex] is a positive real number.
Hence, we can say that the similarities and differences of the rates of change of linear, quadratic, and exponential functions are explained above.
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