Chapter 7.6, Problem 30P

Writing Discuss the differences between exponential functions with a base of 2 and $3,y=2^x$ and $y=3^x$, and quadratic and cubic functions $y=x^2$ and $y=x^3.$

Focus on the shapes of the different graphs and rates of growth.

To determine

To explain:Discuss the difference between exponential functions with a base of 2 and 3, $y=2^x$ and $y=3^x$ , and quadratic and cubic functions $y=x^2$ and $y=x^3.$ Focus on the shapes of the different graphs and rates of growth.

Explanation of Solution

Given information:The given functions are $y=2^x$ , $y=3^x$ , $y=x^2$ and $y=x^3.$

Calculation: The exponential functions have greater rate of change as compare to polynomial functions. The exponential function with greater base have greater rate of change, so $y=3^x$ has greater rate of change as compare to $y=2^x$ .

The polynomial functions with higher degree has greater rate of change. so $y=x^3$ has greater rate of change as compare to $y=x^2$ .

Hence the order of rate of change of the following functions is as follow:

  $y=3^x>y=2^x>y=x^3>y=x^2$

We can verify this with the help of graph also, Plotting all the functions in 1 graph.