Olive is comparing the growth rates of p(x)=5x%0D%0A%0D%0A(%0D%0A%0D%0A)%0D%0A=%0D%0A5%0D%0A%0D%0A and q(x)=20x%0D%0A%0D%0A(%0D%0A%0D%0A)%0D%0A=%0D%0A20%0D%0A%0D%0A using this graph. Based on the graph, Olive concludes that the growth rate of q(x)=20x%0D%0A%0D%0A(%0D%0A%0D%0A)%0D%0A=%0D%0A20%0D%0A%0D%0A is always greater than the growth rate of p(x)=5x%0D%0A%0D%0A(%0D%0A%0D%0A)%0D%0A=%0D%0A5%0D%0A%0D%0A . Where is her mistake?
Her mistake is in assuming that the growth rate of q(x)=20x is always greater than the growth rate of p(x)=5x. This conclusion cannot be made based on the graph alone. While the graph shows that the values of q(x) are always greater than the values of p(x) for any given x, it does not show that the growth rate of q(x) is always greater than the growth rate of p(x). To compare the growth rates, one would need to calculate the derivatives of the functions and compare those values.
