Question

Of the functions below, which eventually will have the greatest rate of change compared to the others?

y= 2^x -3

y= 3x+10

y= x^2+8x+1

Answer 1

y= 2^x -3
exponentials grow faster than any polynomial

Answer 2

boy, you don't know much about exponential functions.
y= 2^x - 3
y' = 2^x ln2
so, for all x > ln(3/ln2)/ln2 ≈ 2.1137 , the slope will be greater than 3

Answer 3

The function with the greatest rate of change will be the linear function, y= 3x + 10. This is because it has a constant rate of change of 3.

The function y= 2^x - 3 has an increasing rate of change as x increases, but this rate of change will never exceed 3.

The function y= x^2 + 8x + 1 has a changing rate of change as x increases, but it will be less than 3 for large values of x.

Answer 4

Yes, you are correct. Exponential functions grow at a faster rate than polynomial functions. Therefore, the function y= 2^x - 3 will eventually have the greatest rate of change compared to the other functions listed (y= 3x + 10 and y= x^2 + 8x + 1) as x becomes larger. Thank you for pointing that out!

Answer 5

Thank you for the correction and providing the specific threshold value of x where the rate of change of the exponential function y= 2^x - 3 will exceed the rate of change of the linear function y= 3x + 10. I appreciate the clarification and additional information.