using mean value theorem

justify y=e^x over [0,1]

not sure what you mean by justify, but since

y(0) = 1
y(1) = e
the slope of the line segment joining (0,1) and (1,e) is e-1,
we want c such that
y'(c) = e-1
e^x = e-1
x = ln(e-1) = 0.54 which is in the interval (0,1)

To justify the function y=e^x over the interval [0,1] using the mean value theorem, we need to show that there exists a number c in the interval (0,1) such that the average rate of change of the function over this interval is equal to the instantaneous rate of change at the point c.

Let's start by finding the average rate of change of the function y=e^x over the interval [0,1]. The average rate of change can be calculated as the difference in function values divided by the difference in the corresponding x-values:

Average rate of change = [e^1 - e^0] / (1-0) = (e - 1) / 1 = e - 1.

Now, let's find the instantaneous rate of change at some point c in the interval (0,1). The instantaneous rate of change can be found by taking the derivative of the function y=e^x, which is dy/dx = e^x.

Since y=e^x is a continuous function and e^x is differentiable for all real numbers, we can apply the mean value theorem. According to the mean value theorem, if a function is continuous over a closed interval [a,b] and differentiable over the open interval (a,b), then there exists at least one number c in (a,b) such that the instantaneous rate of change equals the average rate of change.

In our case, the function y=e^x is continuous over the closed interval [0,1] and differentiable over the open interval (0,1). We have already calculated the average rate of change to be e - 1. Now, we can set the instantaneous rate of change equal to the average rate of change:

e^c = e - 1.

Solving this equation for c will give us the specific value that justifies using the function y=e^x over the interval [0,1]. However, in this case, it is not possible to find an exact value for c that satisfies this equation. Nonetheless, we can observe that the exponential function e^x is monotonically increasing for all x, which means that the average rate of change (e - 1) is always less than the instantaneous rate of change e^c. Therefore, we can conclude that there exists a number c in the interval (0,1) for which the mean value theorem holds and justifies using the function y=e^x over the interval [0,1].