Let y = e^(x/1).
Find the differential dy when x = 5 and dx = 0.5
Find the differential dy when x = 5 and dx = 0.01
To find the differential dy for a given function y = e^(x/1), we can use the derivative of the function. The derivative of e^(x/1) with respect to x can be found using the chain rule, which states that if we have a function f(g(x)), then the derivative is given by f'(g(x)) * g'(x).
In this case, we have y = e^(x/1), so g(x) = x/1 and f(g(x)) = e^g(x). Let's calculate the derivative of y with respect to x.
dy/dx = (d/dx)(e^(x/1))
= e^(x/1) * (d/dx)(x/1)
= e^(x/1) * (1/1)
= e^(x/1)
Now that we have the derivative function dy/dx = e^(x/1), we can substitute the x and dx values to find the differential dy.
When x = 5 and dx = 0.5:
dy ≈ dy/dx * dx
≈ e^(5/1) * 0.5
When x = 5 and dx = 0.01:
dy ≈ dy/dx * dx
≈ e^(5/1) * 0.01
To obtain the approximate values, we can calculate e^(5/1) using a calculator and then multiply it by the given dx value.
Note: The value of e^(5/1) is approximately 148.413.
When x = 5 and dx = 0.5:
dy ≈ 148.413 * 0.5
≈ 74.207
When x = 5 and dx = 0.01:
dy ≈ 148.413 * 0.01
≈ 1.484
Therefore, when x = 5 and dx = 0.5, the differential dy is approximately 74.207, and when x = 5 and dx = 0.01, the differential dy is approximately 1.484.