Given that y = (e^x)/x, x > 0, find the range of values of x where y is a decreasing function of x.

The interval for which y is a decreasing function of x is:

Or as an inequality:

Step-by-step explanation:

We are given the equation:

And we want to find the range of values x for which y is a decreasing function of x.

y is decreasing whenever y' is negative. Find y' using the Quotient Rule:

Differentiate:

y is decreasing whenever y' is negative. Thus:

Multiply both sides by x². This is always positive so we do not need to change the sign:

Factor:

eˣ is always positive. So:

Adding one to both sides produces:

Therefore, y is a decreasing function of x when x is less than one (and greater than 0).

In interval notation:

Or as an inequality:

? = 46

Step-by-step explanation:

Since this is a right triangle, we can use trig functions

sin theta = opp/ hypotenuse

sin ?  = 38 / 53

Take the inverse sin of each side

sin ^-1 ( sin ?) = sin ^-1 ( 38/53)

? = sin ^-1 ( 38/53)

? =45.80579628

To the nearest degree

? = 46