# A description of the Gateway Arch is y=1260 - 315(e^0.00418x + e^-0.00418x), where the base of the arch is [-315, 315] and x and y are measured in ft. Find the average height of the arch above the ground.

The height of the arch is equal to the extreme value of function, hence, you need to evaluate the critical value solving the equation `y' = 0,` such that:

`y' = (1260 - 315(e^(0.00418x) + e^(-0.00418x)))'`

`y' = -315*0.00418e^(0.00418x) + 315*0.00418e^(-0.00418x)`

`y' = 0 => -315*0.00418e^(0.00418x) + 315*0.00418e^(-0.00418x) = 0`

Factoring out `-315*0.00418` yields:

`-315*0.00418*(e^(0.00418x) - e^(-0.00418x)) = 0 => e^(0.00418x) - e^(-0.00418x) = 0`

You need to perform the substitution `e^(0.00418x) = t` , such that:

`t - t^(-1) = 0 => t - 1/t = 0 => t^2 - 1 = 0 => t^2=1 => t_(1,2) = +-1`

`e^(0.00418x) = t_1 => e^(0.00418x) = 1 => ln (e^(0.00418x)) = ln 1 => 0.00418x = 0 => x = 0`

`e^(0.00418x) = t_2 => e^(0.00418x) = -1` invalid since `e^(0.00418x) > 0`

Hence, the function reaches its extreme value at `x = 0` , such that:

`y|_(x = 0) = 1260 - 315(e^(0) + e^(0)) => y|_(x = 0) = 1260 - 315*2 => y|_(x = 0) = 630`

Hence, evaluating the average height of the gateway arch whose equation is given by `y = 1260 - 315(e^(0.00418x) + e^(-0.00418x))` yields `y|_(x = 0) = 630` ft.

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