Calculus is my enemy?

Question

PLEASE HELP ME!!!!!

1. Suppose that 'a' and 'b' are positive constants.. Consider the function: f(x)=e^[-(x-a)^2/2b]
a.) determine the absolute max of this function and the corresponding x-value.
b.) determine the max and min. rates of change of this function and hte corresponding x-values.
c.) give a verbal description of the graph of htis function that includes the significant features.


This has something to do with L'Hopitals Rule, derivatives, increasing and decreasing tests, and inflection points.
But how do I do any of that without any numbers?!?!?!

Answer 1

F(x) = e^[-(x-a)^2 / 2b]

F'(x) = e^[-(x-a)^2 / 2b] * [-(1/2b) * 2(x-a)]

F'(x) = -(x-a)/b * e^[-(x-a)^2 / 2b]

or

F'(x) = -(x-a)/b * F(x)

Set F'(x) = 0

-(x-a)/b * F(x)

This expression is 0 only if x-a = 0, so the extreme occurs at F(a)

F(a) = e^0 = 1

You can check whether this is a max or min from the 2nd derivative evaluated at x = a; also, the maximum rates of change occur when the second derivative is zero.

d/dx -(x-a)/b * F(x) =

-(1/b) * F(x) - (x-a)/b *F'(x) =

-(1/b) * F(x) - (x-a)/b *[-(x-a)/b * F(x)]

F"(x) = [(x-a)^2 / b^2]* F(x) - (1/b) * F(x)

Evaluate this at x = a to get

F"(a) = -(1/b) since the second derivative is negative, this is a maximum value.

To get max/min rates of change, set F"(a) = 0:

[(x-a)^2 / b^2]* F(x) - (1/b) * F(x) = 0

since F(x) is never 0, divide through by it:

[(x-a)^2 / b^2] - (1/b)= 0

(x-a)^2 = b

x-a = ±√b

x = a±√b are the points of max/min rate of change and are the inflection points of the curve. To get the values of these, but x = a±√b into the formula for F'(x):

F'(a+√b)= -√b/b * e^[-(√b)^2 / 2b] = -(1/√b)*e^-(1/2)

F'(a-√b)= √b/b * e^[-(-√b)^2 / 2b] = (1/√b)*e^-(1/2)

Description of curve: there is a peak value of 1 at x = a, the curve slopes downward in both directions from that point. The slope for x > a is negative, and reaches its maximum (absolute) value at x = a√b where there is an inflection point, and the slope reduces in (absolute) value. For x < a, the slope is positive, reaching a maximum at x = a-√b where there is another inflection point. The curve approaches zero as x goes to + or - ∞.