OPTIMIZATION



MAT210 QUIZ 8 KEY

Suppose that the concentration [pic] of a drug in the blood stream [pic] hours after it was taken orally is given by

[pic], [pic].

a) Use a graphing utility to find the hour at which the concentration of the drug in the blood stream is at an absolute maximum. Sketch the graph, and label the approximate position of the absolute maximum.

Answer:

[pic]

b) An inflection point occurs after the concentration has reached a maximum and begins to decrease. Use a graphing utility to find the hour at which the inflection point occurs.

Answer:

To find the inflection point we need to check where [pic] changes concavity. From the graph in part a) we can see that the function is at first concave down and then, around 1.2 changes concavity. To find the exact point where the function changes concavity we need to check the sign of [pic].

First we find [pic]. Using the formula [pic] we have:

[pic]

We can graph the first derivative to check if it makes sense

[pic]

Now we need to check the sing of the second derivative. Since we are asked to use a graphing utility we can simply graph the second derivative using the nDeriv function on our calculator.

Graphing nDeriv(e^(-(x-0.5)^2)(-2x+1),x,x) gives

[pic]

Remark:

We can confirm our answer algebraically by finding the second derivative:

Using the product rule we find

[pic]

Simplifying we get:

[pic]

Since the exponential is always positive we have that the zeros of [pic]are the zeros of [pic].

Solving the quadratic equation gives [pic]. Choosing the positive zero gives [pic].

This is the inflection value since [pic] is negative to the left of this point and positive to the right.

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The graph of [pic]for positive values of t is graphed on the right. Using the maximum function on the calculator we can easily find the absolute maximum at [pic].

From the graph of [pic] on the left we can see that the derivative is positive to the left of x=0.5 and it is negative to the right. This confirms our answer to part a) i.e. the fact that (0.5,f(0.5))=(0.5,1) is a maximum

From the graph of [pic] on the right we can see that the second derivative is negative at first and then positive. Thus the zero of the second derivative is an inflection point. Using the zero function on our calculator we can easily find [pic]as the value of the inflection point.

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