Find the Relative Extrema of a Function

Question

The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is…

(a)

Complete the table and use it to approximate the time when the concentration is greatest…

t	1.5	2	2.5	3	3.5	4	4.5	5
C(t)

(b)

Use a graphing utility (graphing calculator, online graphing tool, etc.) to graph the concentration function and use the graph to approximate the time when the concentration is greatest…

(c)

Use calculus to determine analytically the exact time when the concentration is greatest…

(d)

Find the exact time when the concentration increases at the greatest rate…

(e)

Explain every step in your own words…

Answer 1

(a)

C(1.5) = 0.047

C(2.0) = 0.060

C(2.5) = 0.071

C(3.0) = 0.079

C(3.5) = 0.083

C(4.0) = 0.085

C(4.5) = 0.083

C(5.0) = 0.040

From the table, t=4 is the time of highest concentration

(b)

By inspection from the graph, t=4 is the time of greatest concentration.

(c)

C(t) = 4t/(125+t³)

C’(t) = 4(125-2t³)/(125+t³)

C’(t) = 0 when 125 = 2t³

t = 3.914

C(3.914) = 0.0855

Since C(3.5) and C(4.0) < C(3.914), we know that t=3.914 is a maximum.

(d)

C'(t) specifies the rate of increase, so C''(t) is used to find the maximum rate.

C’’(t) = -24t²(-t³+250)/(t³+125)³

C’’(t) = 0 when -24t²(-t³+250) = 0

t = 0 or -6.288

Since the measurement interval is [0,5], t=0 is the time when the rate of concentration is greatest. That t=0 is a maximum can be shown by inspection of a graph of C’(t) or computing C’(1) = 0.031 < C’(0) = 0.032.

(e)

Part (a): Plugged the values of t into C(t) to get the results for each table entry.

Part (b): I did what was asked for, but could not transfer the graph to the answer section here.

Part (c): Getting the max concentration means finding the max C(t) in the given interval. That is done by taking the first derivative of C(t), setting it to zero to get the critical point, and solving for t. Then, showing that C(t) at points left and right of the CP is lower proved the CP is a max.

Part (d): C'(t) represents the rate of change of concentration with time. Getting the fastest rate of change of the rate of change requires taking the derivative of C'(t) which is C''(t), finding the critical point, and verifying it to be a max.