   Chapter 10.1, Problem 60E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Medication The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by x ( t ) = 2000 t t 2 + 16 (a) For what t-values is x increasing?(b) Find the t-value at which x is maximum.(c) Find the maximum value of x.

(a)

To determine

To calculate: The value of t where the function is increasing if the function is x(t)=2000tt2+16 if the number of milligrams x of a medical in the bloodstream t hours after the dose is taken can be modeled as given above.

Explanation

Given Information:

The provided function is x(t)=2000tt2+16.

Formula Used:

To obtain the relative maxima and minima of a function,

1. Find the first derivative of the function.

2. Now set the derivative equal to 0 then obtain the critical points.

3. Evaluate f(x) at values of x to the left and one to the right of each critical point to develop a sign diagram.

If f(x)>0 to the left and f(x)<0 to the right of the critical value, the critical point is a relative maximum point.

If f(x)<0 to the left and f(x)>0 to the right of the critical value, the critical point is a relative minimum point.

Calculation:

Consider the provided function,

x(t)=2000tt2+16

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

Hence, there will no change in the values of critical values as in the derivative graph.

Take out the first derivative of the function by the power rule,

x=(t2+16)(2000)2000t(2t)(t2+16)2=2000t2+32,000(t2+16)2

Put the value of x=0,

2000t2+32,000

(b)

To determine

To calculate: The value of t where the x-value is maximum if the function is x(t)=2000tt2+16 if the number of milligrams x of a medical in the bloodstream t hours after the dose is taken can be modeled as given above.

(c)

To determine

To calculate: The maximum value of x if the function is x(t)=2000tt2+16 if the number of milligrams x of a medical in the bloodstream t hours after the dose is taken can be modeled as given above.

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