The national debt of a South American country t years from now is predicted to be D(t) = 70 + 18t4/3 billion dollars. Find D'(8). D'(8) = 1 X Interpret your answer. The national debt is increasing Find D'(8). D"(8) = X Interpret your answer. by The rate of growth of the national debt is increasing X billion dollars per year after 8 years. by X billion dollars per year each year after 8 years.
The national debt of a South American country t years from now is predicted to be D(t) = 70 + 18t4/3 billion dollars. Find D'(8). D'(8) = 1 X Interpret your answer. The national debt is increasing Find D'(8). D"(8) = X Interpret your answer. by The rate of growth of the national debt is increasing X billion dollars per year after 8 years. by X billion dollars per year each year after 8 years.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### National Debt Growth Prediction
The national debt of a South American country \( t \) years from now is predicted to be \( D(t) = 70 + 18t^{4/3} \) billion dollars. We need to determine \( D'(8) \) and interpret its meaning.
#### Step 1: Finding \( D'(8) \)
To find the derivative of the debt function \( D(t) \):
\[ D(t) = 70 + 18t^{4/3} \]
Using the power rule:
\[ D'(t) = \frac{d}{dt} \left(70\right) + \frac{d}{dt} \left(18t^{4/3}\right) \]
Since the derivative of a constant is zero:
\[ D'(t) = 0 + 18 \times \frac{4}{3} t^{\frac{4}{3}-1} \]
\[ D'(t) = 18 \times \frac{4}{3} t^{\frac{1}{3}} \]
\[ D'(t) = 24t^{\frac{1}{3}} \]
Substituting \( t = 8 \) into the derivative:
\[ D'(8) = 24 \times 8^{\frac{1}{3}} \]
Since \( 8^{\frac{1}{3}} = 2 \):
\[ D'(8) = 24 \times 2 = 48 \]
#### Interpretation of \( D'(8) \):
The national debt is **increasing** by **48** billion dollars per year after 8 years.
#### Step 2: Finding \( D''(8) \)
To find the second derivative of the debt function \( D(t) \):
\[ D'(t) = 24t^{\frac{1}{3}} \]
Using the power rule again:
\[ D''(t) = \frac{d}{dt} \left(24t^{\frac{1}{3}}\right) \]
\[ D''(t) = 24 \times \frac{1}{3} t^{\frac{1}{3} - 1} \]
\[ D''(t) = 8t^{-\frac{2}{3}} \]
Substituting \( t = 8 \) into the second derivative:
\[ D''](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e0cd55c-6f84-412a-ac81-9a79eaea810f%2F125634e8-487a-4da0-ae05-9df74ee5dd39%2F0bbdc2s_processed.png&w=3840&q=75)
Transcribed Image Text:### National Debt Growth Prediction
The national debt of a South American country \( t \) years from now is predicted to be \( D(t) = 70 + 18t^{4/3} \) billion dollars. We need to determine \( D'(8) \) and interpret its meaning.
#### Step 1: Finding \( D'(8) \)
To find the derivative of the debt function \( D(t) \):
\[ D(t) = 70 + 18t^{4/3} \]
Using the power rule:
\[ D'(t) = \frac{d}{dt} \left(70\right) + \frac{d}{dt} \left(18t^{4/3}\right) \]
Since the derivative of a constant is zero:
\[ D'(t) = 0 + 18 \times \frac{4}{3} t^{\frac{4}{3}-1} \]
\[ D'(t) = 18 \times \frac{4}{3} t^{\frac{1}{3}} \]
\[ D'(t) = 24t^{\frac{1}{3}} \]
Substituting \( t = 8 \) into the derivative:
\[ D'(8) = 24 \times 8^{\frac{1}{3}} \]
Since \( 8^{\frac{1}{3}} = 2 \):
\[ D'(8) = 24 \times 2 = 48 \]
#### Interpretation of \( D'(8) \):
The national debt is **increasing** by **48** billion dollars per year after 8 years.
#### Step 2: Finding \( D''(8) \)
To find the second derivative of the debt function \( D(t) \):
\[ D'(t) = 24t^{\frac{1}{3}} \]
Using the power rule again:
\[ D''(t) = \frac{d}{dt} \left(24t^{\frac{1}{3}}\right) \]
\[ D''(t) = 24 \times \frac{1}{3} t^{\frac{1}{3} - 1} \]
\[ D''(t) = 8t^{-\frac{2}{3}} \]
Substituting \( t = 8 \) into the second derivative:
\[ D''
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