Suppose that T = f(t) gives the temperature T (in degrees Fahrenheit) of a pot of coffee t minutes after being brewed. If f '(20) = -3 and f(20) = 125, use a linear approximation (linearization) to estimate the coffee's temperature after 20.4 minutes.
Suppose that T = f(t) gives the temperature T (in degrees Fahrenheit) of a pot of coffee t minutes after being brewed. If f '(20) = -3 and f(20) = 125, use a linear approximation (linearization) to estimate the coffee's temperature after 20.4 minutes.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Linear Approximation of Coffee Temperature**
In this example, we explore using linear approximation to estimate the temperature of coffee over time. The function \( T = f(t) \) represents the temperature \( T \) (in degrees Fahrenheit) of a pot of coffee \( t \) minutes after being brewed.
**Given:**
- \( f'(20) = -3 \)
- \( f(20) = 125 \)
**Objective:**
Estimate the coffee's temperature at \( t = 20.4 \) minutes using linear approximation.
**Solution Approach:**
Linear approximation (or linearization) allows us to estimate the value of a function near a specific point using the derivative.
The general formula for linear approximation is:
\[ f(t) \approx f(a) + f'(a)(t-a) \]
Here, \( a = 20 \), so:
\[ f(20.4) \approx f(20) + f'(20)(20.4 - 20) \]
Substituting the given values:
\[ f(20.4) \approx 125 + (-3)(20.4 - 20) \]
\[ f(20.4) \approx 125 - 3 \times 0.4 \]
\[ f(20.4) \approx 125 - 1.2 \]
\[ f(20.4) \approx 123.8 \]
Thus, using linear approximation, the estimated temperature of the coffee at 20.4 minutes is 123.8°F.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2d98c66-c867-40dc-8395-8baf72422c58%2Fe4ccb15a-0b76-4766-818e-acacc58d852d%2Fvngmza9_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Approximation of Coffee Temperature**
In this example, we explore using linear approximation to estimate the temperature of coffee over time. The function \( T = f(t) \) represents the temperature \( T \) (in degrees Fahrenheit) of a pot of coffee \( t \) minutes after being brewed.
**Given:**
- \( f'(20) = -3 \)
- \( f(20) = 125 \)
**Objective:**
Estimate the coffee's temperature at \( t = 20.4 \) minutes using linear approximation.
**Solution Approach:**
Linear approximation (or linearization) allows us to estimate the value of a function near a specific point using the derivative.
The general formula for linear approximation is:
\[ f(t) \approx f(a) + f'(a)(t-a) \]
Here, \( a = 20 \), so:
\[ f(20.4) \approx f(20) + f'(20)(20.4 - 20) \]
Substituting the given values:
\[ f(20.4) \approx 125 + (-3)(20.4 - 20) \]
\[ f(20.4) \approx 125 - 3 \times 0.4 \]
\[ f(20.4) \approx 125 - 1.2 \]
\[ f(20.4) \approx 123.8 \]
Thus, using linear approximation, the estimated temperature of the coffee at 20.4 minutes is 123.8°F.
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