The temperature in degrees fahrenheit of a person during an illness is given by the function T(t)= -0.01t^2-0.2t+102.5 where t=0 hours. What was the high point in the person's temperature? When did it occur
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![**Educational Text: Analysis of Temperature in Degrees Fahrenheit**
**Problem Statement:**
The temperature, \( T(t) \), in degrees Fahrenheit during an illness is given by the function:
\[ T(t) = -0.01t^2 + 0.2t + 102.5 \]
**Objective:**
Determine the highest point of the person’s temperature and when it occurs.
**Solution Steps:**
1. **Finding Critical Points:**
- Derive \( T(t) \) to find \( T'(t) \):
\[ T'(t) = -0.02t + 0.2 \]
2. **Set the derivative equal to zero to find critical points:**
\[-0.02t + 0.2 = 0 \]
3. **Solve for \( t \):**
\[ t = \frac{0.2}{0.02} = 10 \]
- Therefore, \( t = 10 \) is a critical point.
4. **Determine if it is a Maximum:**
- Evaluate the second derivative, \( T''(t) \):
\[ T''(t) = -0.02 \]
- Since \( T''(t) < 0 \), \( t = 10 \) is a point of maxima.
5. **Calculate Maximum Temperature:**
- Substitute \( t = 10 \) back into the original equation:
\[ T(10) = -0.01(10)^2 + 0.2(10) + 102.5 \]
\[ T(10) = -1 + 2 + 102.5 = 103.5 \]
**Conclusion:**
The highest temperature of 103.5 degrees Fahrenheit occurs 10 hours after the measurement begins.
**Diagram/Graph Explanation:**
There are no explicit graphs or diagrams provided in this analysis. The mathematical processing involves exploring the behavior of the quadratic function through differentiation for determining maxima.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc82b0f05-2033-4339-b314-7c627e3238ac%2F0d1ae404-3738-4c5e-a072-50154fbf7d42%2F285fplg_processed.jpeg&w=3840&q=75)
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