Problem 78
Question
The Mexican subsidiary of ThermoMaster manufactures an indoor-outdoor thermometer. Management estimates that the profit (in dollars) realizable by the company for the manufacture and sale of \(x\) units of thermometers each week is $$ P(x)=-0.001 x^{2}+8 x-5000 $$ Find the intervals where the profit function \(P\) is increasing and the intervals where \(P\) is decreasing.
Step-by-Step Solution
Verified Answer
The profit function \(P\) is increasing on the interval \((-\infty, 4000)\) and decreasing on the interval \((4000, \infty)\).
1Step 1: Find the first derivative of the profit function.
To find the first derivative, we will differentiate \(P(x)\) with respect to \(x\):
\[P(x)= -0.001x^{2}+8x-5000\]
\[P'(x)= (-0.001 \cdot 2)x+8 = -0.002x+8\]
2Step 2: Find the critical points.
Critical points are the values of \(x\) for which \(P'(x)=0\) or \(P'(x)\) is undefined. In this case, \(P'(x)\) is a linear function, so we only need to find the value of \(x\) for which \(P'(x)=0\):
\[-0.002x+8 =0\]
Now, solve for \(x\):
\[x = \frac{8}{0.002} = 4000\]
The critical point is \(x=4000\).
3Step 3: Determine the intervals of increase and decrease.
To determine the intervals, we need to check the sign of \(P'(x)\) for values of \(x\) in the intervals: \((-\infty, 4000)\) and \((4000, \infty)\).
1. For \(x < 4000\), let's pick an arbitrary point, say \(x=3000\):
\[P'(3000)=-0.002(3000)+8= -6+8= 2\] which is greater than 0.
Since \(P'(3000)>0\), the profit function is increasing on the interval \((-\infty, 4000)\).
2. For \(x > 4000\), let's pick an arbitrary point, say \(x=5000\):
\[P'(5000)= -0.002(5000)+8 = -10+8 = -2\] which is less than 0.
Since \(P'(5000)<0\), the profit function is decreasing on the interval \((4000, \infty)\).
The profit function \(P\) is increasing on the interval \((-\infty, 4000)\) and decreasing on the interval \((4000, \infty)\).
Key Concepts
First DerivativeCritical PointsIntervals of Increase and Decrease
First Derivative
Understanding the first derivative is crucial when discussing various functions, such as the profit function in business applications.
The first derivative of a function, expressed as \(P'(x)\), provides us with the rate at which the function's output changes with respect to changes in its input variable. For the profit function \(P(x) = -0.001x^2 + 8x - 5000\), finding the first derivative \(P'(x)\) means finding out how the profit changes with each additional unit produced.
After differentiating the given profit function, we find that \(P'(x) = -0.002x + 8\), a linear function. This simplicity allows us to easily analyze the change in profit relative to the quantity of units. Positive values of the first derivative indicate an increase in profit, while negative values suggest a decrease. This first derivative is pivotal for identifying trends in the function's behavior and for finding critical points.
The first derivative of a function, expressed as \(P'(x)\), provides us with the rate at which the function's output changes with respect to changes in its input variable. For the profit function \(P(x) = -0.001x^2 + 8x - 5000\), finding the first derivative \(P'(x)\) means finding out how the profit changes with each additional unit produced.
After differentiating the given profit function, we find that \(P'(x) = -0.002x + 8\), a linear function. This simplicity allows us to easily analyze the change in profit relative to the quantity of units. Positive values of the first derivative indicate an increase in profit, while negative values suggest a decrease. This first derivative is pivotal for identifying trends in the function's behavior and for finding critical points.
Critical Points
Critical points are important values where the function’s behavior changes. These occur where the first derivative is zero or undefined.
In our example, \(P'(x) = -0.002x + 8\) has a critical point where this derivative equals zero. To find this, we set the derivative equal to zero and solve for \(x\): \(0 = -0.002x + 8\), yielding \(x = 4000\).
This \(x = 4000\) is the critical point, and it represents a threshold in the profit function. At this level of production, the profit function shifts from increasing to decreasing. This means that producing more than 4000 units starts to decrease the company's profit.
In our example, \(P'(x) = -0.002x + 8\) has a critical point where this derivative equals zero. To find this, we set the derivative equal to zero and solve for \(x\): \(0 = -0.002x + 8\), yielding \(x = 4000\).
This \(x = 4000\) is the critical point, and it represents a threshold in the profit function. At this level of production, the profit function shifts from increasing to decreasing. This means that producing more than 4000 units starts to decrease the company's profit.
Intervals of Increase and Decrease
To identify when profits are rising or falling, we analyze the intervals of increase and decrease by looking at the sign of the first derivative before and after the critical points.
By selecting test points from intervals around \(x = 4000\) and evaluating \(P'(x)\), we can determine the profit improvement advice.
For \(x < 4000\), say at \(x = 3000\), \(P'(3000) = 2\), which is positive. This tells us that within the interval \( (-\infty, 4000) \), the profit function is increasing. However, if we consider \(x > 4000\), such as \(x = 5000\), \(P'(5000) = -2\), which is negative, indicating that on the interval \( (4000, \infty) \), the profit function is decreasing.
By understanding these intervals, businesses can make informed decisions on production levels to maximize profit—a vital concept in both economics and calculus.
By selecting test points from intervals around \(x = 4000\) and evaluating \(P'(x)\), we can determine the profit improvement advice.
For \(x < 4000\), say at \(x = 3000\), \(P'(3000) = 2\), which is positive. This tells us that within the interval \( (-\infty, 4000) \), the profit function is increasing. However, if we consider \(x > 4000\), such as \(x = 5000\), \(P'(5000) = -2\), which is negative, indicating that on the interval \( (4000, \infty) \), the profit function is decreasing.
By understanding these intervals, businesses can make informed decisions on production levels to maximize profit—a vital concept in both economics and calculus.
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