An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Find the derivative of the function: f ′ ( x ) = x + 5 .
Set the derivative greater than zero to find where the function is increasing: 0"> x + 5 > 0 .
Solve the inequality: -5"> x > − 5 .
The function is increasing on the interval ( − 5 , ∞ ) .
Explanation
Analyze the problem We are given the function f ( x ) = 2 1 x 2 + 5 x + 6 and we want to find the interval where the function is increasing. A function is increasing when its derivative is positive.
Find the derivative First, we need to find the derivative of the function f ( x ) . Using the power rule, we have: f ′ ( x ) = d x d ( 2 1 x 2 + 5 x + 6 ) = 2 1 ( 2 x ) + 5 + 0 = x + 5
Solve the inequality Now, we need to find the interval where 0"> f ′ ( x ) > 0 . So we have: 0"> x + 5 > 0 Subtracting 5 from both sides, we get: -5"> x > − 5
Determine the interval This means the function is increasing for all x greater than − 5 . In interval notation, this is ( − 5 , ∞ ) .
Examples
Understanding where a function is increasing or decreasing is crucial in many real-world applications. For example, in economics, a cost function might be modeled as C ( x ) = 2 1 x 2 + 5 x + 6 , where x is the number of units produced. Determining where the derivative C ′ ( x ) is positive tells us the production level at which the cost is increasing. This helps businesses make informed decisions about production levels to optimize profits and manage costs effectively. Similarly, in physics, understanding when a velocity function is increasing helps determine when acceleration is positive, providing insights into the motion of objects.
In a device with a current of 15.0 A for 30 seconds, a total charge of 450 C flows. This equates to approximately 2.81 × 1 0 21 electrons. Thus, 2.81 × 1 0 21 electrons flow through the device in that time.
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