An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Find the x-intercepts by setting f ( x ) = 3 x − 5 x = 0 , which yields x = 0 and x = 9 25 .
Find the y-intercept by setting x = 0 , which gives y = 0 .
Calculate the first derivative using the power rule: f ′ ( x ) = 3 − 2 x 5 .
The x-intercepts are x = 0 , 9 25 , the y-intercept is y = 0 , and the first derivative is f ′ ( x ) = 3 − 2 x 5 .
f ′ ( x ) = 3 − 2 x 5
Explanation
Problem Analysis The function given is f ( x ) = 3 x − 5 x . We need to find the intercepts and the first derivative.
Finding x-intercepts To find the x-intercept(s), we set f ( x ) = 0 and solve for x :
3 x − 5 x = 0
Factor out x :
x ( 3 x − 5 ) = 0
So, x = 0 or 3 x − 5 = 0 .
If x = 0 , then x = 0 .
If 3 x − 5 = 0 , then 3 x = 5 , so x = 3 5 , and x = ( 3 5 ) 2 = 9 25 .
Thus, the x-intercepts are x = 0 and x = 9 25 .
Finding y-intercept To find the y-intercept, we set x = 0 and solve for f ( 0 ) :
f ( 0 ) = 3 ( 0 ) − 5 0 = 0
Thus, the y-intercept is y = 0 .
Finding the First Derivative Now, let's find the first derivative f ′ ( x ) . We have f ( x ) = 3 x − 5 x = 3 x − 5 x 2 1 .
Using the power rule, we get:
f ′ ( x ) = 3 − 5 ⋅ 2 1 x − 2 1 = 3 − 2 x 5
Final Answer Therefore, the x-intercepts are 0 and 9 25 , the y-intercept is 0 , and the first derivative is f ′ ( x ) = 3 − 2 x 5 .
Examples
Understanding intercepts and derivatives is crucial in various real-world applications. For instance, in physics, if f ( x ) represents the position of an object at time x , the intercepts tell us when the object is at the origin, and the derivative f ′ ( x ) gives us the object's velocity at any given time. Similarly, in economics, if f ( x ) represents a cost function, the intercepts can represent fixed costs, and the derivative represents the marginal cost, which is the rate of change of cost with respect to production level.
