Let f be the function defined by f(x)=2sinx+cosx ?
The average rate of change of f over the interval [0,b] is 0.05, where b>0 . Which of the following is an equation that could be used to find the value of b?
A) f(b)=0.05
B) f(b)−f(0)=0.05
C: (f(b)−f(0))/b−0=0.05
D: (f(b)+f(0))/2=0.05
So what’s the answer
d
Well, let's break this down and see which equation makes the most sense here. The average rate of change of a function over an interval is given by the difference in the function values divided by the difference in the inputs. In this case, we're given that the average rate of change is 0.05.
So, to find the value of b, we want to find the equation that represents this information. Let's look at the options:
A) f(b) = 0.05: This equation is just setting the function equal to 0.05, but that doesn't give us any information about the average rate of change. So, A is not the right choice.
B) f(b) − f(0) = 0.05: This equation subtracts the function value at 0 from the function value at b, which is what we want to do to find the difference in the function values. So, this equation makes sense, but it doesn't divide by the difference in the inputs, which we also need to do to find the average rate of change. So, B is not the right choice either.
C) (f(b)−f(0))/b−0 = 0.05: This equation divides the difference in the function values by the difference in the inputs, which is what we need to do to find the average rate of change. So, this equation seems promising. C could be the right choice.
D) (f(b)+f(0))/2 = 0.05: This equation takes the average of the function values at 0 and b, which is not what we're looking for. We want the average rate of change, not the average of the function values. So, D is not the right choice either.
Based on this analysis, it looks like C: (f(b)−f(0))/b−0 = 0.05 is the equation that could be used to find the value of b. Good luck with your calculations!
To find the value of b, we need to determine the equation that represents the average rate of change of f over the interval [0, b].
The average rate of change of a function f over an interval [a, b] is given by the formula:
Average rate of change = (f(b) - f(a))/(b - a)
In this case, the interval is [0, b], so a = 0.
Now let's examine the answer choices:
A) f(b) = 0.05
This equation doesn't represent the average rate of change. It simply sets the value of f(b) equal to 0.05, which doesn't relate to the rate of change.
B) f(b) - f(0) = 0.05
This equation represents the difference in the function values at b and 0, which is then set equal to 0.05. It does not represent the average rate of change.
C) (f(b) - f(0))/b - 0 = 0.05
This equation represents the difference in function values at b and 0, divided by b, and then set equal to 0.05. This is the correct formula for the average rate of change, so this is the correct answer.
D) (f(b) + f(0))/2 = 0.05
This equation calculates the average of the function values at b and 0 and sets it equal to 0.05. This does not represent the average rate of change.
Therefore, the correct answer is C: (f(b) - f(0))/b - 0 = 0.05.
as always, the average rate of change over [a,b] is
(f(b) - f(a)) / (b-a)
So plug in your numbers.