I'm stuck on this problem and I dont know how to even start it...
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
as always, the average rate of change over the interval [a,b] is
(f(b)-f(a))/(b-a)
Looks like C to me
Better review the topic in your text again. It is just the slope of the line joining the two points on the curve.
Well, well, well, it seems you've stumbled upon a math problem. Don't worry, I'm here to help. Let's break it down.
To find the equation that could be used to find the value of b, we need to consider the average rate of change of f over the interval [0, b].
But before we jump into solving it, let's analyze the options.
Option A: f(b) = 0.05. Well, that's not directly related to the average rate of change, so we can scratch that one off.
Option B: f(b) - f(0) = 0.05. Hmmm, we're getting somewhere. This equation compares the difference of f(b) and f(0) to 0.05, which could be the average rate of change. So, let's keep this one for now.
Option C: (f(b) - f(0)) / (b - 0) = 0.05. Ah, the good old slope formula. This equation directly calculates the average rate of change by taking the difference of f(b) and f(0) divided by the difference of b and 0. This one seems promising!
Option D: (f(b) + f(0)) / 2 = 0.05. Well, well, well, what's this? This equation takes the average of f(b) and f(0) and compares it to 0.05. It might not directly give us the average rate of change, so let's set this one aside for a moment.
So, after our careful evaluation of the options, it seems like option C, (f(b) - f(0)) / (b - 0) = 0.05, is the equation we're looking for. It directly calculates the average rate of change of f over the interval [0, b].
I hope that clears things up for you, and remember, math problems may seem tough, but with a little humor and perseverance, you can conquer anything!
To find the value of b, we need to determine the equation that represents the average rate of change of the function f over the interval [0,b].
Average rate of change is defined as the difference between the function values divided by the difference in the corresponding independent variable values. In this case, the formula for average rate of change is:
Average rate of change = (f(b) - f(0))/(b - 0)
Now, we can examine each answer choice to see which equation correctly represents the average rate of change as 0.05:
A) f(b) = 0.05
This equation only gives the function value at b, but it does not include the rate of change component. This is not the correct equation.
B) f(b) - f(0) = 0.05
This equation calculates the difference in function values between b and 0, but it does not include the division by the difference in independent variable values. This is not the correct equation.
C) (f(b) - f(0))/(b - 0) = 0.05
This equation correctly represents the average rate of change by dividing the difference in function values by the difference in independent variable values. This is the correct equation.
D) (f(b) + f(0))/2 = 0.05
This equation calculates the average of the function values at b and 0, but it does not include the division by the difference in independent variable values. This is not the correct equation.
Therefore, the correct equation to find the value of b is:
C) (f(b) - f(0))/(b - 0) = 0.05