How do you use the mean value theorem for #2sinx +sin2x# on closed interval of #[4pi, 5pi]#?

Question

Emilia Aguilar · Accepted Answer

The Mean Value Theorem says that 
if a function #f# is 
continuous on the closed interval #[a, b]# and
differentiable on the open interval #(a, b)#,  then there is a number #c# in #(a, b)# with #color(white)"sssssssssssssss"##f'(c)= (f(b)-f(a))/(b-a)#. (Note:  The amount of detail you'll be asked to provide depends, to some extent, on your teacher's goals for the class.) In the question:
#f(x)=2sin x + sin 2x# the interval is #[4 pi , 5 pi]#  and  #(4 pi , 5 pi)# . It is true that:
#f# is continuous at every real number, so it is continuous on #[4 pi , 5 pi]# 
(#sin# is continuous, and #2x# is continuous so #sin 2x# is contiunuous.  And the sum of continuous functions is continuous.) It is also true that
#f# is differentiable at every real number, so it is differentiable on #(4 pi , 5 pi)#. 
(Differentiable means the derivative exists and #f'(x)=2cosx+2cos2x# exists (is defined) for all values of #x#.) Theerfore, the Mean Value Theorem allows us to conclude that: There is a number #c# in #(4 pi, 5 pi)# with:  #color(white)"sssssssssssssss"##f'(c)= (f(5 pi)-f( 4 pi))/(5 pi - 4 pi )#. Doing some arithmetic we can rewrite this conclusion as; We can conclude that 
There is a number #c# in #(4 pi, 5 pi)# with:  #color(white)"sssssssssssssss"##2cosc+2cos2c#= 0#. That conclude the use of the Mean Value Theorem for #f(x)=2sin x + sin 2x# on the interval is #[4 pi , 5 pi]# (As an additional exercise in solving equations, your teacher or textbook may also ask you to find the value or values of #c# that the theorem assures us are there.)

Elijah Abram · Answer

undefined To use the Mean Value Theorem for $2\sin(x) + \sin(2x)$ on the closed interval $[4\pi, 5\pi]$, follow these steps:

1. Verify that the function $2\sin(x) + \sin(2x)$ is continuous on the interval $[4\pi, 5\pi]$ and differentiable on the open interval $(4\pi, 5\pi)$.
   
2. Calculate the derivative of the function. The derivative of $2\sin(x) + \sin(2x)$ is $2\cos(x) + 2\cos(2x)$.

3. Find the average rate of change of the function over the interval $[4\pi, 5\pi]$. This is done by evaluating the function at the endpoints of the interval and taking their difference, then dividing by the length of the interval. The average rate of change is given by:

$$\frac{f(5\pi) - f(4\pi)}{5\pi - 4\pi}$$

4. Apply the Mean Value Theorem, which states that there exists at least one $c$ in the open interval $(4\pi, 5\pi)$ such that the derivative of the function at $c$ is equal to the average rate of change found in step 3.

So, set the derivative of the function equal to the average rate of change and solve for $c$:

$$2\cos(c) + 2\cos(2c) = \frac{f(5\pi) - f(4\pi)}{5\pi - 4\pi}$$

5. Solve for $c$ to find the value where the derivative equals the average rate of change.