Mean Value Theorem problem help

In summary, we are trying to find the average value of units sold for a new product over the first year, represented by the function S(t)=200(5-(9/(2+t))), where t is the time in months. To find this average, we use the formula f'(c)=(f(b)-f(a))/(b-a), where c is the average value, b is the end of the time interval, and a is the start of the time interval. For part a), we use the interval [1,12] and the original equation to find the average value. For part b), we set the average value
  • #1
physicsman2
139
0

Homework Statement


A company introduces a new product for which the number of units sold S is
S(t)=200(5-(9/(2+t)) where t is the time in months

a) Find the average value of S(t) during the first year
b)During what month does S'(t) equal the average value during the first year


Homework Equations


f'(c)=(f(b)-f(a))/(b-a)


The Attempt at a Solution


well, i have no idea how to do it, but i believe for a), you have to find the derivative of the function(which i have no clue what it is) then use the equation given above to find the average value
For part b), i just have no clue
 
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  • #2
i think i got it but i just need some reinforcement on my answers

for a), you do (f(12)-f(1))/(12-1) using the original equation since the interval for months is [1,12] since i don't think zero would be in that kind of interval
then for b) you would set that answer equal to the derivative of the equation and solve for t

am i right
 
  • #3
Yes, although I suspect you start at t = 0.

For the derivative, I suggest working out the brackets and using the chain rule... do you know how to differentiate 1/u with respect to u?
 
Last edited:
  • #4
i already found the derivative but why would you start at t = 0 if there really is no 0 month if the first month is 1
 
  • #5
can someone please tell me why t = 0 and not 1
 
  • #6
Sure, if you start at t = 1, then what about all that information from t = 0 to t = 1? (the first 30 days).
 
  • #7
thanks it makes more sense
 

1. What is the Mean Value Theorem and how does it apply to problem solving?

The Mean Value Theorem is a fundamental concept in calculus that states that for a function f(x) that is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), there exists a point c within the interval (a,b) such that the slope of the tangent line at c is equal to the average rate of change of f(x) over the interval [a,b]. In problem solving, the Mean Value Theorem can be used to find the average rate of change, determine whether a function is increasing or decreasing, and find the points where a function has a horizontal tangent line.

2. How can I identify when to use the Mean Value Theorem in a problem?

The Mean Value Theorem is typically used in problems that involve finding the average rate of change or the points where a function has a horizontal tangent line. These types of problems often involve functions that are continuous and differentiable on a closed interval. Additionally, the Mean Value Theorem can be applied to a variety of functions, including polynomial, exponential, and trigonometric functions.

3. Can you provide an example of using the Mean Value Theorem in a problem?

Sure, let's say we have a function f(x) = x^2 on the interval [0,4]. We can use the Mean Value Theorem to find the point c within this interval where the slope of the tangent line is equal to the average rate of change of the function. The average rate of change of f(x) over the interval [0,4] is (f(4)-f(0))/(4-0) = (16-0)/4 = 4. The derivative of f(x) is f'(x) = 2x, so we can set 2x = 4 and solve for x to find that c = 2. Therefore, the point (2,4) is where the slope of the tangent line is equal to the average rate of change of the function.

4. What are the limitations of the Mean Value Theorem?

The Mean Value Theorem has a few limitations. First, it only applies to functions that are continuous on a closed interval and differentiable on the open interval. This means that it cannot be used for functions with discontinuities or sharp corners. Additionally, the theorem only guarantees the existence of a point c where the slope of the tangent line is equal to the average rate of change, but it does not provide a method for finding this point.

5. Are there any real-world applications of the Mean Value Theorem?

Yes, the Mean Value Theorem has many real-world applications, particularly in physics and engineering. For example, it can be used to determine the average velocity of an object over a specific time interval or to find the points where the velocity of an object is zero, indicating that it is changing direction. It can also be used in economics to analyze changes in the average rate of change of different variables, such as prices or production rates.

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