Average Value Theorem, Limits, and Slopes.

In summary, the problem has two parts: a) finding the slope of a tangent to a graph and b) finding the limit of a function. The slope of the tangent can be found by finding the limit of the function.
  • #1
ardentmed
158
0
Hey guys,

I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help.

Question:
08b1167bae0c33982682_6.jpg


So for the first one, I just used f(b)-f(a)/ b-a and got 589, 1208, and 1366 respectively via simple substitution.

For 1b, I overaged the values from II and III and got 1287 stores/year.

As for c, I sketched the slope and got a line of best fit. Then I calculated the slope as 5886-1886 / 2002-1998 and got 1000 shops/year. But I'm not too sure about this answer. Are there better ways to compute the slope? As for 2a, I took sample values approaching 9, so x=9.1, .. x=9.0001 and ultimately guessed that the limit is 4.5.

As for b, I couldn't get a definitive answer, but I'm guessing that factoring works. Am I close?
Thanks in advance.
 
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  • #2
1a) i is incorrect, but I agree with the rest. b) Correct. c) You are asked to find the slope of a tangent, so you need a curve...

2.) The image is too dim for me to easily read. I suggest you create a new thread and either obtain a better image or preferably type the problem.

When your problems have multiple parts, it is best to create a separate thread for each.
 
  • #3
MarkFL said:
1a) i is incorrect, but I agree with the rest. b) Correct. c) You are asked to find the slope of a tangent, so you need a curve...

2.) The image is too dim for me to easily read. I suggest you create a new thread and either obtain a better image or preferably type the problem.

When your problems have multiple parts, it is best to create a separate thread for each.

I re-did 1a i and ended up getting 1193 stores/year because:

(5886-3501) / (2002-2000) = 2385/2

~1192.5
 
  • #4
ardentmed said:
I re-did 1a i and ended up getting 1193 stores/year because:

(5886-3501) / (2002-2000) = 2385/2

~1192.5

Looks good. :D
 

FAQ: Average Value Theorem, Limits, and Slopes.

What is the Average Value Theorem and what is its significance?

The Average Value Theorem is a fundamental concept in calculus that states that for a continuous function on a closed interval, there exists a point within that interval where the function's average value is equal to its instantaneous value at that point. This theorem is significant because it allows us to find the average rate of change of a function, which has many real-world applications in areas such as physics and economics.

How do you find the limit of a function?

To find the limit of a function, you first determine the behavior of the function as the input values get closer and closer to a certain value. This can be done by graphing the function or by using algebraic techniques such as factoring and simplifying. If the function approaches a single value as the input values approach the given value, then that value is the limit of the function at that point.

What are the different types of limits and how are they evaluated?

The three types of limits are left-sided, right-sided, and two-sided limits. Left-sided limits are evaluated by approaching the given value from the left side of the graph, right-sided limits from the right side, and two-sided limits from both sides. To evaluate a limit, you can use techniques such as direct substitution, factoring, and rationalizing the denominator.

How do you determine the slope of a function?

The slope of a function is a measure of its steepness at a given point. To determine the slope of a function, you can use the slope formula, which is the change in y over the change in x. This can also be interpreted as the rise over run on a graph. Alternatively, you can use the derivative of the function to find the slope at a specific point.

What is the relationship between the Average Value Theorem and slopes?

The Average Value Theorem and slopes are closely related as they both involve finding the average rate of change of a function. The Average Value Theorem uses the slope of a secant line to find the average rate of change over a given interval, while the derivative of a function gives the instantaneous rate of change, or slope, at a specific point. Additionally, the Mean Value Theorem, which is a special case of the Average Value Theorem, states that the slope of the tangent line at a given point is equal to the slope of the secant line passing through the endpoints of the interval.

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