Math 260 Week 1 Lab
Name______
Part I - Directions: For each category of problem type you are provided several problem questions and the correct answer. Look at the example questions and answers you are provided and determine how the answers were arrived at. Then, devise a rule you could use that could be used to solve any such problem of that type. Once you have completed all 4 categories use the rules/processes you have devised to solve the 4 additional problems at the end of the worksheet.
Category 1:
1)Q: 2) Q: 3) Q:
A: A: A:
What Rule can you discover that allows you to find the limit for any problem like those above? If example 3 had been written as what issue would have arisen? What does that mean for the limit?
1)
2)
3)
The answer is wrong, to get the question would have to be .
the problem with this equation is that solving for the limit directly results in division y zero. This means that another method (factoring for example) is necessary to find the limit of the equation, because the function approaches the limit, but never reaches it.
Category 2:
1)Q: 2) Q: 3) Q:
A: A: A:
What process can you develop that can be used to solve any category two problem? What issues
arise when you try to use the rule developed for category one problems? If you were to graph these functions what feature might exist in the graph at the limiting value? Why?
1)
2)
3)
This method does not work with the problems in category one because they cannot be factored.
If these functions were to be graphed a discontinuous point in the graph, such that the graph is undefined only at that position.
Category 3:
1)Q: 2) Q: 3) Q:
A: A: A:
What seems to be the most important factor when dealing with limits as x approaches infinity? What do the degrees of the polynomials in the numerator and denominator have to do with it? What rules or process can you devise to solve any category 3 problem? If you were to graph these functions, what feature would appear in the graph at the limiting value? Why?
The most important factor for this category seems to be the degree of the polynomials in the function.
It would appear that if the degree of the polynomials in the numerator equal the degree of the polynomial in the denominator, then the coefficient of the degree term defines the limit. If the degree of the denominator is greater than the numerator, then the limit is 0. If the degree of the numerator is greater than the degree of the denominator, then the limit is ∞.
If these functions were to be graphed then an asymptote that approaches the limit would appear, because the function will continue to approach the limit, but never actually reach it.
Category 4:
1)Q: 2) Q: 3) Q:
A: does not exist A: A:
What happens when you try to apply the method for category 1 problems to these limits? What seems to dictate when a limit such as this does not exist and when it is either infinity or negative infinity? What process can we use to determine the answer for any category 4 problem? If you were to graph these functions, what feature would appear in the graph at the limiting value? Why?
Applying category one method:
It seems to be that when the denominator has a higher power than the numerator, the function approaches positive or negative ∞. If the coefficient of the numerator is positive, then ∞ will be the limit, otherwise if the coefficient of the numerator is negative, then -∞ will be the limit.
When graphed at a limiting value the function asymptotes from both sides of the limiting value as the function approaches positive or negative ∞.
For these problems find the limit.
1) 3)
Category 3Category 1
2) 4)
Category 2Category 4
∞
Part II:
Given the piecewise function defined below and its graph determine the following:
a)Where is this graph discontinuous? Which discontinuities are removable and which are not?
The graph has discontinuities at x = -2 and x = 4. At x = -2 the limit of the both functions defined for x < -2 and x > -2 is 9, so this discontinuity is removable by removing the statement f(x = -2) = 2, and expanding the range for or to include x = -2. At x = 4 the discontinuity is removable by defining that f(x = 4) = 9, the limit of the function currently defined.
b)Does the limit of this function exist as x approaches -2? Why or Why Not? If so, what is it?
No, the limit is 9. f(x = -2) = 2, therefore the limit does not exist.
c)Is this graph continuous at -2? Why or Why not? If it is not continuous what exists in the graph at x = -2?
No it is not continuous, because the limit of and is 9 at x = -2, yet the function is defined such that f(x = -2) = 2.
d)Does the limit of this graph exist as x approaches zero? Why or Why Not?
Yes, the limit as x approaches 0 is 0, which exists for this graph.
e)Is this graph continuous at 4? Why or Why not?
No, the graph is undefined for x = 4.
f)What is the limit of this function as x approaches 4 from the left? How about the right? What exists in the graph at x = 4?
An asymptote from both sides exists with a discontinuity at x = 4.
g)Does there appear to be a limit as x approaches positive infinity? If so, what is it? What exists in the graph as x approaches infinity?
Yes, using the rule from Category 3 of Part I, the limit is 1. As x approaches infinity the graph asymptotes to y = 1.
h)Using set notation describe where this graph is continuous.
{-∞≤ x <-2, -2< x ≤∞}
Part 3:
a)What is a derivative? What information can it provide us?
A derivative is the instantaneous slope of a function. It provides us with the slope at a given point, and where the derivative equals 0 is a minimum or maximum value for the function.
b)What is the limit definition of the derivative? From what basic formula relating to lines is it developed? How is that related to part a?
Where the derivative is the slope of the line tangent to a function, as Δx approaches 0 the slope becomes closer to a specific point on the line. By taking the limit the as Δx approaches 0, the instantaneous slope for a specific point on the function can be found.
c)Use the limit definition of the derivative to find the derivative of the following function.
d) Given that the generic form of an objects path through time in relationship to earth is
Do the following:
a)Find the model for the Velocity of the object through time where the initial velocity is 50ft/sec and the initial height is 600ft.
b)Find the average velocity of the object between t = 1 and t = 4 seconds
c)Find the instantaneous velocity of the object at t = 2 seconds
e) Remembering, that the derivative is equal to the slope of the tangent line to the curve at any point on the curve, do the following:
a)Plot and in your graphing calculator
b)How does f’(x) reflect the changes in the curve of f(x)?
f’(x) is equal to the slope of the curve f(x) for the same x value.