Math 1500: Tips for Assignment 4

Published: Fri, 02/26/16

Don't have my book or audio?  You can download a free sample of my book and audio lectures containing Lessons 1 and  2:
Did you read my tips on how to study and learn Math 1500?  If not, here is a link to those important suggestions:
Did you read my tips I sent earlier on making sign diagrams and finding range? 
Here are the links:
Did you see my tips for Assignment 1? Click here.
Did you see my tips for Assignment 2? Click here.
Did you see my tips for Assignment 3? Click here.
Tips for Assignment 4
These are tips for the first assignment in the Distance/Online Math 1500 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course.  These assignments are very good (and challenging) practice.  The first assignment is a great way to build and review key skills that will be helpful for this course.

Here is a link to the actual assignment, in case you don't have it:
Remember, it is almost a certainty that, for every hour you spend studying my lessons, that will save you at least one hour, if not more, in time spent struggling on the assignment.  You are not saving time to start working on an assignment unprepared!  You are really increasing the time you will need to do an assignment.

You should thoroughly study Lesson 9: Curve-Sketching before attempting this lesson.
Question 1
This question is asking for the critical numbers.  That means they want the critical points and singular points.  All the points where the derivative is either zero or undefined.  That is the top and bottom zeros of the first derivative are the critical numbers (if the bottom zero is a vertical asymptote, it is not a critical number). 

Make sure you give both the x and y coordinates of your critical numbers, even though it would be fine to just give the x values in this question.

Part (b)
In order to compute g'(t), you have to first deal with the absolute value function.  First, find the zero for this function.  It is obvious that the zero is t = 3.  When t < 3, g(t) is negative.  When t > 3, g(t) is positive.  That means that you can now define g(t) as a piecewise function using the squiggly bracket { like we see all the time in Lesson 3: Continuity:

g(t) =
  • -(2t - 6) when t < 3
  • 0 when t = 3
  • 2t - 6 when t > 3
Recall: In Assignment 2, question 9, we proved that an absolute value function is not differentiable at its zero, so g'(t) does not exist at t = 3.  (I think we may have just found a critical number!).  But, you can now compute and state g'(t) for t < 3 and t > 3. 
Question 2
Similar to my Lesson 9, question 5.  Make sure you include the sentences I box in in your answer as that is necessary to justify your conclusions.

You will get a negative exponent when you do the derivative.  Pull that down to the denominator and then get a common denominator to properly identify the top and bottom zeros.  You may find it easier to just guess at what the top and bottom zeros are.  Be organized.  Sub in x=0, 1, -1, 2, -2, ...  There is no shame in just using trial and error to find zeros when things are difficult to factor.  Facts are facts.  If you find an x-value that causes either a top or bottom zero, then there is no doubt. 

Hint: There are two critical numbers in [-1, 1], one critical point and one singular point.
Question 3
This is a Mean Value Theorem question.  Click the link below for the procedure to follow to "verify" the Mean Value Theorem:

Be sure to point out that the given function is a polynomial, and so it is certainly continuous and differentiable, therefore the Mean Value Theorem applies.

Hint: You should get two answers for c, but only one of them will be within [-1, 0].
Question 4
Don't forget to make your x, y table of values in this problem to properly determine the (x, y) coordinates they request.

Part (a)
Recall the tips I gave you about finding the domain of a function back in Lesson 1, questions 2 and 3.  Make sure you are clear about the domain here!

Do part (c) before you do part (b)!

Part (c)
Since they are asking you to show your work, I suggest that you do the one-sided k/0 limits for all the bottom zeros of f(x) here to prove if y is going up to infinity or down to negative infinity as it approaches each bottom zero from each side.

Be sure to state the equation of the vertical asymptote properly.  For example, if there is a vertical asymptote at 10, don't say "10," say "Vertical Asymptote at x = 10."

Otherwise, the rest of this problem is doing a complete first derivative analysis as I teach in Lesson 9.

Part (b)
Vertical Asymptotes are not critical numbers! Your crtical numbers are the critical points (top zeros of f'(x)) and singular points (bottom zeros of f'(x) that are not already known to be vertical asymptotes).

Parts (d) and (e)
You will use a sign diagram for f'(x) to identify increase and decrease.
Make sure you state the intervals.  Don't use the union symbol "U" if there are two or more intervals of increase/decrease.  Some profs object to uniting intervals with the union symbol for some reason.

For example, if you establish f(x) is increasing from -5 to -1, and also from 3 to 10, just write these intervals in a list separated by commas, always using round brackets. 

Thus, f(x) is increasing on (-5, -1), (3, 10).

Part (f)
By "first derivative test," they simply mean, use the sign diagram you made for increase/decrease earlier in the problem to visualize whether the critical points are local mins or max.

Part (g)
I discuss and illustrate the second derivative test in Lesson 9, Practise Problem 20.  Note that you are not asked to do any analysis with the second derivative (such as concavity and inflection points), you are just asked to use the second derivative test to check the critical points.  Obviously, you should reach the same conclusion as you did in the previous part.

That hint about the polynomial is basically telling you that part of f" is always a positive amount (and, therefore, it has no zero, either), which may be helpful since all you need to know when performing the second derivative test is the sign of f" at the critical point.  If you prefer, you could make a complete sign diagram for f" and then that will tell you its sign at each critical point.  In fact, the concavity symbols for each region will make it easy for you to visualize if the critical point is a local max or local min.
Question 5
A classic curve sketch problem. Make sure you have done my Lesson 9, questions 3 and 4, and Practise Problems 1-16 in order to be thoroughly prepared for this problem.

Since you are asked to show all your work, I suggest that you actually compute the k/0 limits and solve the infinity limits formally, as I teach in Lesson 2, to ensure you are not penalized.

Again, that hint about the polynomial is basically telling you that part of f" is always a positive amount (and, therefore, it has no zero, either), which will be helpful when you make a complete sign diagram for f" to identify the concavity.