Math 1500: REVISED Tips for Assignment 1 (includes a copy of the assignment)

Published: Fri, 09/30/16

There was a typo in my hints for question 4 that has caused confusion. There is a hole at (-1,-1), not (-1,0) as I erroneously stated.
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Did you read my tips on how to study and learn this course?  If not, here is a link to those important suggestions:
Tips for Assignment 1
Here is a link to the actual assignment, in case you don't have it handy:
Study Lesson 1 (Skills Review), Lesson 2 (Limits), Lesson 3 (Continuity), and Lesson 4 (Definition of Derivative) from my Intro Calculus book to prepare for this assignment.

Do note that my free sample above does include both Lessons 1 and 2 in my book and audio lectures.
Question 1
This question uses the skills that I teach in Lesson 4.  It is actually the definition of derivative but without doing the limit as h goes to 0.  I suggest you study that lesson to learn the necessary skills (such as simplifying Triple Deckers).  Do not use limits, of course, but make sure you have simplified to the point of factoring h out of the top and canceling with the h below.  Also note that the a in their questions is just like the x in my examples.  You are subbing a+h in place of x in the function, and a in place of x.

In other words, do exactly what I do in my Lesson 4, questions 1(b) and 2(a) and (b), but never even mention the limit as h approaches 0, and, therefore, don't sub h=0 in at the end.  And use a instead of x.  You should be able to simplify the numerator in each problem and factor h out to cancel with the h in the denominator.  Once you cancel the h factors in the numerator and denominator you are done (since you are not doing the limit as h approaches 0).
Question 2
Part (a)
You can google complete the square if you want to be reminded of how to convert this given quadratic equation into the parabola form they request, but I don't think that you will ever have to do something like this again.  The h and k they are referring to are the x and y coordinates, respectively, of the vertex of the parabola.  In general, for any quadratic equation (ax^2 + bx + c), the x coordinate of the vertex will be -divided by (2a).  Which is to say, h = - b / (2a).  Once you know h, you can sub it into the given quadratic equation to compute k, the y-coordinate of the vertex.

Once you know h and k, sub them into the given format and you have answered part (a).

I recommend that you sketch a graph of this parabola and include it in your solution to help visualize the problem.  Just make a quick table of values.  Plot the vertex you just found, and choose one or two other points on each side of the vertex.  Note, that you could also factor the given trinomial function to identify its zeros. Those would be the x-intercepts of the parabola.  You could even find the vertex that way since it must be midway between the x-intercepts.

Part (b)
The vertical line test checks if a graph is a function or not.  If a vertical line can never pass through more than one point on the graph, the graph represents a function

The horizontal line test checks if a graph's inverse would be a function.  If a horizontal line can never pass through more than one point on the graph, then its inverse would be a function.

A graph is a one-to-one function if and only if it passes both the vertical and horizontal line tests.  That proves that for any one value of x there is only one value of y on the graph.  The function is one-to-one.

Sketch the graph of the parabola, and it is obvious that it fails the horizontal line test.  So you must restrict the domain to make it one-to-one.  The most logical thing to do is to restrict the domain to all the x values starting from the vertex and to the right.

Part (c)
Be sure to use the (h, k) form they had you generate in part (a).  Much easier to do the algebra with that one.

I show you how to find the inverse of a function in Lesson 8 of my book (just after question 4 in the lecture, page 230).  I like to immediately have x and y change places, then proceed to isolate y. 

Once you have isolated y, make sure you replace y with f^-1 (x), the f-inverse symbol.

Since an inverse changes the x and y around, it is sort of rotating a graph sideways (and flipping it).  What was vertical becomes horizontal, and vice-versa.  Thus, a graph that passes the horizontal line test, will have an inverse that passes the vertical line test, proving that the inverse will be a function.  Only one-to-one functions will have inverses that are also function.

Part (d)
The domain and range for f(x) should be obvious from the sketch you made earlier in the question (but, remember, you must use the restricted domain you identified in part (b) for the one-to-one function). 

The domain and range of the inverse function are easy.  The domain of f is the range of f-inverse, and the range of f is the domain of f-inverse.  Just reverse the answers you just stated.
Question 3
Make sure you have read the Logs and Exponentials section of Lesson 1 in my book (starts on page 23).

Part (a)
Use a log law to combine the left side into one logarithm, then convert the log to an exponential.  Be careful!  If you do things properly, you will have to use the quadratic formula and get two answers for x, but be sure to check both of those answers to see if either one must be discarded because you can only do the logarithm of positive numbers.  Note that it is not important whether a solution is positive or negative. Rather, does subbing the solution into the original equation cause the log of a negative or log of 0. Make sure you discuss all this or you will be penalized.

Part (b)
You have to use logs here (use ln, the natural log).  Challenging algebra.  Once you have used logs and simplified by using log laws (watch your brackets), collect all the terms with x to one side, and factor x out in order to isolate it.

Part (c)
This is actually a quadratic equation.  If e^x is t, then e^(2x) is t^2 (t-squared).  Solve for t first.  Once you know what t is (there are two answers), solve for x since t is e^x.  But one of those answers will have to be discarded!
Question 4
Part (a)
To sketch each piece of this function, merely plot 2 or 3 points for the domain of each piece and connect the dots.  For example, graph y= x^2 - 2 for the region of (-infinity, -1) by plotting 3 points in that region.  For sure, one of those points should be the endpoint at -1.  However, since that region is up to but not including the endpoint, plot a "hole" at that location rather than a dot.  Which is to say, there will be a hole at (-1, -1).

If the domain of a piece includes the endpoint, plot a dot.  If the domain is up to but not including the endpoint plot a hole at the y value that it almost reaches.

Part (b)
Once you have graphed all four pieces of this function, it is a simple matter to read off the solutions for all the limits, similar to what I do with my opening example in Lesson 2 of my book.  You might find my audio discussing the start of Lesson 2 helpful here.  That is included in the free sample at the top of this message.

You could also solve the limits algebraically using the procedure I show in Lesson 3, but the fact that they have asked you to draw the graph suggests they intend for you to merely read the limits off of the graph, no work. You can use the algebraic method to check your answers though.

Remember: It is irrelevant what is AT x=a when doing the limit as x approaches a
  • For x approaching a-, find the piece of graph that is immediately to the left of a, and find what y-value that piece connects to (the endpoint of that piece, be it a hole or a dot, is the limit).
  • For x approaching a+, find the piece of graph that is immediately to the right of a, and find what y-value that piece connects to (the endpoint of that piece, be it a hole or a dot, is the limit).
  • If and only if the limits for a- and a+ agree, the limit approaching a exists and has that value.
Part (c)
You can answer this question by merely observing the graph you have drawn and saying whether or not the graph is broken or has a hole in it at the x-values in question. If you want to be thorough, you can mention the definition of continuity to justify your answer.
Question 5
This is a good run-through of limits.  Study Lesson 2 thoroughly to prepare for this question.  This is the most important question on this assignment, in my opinion.  Many of these limits could appear on your final exam.

Part (a)
As always when solving a limit, sub in the given x and see what happens first.  Only once you know which of the three possibilities you are dealing with, do you know what to do next.

Part (b)
Classic 0/0 limit problem like my question 1.  However, the x-cubed polynomial on top makes it difficult to factor this expression.  You should have realized that 1 is a zero on top and bottom, so (x-1) is a factor on top and bottom as my factoring tip in Lesson 2 discusses.  This will require polynomial long division.  You may want to Google that term to get an idea how to do this.  Here is a video I found on YouTube that explains things well:


In your case, you need to divide x-1 into x^3 - 2x^2 - 4x + 5 to see what (x-1) multiplies with to factor the top properly.  Check your answer.  If you have factored the top properly, you should be able to multiply it back together to get x^3 - 2x^2 - 4x + 5.

Part (c)
Standard conjugate problem like my questions 2 and 3.

Part (d)
A 0/0 problem that will transform into a K/0 limit problem like my questions 6 and 7.

Part (e)
Standard absolute value limit like my questions 4 and 5.

Part (f)
Read the Squeeze Theorem section at the end of the lecture in Lesson 2 of my book.  My examples and question 17, should be quite helpful here.

Parts (g) and (h)
These are rather challenging infinity limits like I teach in Lesson 2.  Be sure to look at all of my questions 10 to 14.  You may also find Practise Problems 64 to 74 (especially 74) helpful, too.
Question 6
Classic continuity question like my Lesson 3, questions 1 to 3.  Make sure you use the correct piece for f(3), limit as x approaches 3-, and limit as x approaches 3+.  Hint, each piece will be used once, and only once.

Remember the log review from Lesson 1.  That natural log should not cause you any consternation.  After all, what is ln(1)?
Question 7
This uses the Intermediate Value Theorem like my Lesson 3, questions 4 and 5.  First, pull everything over to the left side of the equation, and define the left hand side as your function f(x).  Make sure you formally state, "Let f(x) = ...."

Then prove f(x) has at least one zero on (-1,0).  Remember, that if the sign changes, you know there must be at least one zero. You can never know from this theorem alone exactly how many zeros there are. Make sure you state that f(x) is continuous (why is it continuous?). Be sure that you say "by Intermediate Value Theorem" as your justification.