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:

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:

Note that you need to study Lesson 1 (Skills Review) and Lesson 2 (Limits) from my Intro Calculus book to prepare for this assignment.  I also recommend that you study Lesson 4 (Definition of Derivative) as the algebra skills that are taught in that lesson will assist you with question 1 below.

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(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 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 -b  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.

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.

Make sure you have read the Logs and Exponentials section of Lesson 1 in my book (starts on page 23).

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 get two answers for x, but then will explain that one of those answers must be discarded because you can only do the logarithm of positive numbers.  But make sure you discuss all this or you will be penalized.

You have to use logs here (I 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.

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!

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+1)^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, 0).

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.

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.

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.

Don't ever let a weird looking question intimidate you.  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.

Classic 0/0 limit problem like my question 1.