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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 this course? If not, here is a link to those important suggestions: Here is a link to the actual assignment, in case you don't have it: 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.
Of course, always seek out assistance from my book, your course notes, etc. if you ever hit a question you don't understand, but try not to be learning things as you do an assignment. Learn first, then put your learning to the test. 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.
NOTE: They are NOT saying there will be no h in your final answer. They are just warning you that you will be
able to get rid of some of the h's. As I teach in Lesson 4, you will always be able to factor h out of the top and cancel with the h in the bottom if you have done your algebra properly.
In other words, do exactly what I do in my Lesson 4, #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).
Be sure to study my Lesson 1, pages 5-20 to prepare for this question. You may find my audio on making sign diagrams helpful here to properly understand what I am teaching you about how to make a sign diagram. I have given you the entirety of my audio for
Lessons 1 and 2 in the free samples above. Here is my audio introducing sign diagrams:
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 -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.
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 #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.
Be warned! If you are doing your math right, you should get two answers for the inverse (which is impossible) because you will have to take a square root. WHENEVER YOU HAVE TO USE A SQUARE ROOT TO HELP SOLVE A PROBLEM, YOU MUST REALIZE THERE ARE TWO POSSIBLE ANSWERS: THE POSITIVE SQUARE ROOT OR THE NEGATIVE SQUARE ROOT.
However, only one of those answers fits your restricted domain from part (b). Show both answers, then explain why you are discarding one of them.
For example, if you were solving the equation x^2 = 9 (nothing like this particular problem), you would square root both sides and get x = + square root of 9 or x = - square root of 9. So x= 3 or x= -3.
Make sure you have read the Logs and Exponentials section of Lesson 1 in my book (starts on page 23).
You may also find my audio working through my Lesson 1, #4 helpful (included in my free samples
above):
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!
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 - 3 for the region of (-2, 1] by plotting 3 points in
that region. For sure, two of those points should be the endpoints at -2 and 1. However, since that region is up to but not including the endpoint at -2, plot a "hole" at that location rather than a dot. Which is to say, there will be a hole at (-2, -3). On the other hand, the region is up to and including 1, so you will plot a dot at (1,6).
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.
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 - 3x^2 - 4x + 6 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 - 3x^2 - 4x + 6.
Part (c) Standard conjugate problem like my
questions 2 and 3. Watch your bracketing!
Part (d) 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.
In
part (g), you can use a conjugate if you want to, but it is not necessary (and just makes the question messier).
In part (h), you must use a conjugate before you do anything else.
Classic continuity question like my Lesson 3, #1 to 3. Make sure you use the correct piece for f(2), limit as x approaches 2-, and limit as x approaches 2+. 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)?
This uses the Intermediate Value Theorem like my Lesson 3, #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 (1,2). 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.
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