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Hey Mr. Campbell, could you help me solve questions 15 and 17 in the attached link below.  These are from Spanning Sets and Linear Combinations. The main issue I'm having is the wording of the questions. For both questions its stated that the vectors a and b span R^2, which implies they are non-collinear. But if thats the case, how can (Scalar Value)a = (Scalar Value)b and (Scalar Value)a + (Scalar Value)b = 0 ?
Thank You.

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I went to do the Unit 5 assignment today, but for some reason it was already handed in! About 2/3rd of the questions are unanswered and the ones that are have answers I did not put in. What should I do? I didn't even start the assignment yet.
Collin

Hello Collin, I have a feeling this issue may already be resolved. I did have to re-post the assignment a while back due to technical difficulties. I checked to see if you submitted the assignment and my records show that you haven't. Collin try to re-submit the assignment again, after you have had the chance to answer all the question that were left blank. Good luck.

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Hi! Mr. Campbell, How are you! This is less of a math question. I justed wanted to ask, is it ok if we watched the video lessons later than the day that it is posted, because I have an english assignment due this week, and I am getting quite stressed out with all the due dates. are you marking for completion on the day of the post or anything, because I don't want to lose marks in this course because of my other courses.

Hello Andrew.  That is perfectly fine with me.  I leave it up to you to manage your time while learning from home.  I am only making each video lesson an assignment to track student engagement and participation.  I am not too worried when you get it completed.  As long as you stay caught up with the lessons I am giving you guys, I will leave it up to you to set your own schedule for when you watch them.  This is why I don't put due dates on them.  Andrew good luck with your English assignment and take care and stay safe.

Thank you Mr.Campbell!

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The example question that we did for curve sketching required finding the critical values. I see that you also included the VA as the critical values, but I think they are not included as part of it. For example, on the formative quiz, there was a question asking for the number of critical values, but there is none. Do VAs count as critical values?

Hi Sophie. As far as when f'(x)=0, the VA's are not critical numbers and that is what the quiz was referring to. However, when setting up a first derivative interval table you are attempting to determine intervals of increase and decrease. This is where VA's are critical values since a function can change from increasing or decreasing (or vice verso) on either side of a VA. So VA's do count as critical values when setting up first and second interval tables. The quiz had a preset answer key and I have since changed the answer to include the VA's as critical values. I agree they should be, even if they do not correspond to when f'(x)=0. Good question Sophie, sorry for the confusion. I hope that helps.

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Why don't we calculate end behaviors to graph the rational function? such as x--> +∞ or x--> -∞

Hi Jessie. We establish the end behaviour of a function by comparing the degree of the numerator to that of the denominator, such as the existence of a horizontal asymptote. Then the simplest way to determine how the function approaches this asymptote is by calculating the value of the function as x gets large (x--> +∞ or x--> -∞).  By substituting a large value of x into the function, such as f(10000), we can see if the function approaches the H.A. from above or below.  For a video outlining the process of finding limits at infinity see:  https://www.youtube.com/watch?v=sjLFl7Z8W_I.
Jessie, I hope this helps you.

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The homework question 2d) on page 173 shows the graph of f''(x). It's a third degree function. Theres a double root and a single root.
Why is it that only the single root is a POI of f(x)?

Hi Chantelle.  Remember when we looked at y=x^4, and found y''? When we set y''=0 to find any POI's we got x=0 but it was not a POI.  This is when I mentioned that f''(x)=0 may indicate a POI.  In 2d) on page 173 the function goes from concave down to concave down on either side of x=-1.  For it to be a point of inflection it would have to change from concave down to concave up or vice versa for it to be a POI.  So remember the concavity must change at the point where f''(x)=0 for it to be a point of inflection.  Chantelle, I hope that helps.

Yes, thanks Mr. Campbell!!

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How would you find the end behaviours for this function (x^3+x)/(x^2-1) using the calculus method.

Hi Steve.  This one is a little tricky, since the degree in the numerator and the denominator decreases by 2 between the two consecutive terms. You know that for polynomial functions that as x gets large the function behaves like the leading term.  So you just have to consider it as the function y=x^3/x^2 and then evaluate the limit by dividing by the highest power of x (ie. x^3). Once you simplify by invert multiplying, you will get y=x. Which is the equaton of the linear oblique asymptote.

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On page 47, when it refers to infinite limits, if we use limit notation should we write the limit DNE and in brackets if it's going to infinity or not? Or would you prefer us to just write the limit equal to infinity?

Hi Sahir. Actually writing DNE (infinity) would be the preferred way to write it. Good point. But don't worry if you use the method of n>m and long division you won't need to answer the limit directly since you will be just identifying the linear oblique asymptote.

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Are all absolute max/mins always also local max/mins?

Hi Joline.  You are the very first person to ever use our Math Help.  Bravo!  Yes absolute max/mins are also considered local max/mins since they are the max or min on an interval.