Turn in #2 is due Monday, Sept. 15, 2014!
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I'm stumped on number 9 in the homework on page 91. Is there a way to apply the identity lim x>0 (sinx/x)=1?
ReplyDelete-Rachel Hersch
Yup! Try rewriting cscx as 1/sinx and then simplifying fractions. You should be able to split up the limits and apply that identity. Hope that helps :)
Delete-Safia Sayed
I have a question that relates to number 11 on the book work tonight. When you are just trying to find either the left or right of a limit, is there anyway you can find that limit algebraically?
ReplyDeleteThanks!
~Katie Weitzel~
If the function is continuous approaching from a given direction, you can evaluate the limit via substitution (if the function is defined at the chosen x value). Otherwise, look at the graph.
DeleteWhen you have a hole and also a point at that x-value already, should you fill the hole when asked to remove any discontinuities?
ReplyDelete-Eva A-L
Yes, your goal is to redefine the function so that it is continuous there.
Deleteeven if it is on the end of that interval of the function?
Delete-Eva A-L
Yes, the process is also valid for the end points of closed intervals.
DeleteWhat are the differences between sharp corners and cusps? It seems like the only difference is what triggers them (absolute values or rational exponents) I was wondering if there is something else that makes them unique?
ReplyDeleteThanks,
Rachel Hersch
Terrific question! For all practical purposes, there is no difference. They both cause sharpness (non-smooth) so they are points where a function is not differentiable. As to what is causing each, a corner can only come from an absolute value or a piece wise function. Cusps can be found in functions with rational exponents. A corner will be created when the derivative function is approaching different values from either side, while a cusp will actually have derivatives approaching infinite values ( vertical tangency), but in opposite directions(+infinity from one side, - infinity from the other).
DeleteHope this helps clarify! Thanks for the great question!
Can a function be differentiable at only certain intervals or do we look at the function as a whole?
ReplyDeleteThanks! Allison Honet
I'm not sure I understand your question. A function can be differentiable on its entire domain (think polynomials) or for specific intervals within its domain. Usually, we identify x values for which a function's derivative doesn't exist and use these to define intervals for which the function IS differentiable. Hope this helps!
DeleteCan you use IVT on an open interval?
ReplyDelete-Laura G.
IVT requires that a function is continuous on a CLOSED interval. The "c" value, however, will belong to the open interval unless you are trying to show the function is equal to one of its endpoint values (ie. f(a) or f(b)).
DeleteIn question 1c on the turn in, do we assume that it is a closed interval like 1a?
ReplyDeleteSarah Fried
I think it is a typo and that it should be the closed interval so you can use IVT...but I'm not sure
Delete-Eva A-L
Not a typo, see the thread above.
DeleteWhat is a one-sided derivative?
ReplyDeleteSarah fried
Is there a difference between solving for one-sided derivatives and regular derivatives?
Delete-Allison Honet
For something to be differentiable, it needs to meet the same limit on both sides. In the case of one sided derivatives, I think you have to solve the derivative as you would normally but if for example in a piece wise function, the limits are not the same approaching from the left and right sides, then the function isn't differentiable.
Delete-Kelsey Nowak
If we use the second definition of derivative, you can evaluate it from one side. Not usually helpful, but sometimes we are asked to do so...
DeleteCan you use the sum and difference rule to differentiate other expressions besides polynomials term by term?
ReplyDeleteThanks,
Rachel Hersch
Sure. Any sum of functions. Say, a sum of periodic trig functions like f(x)=5cosx +6sinx -8tanx. Towards the end of the course in third term we will study series, which is a sum of terms that are each functions of x, so the rule we will be particularly useful to us then!
DeleteIf you add two differentiable functions can you assume the sum is differentiable?
ReplyDeleteThanks,
Rachel Hersch
Not an assumption, it is fact. It is why the sum/difference rule works :-) great question!
DeleteThis comment has been removed by the author.
DeleteSo is this why the product rule works? (Because the product of two differentiable functions is differentiable.)
Delete-Laura G.
How can you determine if there is a horizontal tangent? Do you have to graph the function? Is there a way to determine it algebraically?
ReplyDelete-Laura G.
If a tangent line is horizontal, it's slope is? This means the derivative function at that point is equal to...?
DeleteHow should one approach #22 on pg. 120 "devise a rule for d/dx(1/f(x))." What is it even asking? To just solve for the derivative using the quotient rule?
ReplyDelete-Eva A-L
Yes, apply quotient rule with the numerator function being the constant function 1.
DeleteWhat was the graphing app which you recommended we could use?
ReplyDelete-Marie Suehrer
It is called "winplot". If you google it, you can download for free, and there are also "how to" videos/pages available as well.
DeleteI am really stuck on the last question on the Turn-In (number 6). I tried finding the derivative of e^x and got xe^(x-1) and then I cannot seem to simplify anything from there. What are we supposed to do with the identity given to us?
ReplyDelete~Katie Weitzel
Use the limit definition of derivative and take out an e^x in numerator. You should be able to take it from there.
Delete-Allison Honet
Just clarifying- When using the quotient rule should we expand the denominator or keep it simplified?
ReplyDeleteThanks!
Allison Honet
Keep it simplified so that it would be easier to find the 2nd or 3rd derivative if you had to. I think it would also be easier to see holes and asymptotes in that form too.
Delete-Laura G.
You don't need go expand it. Mr. Wilson said that if you don't expand it, it will be easier to work with when we learn how to take the second derivative
Delete~Katie Weitzel