Week+10+Oct+20-24

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Monday Oct 20th
First task was to go over the homework worksheet from Friday. This worksheet and the solutions can be viewed in the slideshow posted on last week's page. After this, we practiced a few 'strange" chain rule problems in preparation for a complicated step within the derivations of the derivatives of sine inverse and cosine inverse.The pratcice problems were similar to these: 1) If y = sinx, find dy/dx. Solution: dy/dx = cosx

2) If y = sin t, find dy/dx. Solution: dy/dt = cos t

3) If y = sin t find dy/dx : Solution: dy/dx = (dy/dt) (dt/dx) = cos t (dt/dx)

4) If y = sin x find dy/dt : Solution dy/dt = (dy/dx)(dx/dt) = cosx (dx/dt)

Unfortunately we took up a lot of class time with the homework so we did not get through the derivatives of all the trig. functions. The derivations of the derivatives of the inverse sine and inverse cosine functions are shown below.

Due to lack of time, we were unable to derive the derivatives of the other four inverse trig. functions. All sx are shown in the table below.

Tuesday 21st Oct
Today we worked on the derivative of the inverse secant function (see the graphic below). To do this, we first reminded ourselves of the relationship between the inverse cosine and inverse secant functions so we could looked at the graph of inverse secant on the calculator. This relationship is that sec¯¹(x) = cos¯¹(1/x).

We worked some more derivative problems, including a few we had for homework on Monday.

Wednesday 22nd October
We reviewed for tomorrow's test using the Senteo system and the Smartboard. The problems and answers are shown in the slide-show below. media type="custom" key="2251233" Do not forget to memorize the derivative of all the functions we have worked with so far, as well as the Product Rule, Quotient Rule, Chain Rule and Power Rule. All rules are summarized on the page Derivative Rules.

Thursday 23rd October
Test on sections 4.1 - 4.5

Friday 24th October
Class began with the posting of two questions. 1) True or False: If function f is continuous at x = c then f is differentiable at x = c. Since we didn't really know what differentiable meant, we could re-word the second part of the statement as " f ' (c) exists " Between us, we came up with several different functions which indicated that the statement was false. Examples of graphs of functions which were counterexamples to the statement are shown below.

The conclusion here is that CONTINUITY does NOT imply DIFFERENTIABILITY.

2) True or False? If f is differentiable at x = c, then f is continuous at x = c. This proved more difficult to find a counterexample to this statement; we were trying to think of a function which had a tangent line at a point x = c but did not have a function value at x = c. Thinking a little more prompted the question " how can you have a tangent line to a curve at a point which does not exist?" Of course, you can't!

The conclusion here is that DIFFERENTIABILITY DOES imply CONTINUITY.