Web resources available

This quiz tests the work covered in Lecture 15 and corresponds to Section 3.4 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There are more web quizzes at Wiley, select Section 4.

There is a film at http://www.calculus-help.com/funstuff/tutorials/derivatives/deriv05.html which covers the chain rule, although it uses the derivative of trigonometric functions which you haven’t covered yet, but don’t let that stop you.

Section 3.5 of The Mathematics Learning Centre’s booklet on differentiation Introduction to Differential Calculus covers differentiating using the chain rule.

There is an applet at http://www.scottsarra.org/applets/calculus/FunctionComposition.html which shows you what the composition of two functions looks like and shows the tangent at each point. There are plenty of exercises with solutions at the following sites but you may not know how to do some of them yet as they involve trigonometric and logarithmic functions.
http://www.math.ucdavis.edu/ kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html
http://archives.math.utk.edu/visual.calculus/2/chain_ rule.2/

Question 1

a)		If   then the inside function is
b)		If   then the inside function is
c)		If   then the inside function is
d)		If   then the outside function is
e)		If   then the outside function is

 

There is at least one mistake.
For example, choice (a) should be false.

If we have to raise all of to the power so is the inside function.

There is at least one mistake.
For example, choice (b) should be true.

There is at least one mistake.
For example, choice (c) should be false.

needs to be differentiated using the chain rule.
The inside function should be and the outside function should be

There is at least one mistake.
For example, choice (d) should be true.

There is at least one mistake.
For example, choice (e) should be true.

Since   and the statement is correct.

Your answers are correct

1. False. If we have to raise all of to the power so is the inside function.
2. True.
3. False. needs to be differentiated using the chain rule.
   The inside function should be and the outside function should be
4. True.
5. True. Since   and the statement is correct.

Question 2

a)		If   then
b)		If   then
c)		If   then
d)		If   then
e)		If   then

 

There is at least one mistake.
For example, choice (a) should be true.

This is a straight forward application of the chain rule.

There is at least one mistake.
For example, choice (b) should be true.

Since we have

There is at least one mistake.
For example, choice (c) should be false.

Since we have

There is at least one mistake.
For example, choice (d) should be false.

There is at least one mistake.
For example, choice (e) should be true.

Since we have

Your answers are correct

1. True. This is a straight forward application of the chain rule.
2. True. Since we have
   
3. False. Since we have
   
4. False.
5. True. Since we have
   

Question 3

a)		The derivative is never zero.		b)
c)				d)

 

Not correct. Choice (a) is false.

Try again, there is a solution.

Not correct. Choice (b) is false.

Try again, you have not substituted into the formula for correctly.

Not correct. Choice (c) is false.

Try again, you do not have the correct value for

Your answer is correct.

Hence when and

Question 4

a)		No interval,  is always concave down.
b)		is always concave up.
c)		is concave up when
d)		is concave up when

 

Not correct. Choice (a) is false.

Try again, you need to find the second derivative and find out when it is positive.

Not correct. Choice (b) is false.

Try again, you need to find the second derivative and find out when it is positive.

Your answer is correct.

From Question 3 we know that so we have

The second derivative is zero at and it is positive when and negative when

Not correct. Choice (d) is false.

Try again, you have found the correct place where the concavity changes.
You need the second derivative to be positive.