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School of Mathematics
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The Chain Rule Quiz

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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

The chain rule is applied to composite functions $h(x) = f(g(x))$ where we describe $f(x)$ as the outside function and $g(x)$ as the inside function.
Each of the following functions can be composed of two simple functions which can be differentiated without using the chain rule.
Which of the following statements are true and satisfy the above condition?

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

If we have to raise all of $x2 + 5$ to the power $e$ so $x2 + 5$ 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.

$g(x) = (x3 + 2x + 1)2$ needs to be differentiated using the chain rule.
The inside function should be $g(x) = (x3 + 2x+ 1),$ and the outside function should be $-1 f(x) = x2 .$

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 $------------- h(x) = 3∘(2x2 +3x + 1)2 = (2x2 + 3x+ 1)23$ and the statement is correct.

Your answers are correct

False. If we have to raise all of $x2 + 5$ to the power $e$ so $x2 + 5$ is the inside function.
True.
False. $g(x) = (x3 + 2x + 1)2$ needs to be differentiated using the chain rule.
The inside function should be $g(x) = (x3 + 2x+ 1),$ and the outside function should be $-1 f(x) = x2 .$
True.
True. Since $------------- h(x) = 3∘(2x2 +3x + 1)2 = (2x2 + 3x+ 1)23$ and the statement is correct.

Question 2

Which of the following have been differentiated correctly?

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 $h(x) = (x2 + 5)12$ we have
$1 1 x h′(x) = 2x-(x2 +5)- 2 = √-2---. 2 x + 5$

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

Since $h(x) = (x3 + 2x+ 1)-2$ we have
$2 h′(x) = (3x2 + 2)(- 2)(x3 + 2x+ 1)-3 =---2(3x-+-2) . (x3 +2x + 1)3$

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

$′ 2 4 x 2 2 4 x 2 h (x) = (8x+ e )3(2x + e) = 3(8x + e)(2x + e ) .$

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

Since $2 h(x) = (2x2 +3x + 1)3$ we have
$h′(x) = (4x+ 3)2(2x2 + 3x+ 1)- 13 =-√-2(2x+-3)-. 3 3 32x2 + 3x +1$

Your answers are correct

True. This is a straight forward application of the chain rule.
True. Since $h(x) = (x2 + 5)12$ we have
$1 1 x h′(x) = 2x-(x2 +5)- 2 = √-2---. 2 x + 5$
False. Since $h(x) = (x3 + 2x+ 1)-2$ we have
$2 h′(x) = (3x2 + 2)(- 2)(x3 + 2x+ 1)-3 =---2(3x-+-2) . (x3 +2x + 1)3$
False. $′ 2 4 x 2 2 4 x 2 h (x) = (8x+ e )3(2x + e) = 3(8x + e)(2x + e ) .$
True. Since $2 h(x) = (2x2 +3x + 1)3$ we have
$h′(x) = (4x+ 3)2(2x2 + 3x+ 1)- 13 =-√-2(2x+-3)-. 3 3 32x2 + 3x +1$

Question 3

At which of the following points is the derivative of $y = te-2t$ zero?

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 $y$ correctly.

Not correct. Choice (c) is false.

Try again, you do not have the correct value for $t.$

Your answer is correct.

$dy = e-2t - t(2)e-2t = e-2t(1 - 2t) dt$
Hence $dy -- = 0 dt$ when $1 t = - 2$ and $1 y = 12e-1 = --. 2e$

Question 4

On which of the following intervals is $-2t y = te$ concave up?

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 $dy = e-2t - 2te-2t dt$ so we have
$2 d-y2 = - 2e- 2t - 2(e-2t - 2te- 2t) = e-2t(- 2- 2 + 4t) = e- 2t(4t- 4). dt$
The second derivative is zero at $t = 1$ and it is positive when $t > 1$ and negative when $t < 1.$

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.