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MathQuiz 4.3
University of Sydney
School of Mathematics
and Statistics
© Copyright 2004-2006

The Derivative at a Point Quiz

Question

Web resources available

This quiz tests the work covered in Lecture 8 and corresponds to Section 2.2 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There is a web quiz at Wiley. It is the same quiz for each section in Chapter and you should probably wait until the end of lecture 9 before you attempt it.

Be aware that it doesn’t seem to accept the written answers so you will have to check whether your answers are correct when they print the correct answer. Questions 11 and 12 were illegible on 14/11/05.
The Mathematics Learning Centre has a booklet on differentiation Introduction to Differential Calculus which covers all of the topics for the next few lectures. In particular, Chapter 2 of the booklet covers this topic.

The site http://www.math.uncc.edu/~bjwichno/fall2004-math1242-006/Review˙Calc˙I/lec˙deriv.htm covers some of the material in Section 2.1-2.3

Question 1

The table of values for $f(x) = lnx$ that is log base $e$ rounded to 4 decimal places is written below.

$x$	1	1.5	2	2.5	3
$lnx$	0	0.4055	0.6931	0.9163	1.0986

Which of the following is the average rate of change of $f(x)$ between $x = 1$ and $x = 3?$

Not correct. Choice (a) is false.

Try again, you seem to have said that $ln 1 = 1 .$

Not correct. Choice (b) is false.

Try again, you seem to have used the value of $ln 1.5$ instead of $ln1 .$

Not correct. Choice (c) is false.

Try again, you have found the correct change in the value of $ln x$ but you need to divide it by the change in $x.$

Your answer is correct.

$1.0986---0 2 = 0.5493.$

Question 2

The table of values for $f(x) = lnx$ that is log base $e$ rounded to 4 decimal places is written below.

$x$	1	1.5	2	2.5	3
$lnx$	0	0.4055	0.6931	0.9163	1.0986

Which of the following is the average rate of change of $f(x)$ between $x = 1.5$ and $x = 2.5 ?$

Not correct. Choice (a) is false.

Try again, you seem to have the correct change in $lnx$ but the change in $x$ is 1 not 2.

Not correct. Choice (b) is false.

Try again, that is the average rate of change between $x = 1$ and $x = 3.$

Your answer is correct.

$0.9163--0.4055 1 = 0.5108 .$

Not correct. Choice (d) is false.

Try again, that is the instantaneous rate of change of $ln x$ at $x = 2 .$

Question 3

Consider the graph of $y = f(x)$ below

Which of the following statements are correct? There may be more than one correct answer.

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

Try again, the function is zero at B and G.

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

There is a turning point at D so the derivative is zero there.

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

The function has negative slope at both E and F and seems to be of about the same steepness. So the derivative is about the same at E and F.

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

The function has the same value at C and E but the derivative is positive at C and negative at E.

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

The function is negative at A but the derivative is positive.

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

The function crosses the $x$ -axis at B and G and is therefore zero at those points.

Your answers are correct

False. Try again, the function is zero at B and G.
True. There is a turning point at D so the derivative is zero there.
True. The function has negative slope at both E and F and seems to be of about the same steepness. So the derivative is about the same at E and F.
False. The function has the same value at C and E but the derivative is positive at C and negative at E.
False. The function is negative at A but the derivative is positive.
True. The function crosses the $x$ -axis at B and G and is therefore zero at those points.

Question 4

Consider the graph of $y = f(x)$ below

Which of the following statements are correct?

Not correct. Choice (a) is false.

Try again, $f(1) > f(4).$

Your answer is correct.

The gradient of the function is negative and quite steep at $x = 1.$
The gradient of the function is negative and much less steep at $x = 4$ so
$f′(1) < f′(4).$

Not correct. Choice (c) is false.

Try again, remember that the function has negative gradient at all points shown.

Not correct. Choice (d) is false.

Try again, we don’t need a scale to consider whether these statements are true of false.