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The Derivative Function Quiz

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This quiz tests the work covered in Lecture 9 and corresponds to Section 2.3 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 you attempt it now.

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, Chapters 2 and 3.1 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

There is an applet that lets you sketch the derivative of a given function at http://www.ltcconline.net/greenl/java/Other/DerivativeGraph/classes/DerivativeGraph.html After you have mastered the topic you might like to try the tests at http://www.univie.ac.at/future.media/moe/tests/diff1/defabl.html and http://www.univie.ac.at/future.media/moe/tests/diff1/poldiff.html and the puzzle at http://www.univie.ac.at/future.media/moe/tests/diff1/ablerkennen.html

Question 1

Which of the following is the derivative of   f(x) = 3x + 2?

a)
f′(x) = 2.
  b)
f ′(x) = 3.
c)
f′(x) = 5.
  d)
There is not enough information to answer the question.

 

Not correct. Choice (a) is false.
Try again, if f(x) = mx + b then the derivative is m.
Your answer is correct.
Since the derivative of f(x) = mx + b is m, f′(x ) = 3.
Not correct. Choice (c) is false.
Try again, if f(x) = mx + b then the derivative is m.
Not correct. Choice (d) is false.
Try again, if f(x) = mx + b then the derivative is m.

Question 2

Which of the following graphs of y = f(x) satisfy the following three conditions:

  • f′(x ) < 0 for x < - 1
  • f′(x ) > 0 for - 1 < x < 2
  • f′(x ) = 0 for x > 2.
a)
PIC
   b)
PIC
c)
PIC
   d)
PIC

 

Your answer is correct.
Not correct. Choice (b) is false.
Try again, this graph is of a function that is negative, positive and zero in the required region, not of a function whose derivative satisfies these conditions.
Not correct. Choice (c) is false.
Try again, your graph has the wrong sign for its derivative.
Not correct. Choice (d) is false.
Try again, you may need to review what having positive or negative derivative means.

Question 3

Consider the graph below.
PIC
Which of the following is the matching derivative function?

a)
PIC
   b)
PIC
c)
PIC
   d)
PIC

 

Your answer is correct.
The graph has turning points at x = 1 and x = 3 so the graph of the derivative must cut the axis at these points. The graph moves from positive gradient to negative gradient and back to positive so the graph of the derivative is positive then negative and then positive again.
Not correct. Choice (b) is false.
Try again, this graph has the correct zeros but is not positive where the graph has positive gradient etc.
Not correct. Choice (c) is false.
Try again, you do not have the zeros in the correct spots.
Not correct. Choice (d) is false.
Try again, this graph does not have the correct zeros and is not positive where the graph has positive gradient etc.

Question 4

Which of the statements below correctly match the function with its derivative?

PIC PIC PIC PIC PIC PIC
a)
C is the graph of the derivative of B
F is the graph of the derivative of C
D is the graph of the derivative of A
E is the graph of the derivative of D
   b)
A is the graph of the derivative of C
C is the graph of the derivative of F
B is the graph of the derivative of D
D is the graph of the derivative of E
c)
C is the graph of the derivative of A
E is the graph of the derivative of C
D is the graph of the derivative of B
F is the graph of the derivative of D
   d)
C is the graph of the derivative of A
F is the graph of the derivative of C
D is the graph of the derivative of B
E is the graph of the derivative of D

 

Not correct. Choice (a) is false.
Try again, looking carefully at the gradients of graphs A and B before the first turning point.
Not correct. Choice (b) is false.
Try again, graph A represents a function with two turning points so its derivative must have 2 zeros. Look at the graphs again.
Not correct. Choice (c) is false.
Try again, graph C has negative gradient and then positive gradient so its derivative cannot be graph E.
Your answer is correct.
Graph A has positive gradient to almost 1 and negative gradient to a bit more than 3 and then positive gradient. This matches graph C.
Graph C has negative gradient to a bit more than 2 and then positive gradient. This matches graph F.
Similarly for the other 3 graphs.
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