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The Second Derivative Quiz

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This quiz tests the work covered in Lecture 11 and corresponds to Section 2.5 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There is an applet at http://www.walter-fendt.de/m11e/deriv12.htm which allows you to put in the formula for a function and then it draws the first and second derivative functions.

There is a good summary of the second derivative at http://archives.math.utk.edu/visual.calculus/3/graphing.14/ although some of the language is technical. There are also difficult, but useful quizzes at http://archives.math.utk.edu/visual.calculus/3/graphing.2/index.html and http://archives.math.utk.edu/visual.calculus/3/graphing.3/index.html.


 

Question 1

 
 
Consider the graph of y = f(x) below.
PIC
Which of the following statements are correct?
a) dy d2y --> 0 and --2 < 0 at A. dx dx   b) dy d2y ---< 0 and--2 < 0 at A . dx dx
c) -dy d2y dx > 0 and dx2 > 0 at A .   d) dy- d2y dx < 0 and dx2 > 0 at A .

 

Not correct. Choice (a) is false.
Try again, the function is decreasing at A .
Your answer is correct.
The function is decreasing and concave down at A , so
-dy d2y dx < 0 and dx2 < 0 at A .
Not correct. Choice (c) is false.
Try again, the function is decreasing at A.
Not correct. Choice (d) is false.
Try again, the function is concave down at A.
 

Question 2

 
 
Let P(t) be the number of alien sightings at time t. We are told that both P′(t) and P′′(t) are positive.
Which of the statements below best reflect this?
a) There are more and more alien sightings every year.
b) There are more alien sightings this year but less than we expected given the the increases over the past few years.
c) There are less alien sightings this year but more than we expected given the the decreases over the past few years.
d) There are fewer and fewer alien sightings every year.

 

Your answer is correct.
The number of sightings are increasing so P′(t) > 0 and the rate of increase is increasing so P′′(t) > 0.
Not correct. Choice (b) is false.
Try again, since P′′(t) > 0 the rate of sightings must be increasing.
Not correct. Choice (c) is false.
Try again, since P′(t) > 0 the number of sightings must be increasing.
Not correct. Choice (d) is false.
Try again, since P′(t) > 0 the number of sightings must be increasing.
 

Question 3

 
 
Consider the graph of y = f(x) below.
At which point on the graph is f′(x) > 0 and f′′(x) < 0. PIC
a) A
b) B
c) C
d) D
e) None of the above.

 

Not correct. Choice (a) is false.
Try again, ′ f(x) < 0 and ′′ f (x) > 0 at A.
Not correct. Choice (b) is false.
Try again, ′ f(x) > 0 and ′′ f (x) > 0 at B .
Your answer is correct.
f is increasing and concave down at C so ′ f (x) > 0 and ′′ f (x) < 0 at C .
Not correct. Choice (d) is false.
Try again, ′ f (x) < 0 and ′′ f (x) < 0 at D .
Not correct. Choice (e) is false.
There is a correct answer. You need to find a point where f is increasing and concave down.
 

Question 4

 
 
A function f is decreasing for x ≥ 2 and ′ f(2) = 20, f (2) = - 2 and ′′ f (x) > 0 for x ≥ 2.

Which of the following is a possible value for f(4)?
a) f(4) = 24.   b) f (4) = 21.
c) f(4) = 18.   d) f(4) = 16.

 

Not correct. Choice (a) is false.
Try again, since f is decreasing f(4) < 20.
Not correct. Choice (b) is false.
Try again, since f is decreasing f(4) < 20 .
Your answer is correct.
Since f is decreasing f(4) < 20 and since ′′ f (x) > 0 the function cannot continue to decrease at the same rate as it was decreasing at   x = 2   therefore f (4) > 16 .
Not correct. Choice (d) is false.
Try again, since ′′ f (x) > 0 the function cannot continue to decrease at the same rate as it was decreasing at   x = 2   therefore f (4) > 16 .