CT4.1.A function, f(x), depends upon x as: f(x) = 10 x + 5Measurement of x shows that the central value and

uncertainty are such that:

2.0 0.3x x

The uncertainty in f is:A)±15B)±5C)±0.3D)±3E)None of these

CT4.1.A function, f(x), depends upon x as: f(x) = 10 x + 5Measurement of x shows that the central value and

uncertainty are such that:

2.0 0.3x x

The uncertainty in f is:A)±15B)±5C)±0.3D)±3E)None of these

There is a fixed uncertainty x in any measurement of x. At what value of x is f the largest?

x

f(x)

x1 x2 x3 x4 x5

A) x1 B) x2 C) x3 D) x4 E) x5

CT 4.2

There is a fixed uncertainty x in any measurement of x. At what value of x is f the largest?

x

f(x)

x1 x2 x3 x4 x5

A) x1 B) x2 C) x3 D) x4 E) x5

Answer: x3 , where the magnitude of the slope

is greatest.

f / x

CT4-3.

A BP(A,B,C)

C=

A, B, and C have uncertainties A, B, and C, respectively.What is the correct formula for P?A)

B)

C)

D)

E) None of these

222 )()()(C

C

B

B

A

A

222 )()()( CBA

222 )()()( CBA

222 )()()(C

C

B

B

A

A

C

AB

CT4-3.

A BP(A,B,C)

C=

A, B, and C have uncertainties A, B, and C, respectively.What is the correct formula for P?A)

B)

C)

D)

E) None of these

222 )()()(C

C

B

B

A

A

222 )()()( CBA

222 )()()( CBA

222 )()()(C

C

B

B

A

A

C

AB

CT4.1. A function, f(x), depends upon x as:f(x) = m x + bMeasurement of x shows that the central value and uncertainty are such that:

2.0 0.3x x The uncertainty in f is:

A)±bB)±0.3mC)±0.3b

CT5-7. You purchase a metallic meter stick and use it to measure the length of various objects. The manufacturer has accurately

manufactured the stick at a factory where the temperature is 20C. You use the stick to measure objects is a laboratory at 30C. The error that

you will make in your length measurements due to this temperature difference is:

A) a systematic error because the stick has thermally expanded

B) a statistical error because the stick has thermally expanded

C) both

CT5-8. You purchase a meter stick and use it to measure the length of various objects. The manufacturer has accurately manufactured the

stick at a factory where the temperature is 20C and has marked the stick with tick marks every millimeter. You use the stick to measure

objects in a laboratory at 20C. The error that you will make in your length measurements will have

a systematic error because the tick marks have finite width

a statistical error because the tick marks are only every millimeter.

both

CT7-1. Consider g(x,y) = x/y. What is the correct formula for g?(A)

x yd +d2 2

(B)

x y

x y

æ öæ öd d ÷ç÷ç + ÷÷ çç ÷÷÷ç ç ÷çè ø è ø

22

(C)

x yg

x y

æ öæ öd d ÷ç÷ç× + ÷÷ çç ÷÷÷ç ç ÷çè ø è ø

22

(D)

x y

x y

æ öæ öd d ÷ç÷ç - ÷÷ çç ÷÷÷ç ç ÷çè ø è ø

22

E) None of these

CT4-3.

A BP(A,B,C)

C=

A, B, and C have uncertainties A, B, and C, respectively.What is the correct formula for P?A)

B)

C)

D)

E) None of these

222 )()()( CC

PB

B

PA

A

P

222 )()()( CBA

222 )()()( CBA

222 )()()( CC

PB

B

PA

A

P

C

AB

CT5-10. You measure the time required for a ball to drop from a 5 meter height near the surface of some planet. You do the experiment eight times and get the following results:

Trial Time to drop (sec.)

1 1.05

2 1.11

3 0.95

4 0.99

5 1.02

6 0.59

7 0.96

8 1.10

A)Sure, looks GREAT to me.B)Hmmm. Something looks funny.C)Both A and BD)Neither A or B

x

frequency

A quantity x is measured N times (N >> 1), and in each measurement there is some random error. The individual measurements are xi

= x1, x2

, x3 … x

N

The average value is determined to be xCT6-2. What can you say about the value of

( )ix x-

?A) It is zero regardless of the number of trialsB) It is always positive.C) It can be either positive, negative or zero.CT6-3. What can you say about the average value of

( )ix x-

?

( )ii

x x?

N

-=

å

A) It is zero regardless of the number of trialsB) It is always positive.C) It can be either positive, negative or zero.

x

frequency

A quantity x is measured N times (N >> 1), and in each measurement there is some random error. The individual measurements are xi

= x1, x2

, x3 … x

N

The average value is determined to be xCT6-4. What can you say about the value of

( )ix x-2

?A) It is zero regardless of the number of trialsB) It is always positive.C) It can be either positive, negative or zero.CT6-5. What can you say about the average value of

( )ix x-2

?

( )ii

x x?

N

-=

å 2

A) It is zero regardless of the number of trialsB) It is always positive.C) It can be either positive, negative or zero.

x

frequency

A quantity x is measured N times (N >> 1), and in each measurement there is some random error. The individual measurements are x i

= x1,

x2, x3

… xN

The average value is determined to be xCT6-6. What can you say about our estimate of

2

1ix x

N

as the number of trials increases?A) It stays pretty much constant as N increases.B) It decreases as N increasesC) It increases as N increases

CT6-7. What can you say about mean

N

as the number of trials increases?A) It stays pretty much constant as N increases.B) It decreases as N increasesC) It increases as N increases

CT6-8.

A quantity x is measured 20 times and the best estimate of x is found to be x = 17 4

How many measurements would be needed in order to cut the uncertainty in half; that is, reduce the uncertainty to x = 2?

A) 40 B) 60 C) 80 D) 120 E) No idea – I’m lost

CT8-4. Consider a plot of the normal (or Gaussian distribution function)

( )x x /( )x,f (x) e- - ss =

s p

2 221

2What is the total area under the curve,

f (x)dx+¥

- ¥ò?

x

f(x)

x x+x-

(A) 1 (B)

s p

1

2(C) Answer depends on

x and .(D)

s p2

12

x

f(x)

x x + t

tx t

x

f (x)dx %+ s

´ò 100

1.0 34.1%1.5 43.3%2.0 47.7%2.5 49.4%3.0 49.9%4.0 49.997%

CT8-5. In a distant foreign country, the distribution of heights of adult males is found to be gaussian with a mean of 5’10” and a

standard deviation of 3”. What fraction of the males is taller than 6’4”?

(A) less than 1% (B) about 5%

(C) about 16% (D) about 2.5%

(E) none of these

s

CT4.2. A function, f(x), depends upon x in degrees as:f(x) = sin(πx/180)Measurement of x shows that the central value and uncertainty are such that:

200 100x x The uncertainty in f is:

A)±1B)±100C)±πD)None of these