• Langara College

  MATH 1271

MATH 1271

SAMPLE EXAM 2 SOLUTIONS

NAME: ________________________ STUDENT NUMBER: __________________

1. There are 9 questions on 4 pages – CHECK YOURS NOW.

2. The time allotted to write the examination is 60 minutes.

3. Simple scientific or financial calculators are permitted only.

4. Answer each question in the space provided – use the back of pages as needed.

5. Give exact answers where possible.

6. For full marks all relevant work should be included.

7. Good luck!

MATH 1271 Sample Exam 2 Solutions Page 1 of 4

1. Find the area of the region in the fourth quadrant that

is bounded by the x-axis and the parametric curve

2 3

x t t y t t = + = - , . [5]

Solution:

y t = ⇒ = - 0 1,0,1

( ) 2 1 ( ) 0 0 x x A y dx = = ⇒ = - ∫ ( )( ) 1 3 0 31 2 1 60 = - - + = ∫ t t t dt

2. Let ℜ be the region bounded by the curves y x = -1

and y x = - 2 1.

a) Write a definite integral for the area of ℜ . [4]

b) Find the volume of the solid obtained by rotating ℜ

about the line x = 2 . [4]

c) Find the volume of the solid with base ℜ and

cross-sections perpendicular to the x-axis that are

rectangles with height twice as long as the base. [4]

Solution:

a) x x x - = - ⇒ = 1 1 0,1 2

( ) ( ) 1 2 0 1 1 1 6 ⇒ = - - - = A x x dx   ∫   (Alternative: 0 1 A y y dy 1 ( 1) - = + - +   ∫   )

b) Slice @ y is a washer: r y y r y out = - + = - = - + 2 1 1 , 2 1 ( ) in

( ) ( ) 0 2 2 1 1 2 1 2 V y π π y dy π -   ⇒ = - - - + = = ∫     L

Shell @ x : r x h x x = - = - - - 2 , 1 1 ( ) ( 2 )

( )( ) 1 2 0 ⇒ = - - V x x x dx ∫ 2 2 π

c) Cross-section @ x : base 1 1 , height 2 base , area 2 base = - - - = = (x x ) ( 2 ) ( ) ( )2

( ) 1 2 2 0 1 2 15 ⇒ = - = = V x x dx ∫ L

MATH 1271 Sample Exam 2 Solutions Page 2 of 4

3. The density of cars heading north on a highway at distance x km from a border

crossing (at a certain time) is approximated by f x x ( ) = + 200 / 1 ( 2 ) cars per km.

Approximate the average number of cars per km within 3 km of the crossing on the

south side. [5]

Solution:

[ ] 3 3 2 0 0 1 200 200 arctan 83 3 1 3 f dx x x = = = ∫ +

4. A random variable has probability density function f x x x ( ) = ≤ ≤ cos , 0 / 2 π .

a) Find the mean µ of this random variable. [4]

b) Find the probability that the random variable will be less that its mean. [3]

Solution:

a) [ ] /2 /2 /2 0 0 0 cos sin sin 1 2 x x d x x x x dx π π π π µ = = - = - ∫ ∫

b) ( ) /2 1 [ ] /2 1 ( ) P X 0 cos sin sin / 2 1 54% x dx x 0 π π µ π - - < = = = - ≈ ∫

MATH 1271 Sample Exam 2 Solutions Page 3 of 4

5. The weather on a February day in Vancouver is considered unseasonable if

Vancouver gets less than 2 mm or more than 26 mm in one day. Suppose that the

amount of rainfall (in mm) that Vancouver gets on a February day is normally

distributed with mean µ =14 mm and standard deviation σ = 4 mm.

a) What is the random variable, X , in this problem. What is its probability density

function? [2]

b) Find the probability that on one randomly chosen day in February, the weather in

Vancouver is unseasonable. [4]

Solution:

a) X = amount of rain in Vancouver on a randomly chosen day in February

Its p.d.f. is ( 14)2 1 32 ( ) 4 2 x f x e π - - =

b) ( ) ( ) 26 ( 14)2 32 2 1 2 or 26 1 2 26 1 4 2 x P X X P X e dx π - - < > = - < < = - ∫

Make the substitution 14 4 x z - = .

( ) ( ) 3 3 2 2 2 2 3 0 1 1 2 or 26 1 1 2 1 2 .4986 .28% 2 2 z z P X X e dz e dz π π - - - < > = - ∫ ∫ = - ≈ - =

6. A certain company determines that the rate of change of monthly profit P, as a

function of monthly advertising expenditure x, is proportional to the difference

between a maximum amount of $11,000 and P. Furthermore, if no money is spent on

advertising, the profit is $1,000; if $100 is spent on advertising, the profit is $6,000.

Write and solve a suitable initial-value problem to determine the profit if $200 were

spent on advertising. [6]

Solution: