MATHEMATICS 1M03 Final Examination - Practice Version
Name: _______________________________
Student Number:____________________
MATHEMATICS 1M03 Winter Session, 2011
Final Examination - Practice Version Please Read the Instructions: They will be the same as on the actual examination!
FINAL EXAMINATION DAY CLASS DURATION OF EXAMINATION: 3 hours MAXIMUM GRADE: 27 MCMASTER UNIVERSITY EXAMINATION Thursday, April 8, 2010 THIS TEST INCLUDES 12 PAGES AND 25 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSURING YOUR COPY OF THE TEST IS COMPLETE. BRING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR. Instructions: 1. Only the Casio FX-991 calculator is allowed to be used on this test. 2. Make sure your name and student number is at the top of each page. 3. Each question is worth one mark. 4. A blank answer is an automatic zero for any question, even if the correct solution is circled on the question itself. 5. Unless specifically stated otherwise in the question, incorrect or multiple answers are also worth zero marks. No negative marks or part marks will be assigned. 6. In the event of a discrepancy between instructions provided in a question and these instructions, the instructions in the question supersede those written here. 7. Rough work materials will be provided. All rough work must be handed in with the test, but any solutions written on the rough paper will NOT be graded. 8. Good Luck! _________________________________________________ PLEASE READ THE OMR INSTRUCTIONS ON PAGE #2
Page #2 of 12 Name: _______________________________
Student Number:____________________
OMR EXAMINATION INSTRUCTIONS NOTE: IT IS YOUR RESPONSIBILITY TO ENSURE THAT THE ANSWER SHEET IS PROPERLY COMPLETED: YOUR EXAMINATION RESULT DEPENDS ON PROPER ATTENTION TO THESE INSTRUCTIONS. The scanner which reads the sheets senses shaded areas by their non-reflection of light. A heavy mark must be made, completely filling the circular bubble, with an HB pencil. Marks made with a pen or a felt–tip marker will not be sensed. Erasures must be thorough or the scanner may still sense a mark. Do not use correction fluid on the sheets. Do not put any unnecessary marks or writing on the sheet. 1. Print your name, student number, course name, section number, instructor name, and the date in the spaces provided at the top of Side 1 (red side) of the sheet. Then you must sign in the space marked SIGNATURE. 2. Write your student number in the space provided and fill in the corresponding bubble numbers underneath. 3. Mark only one choice from the alternatives (1, 2, 3, 4, 5, or A, B, C, D, E) provided for each question. The question number is to the left of the bubbles. Make sure that the number of the question on the scan sheet is the same as the question number on the test paper. 4. Pay particular attention to the Marking Directions on the form. 5. Begin answering the questions using the first set of bubbles, marked “1”.
134
Continued on page #2
Page #3 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #1. Evaluate:
e
∫
x ln x dx
1
a) (2e2 + 1) / 4
b)
1 2
x 2 ln e −
1 4
c) (e2 + 1) 4
e) (e2 + 1)
d) 1/4
_____________________________________ #2. Evaluate:
∞
∫
2e x
x 2 0 (1 + e )
b) − 2/e
a) 1
dx
c) 0
d) 1/e
e) 2
_____________________________________ #3. A scientist discovers a new radioactive element, "Zorkium" with a half-life of 4 days. If he has 10g of pure Zorkium when he announces his discovery to the world, how much will he have left in 8 days when he receives his Nobel Prize?
a) 1.25g
b ) 2.5g
c) 3.33g
d ) 6.66g
e) 8g
_____________________________________
Continued on page #4
Page #4 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #4. At what point(s) does the function:
f ( x, y ) = x 4/3 y 2/3 attain it's maximum value when restricted to the curve:
a) (1,‒1),(‒1,1)
x 2 + y 2 = 12
b) ( 2√2, ‒2 ),( ‒2√2, 2 ) d) ( 2√2, 2 ) only
c) ( ‒2√2, ‒2 ),( 2√2, 2 )
e) ( 1, 1 ) only
_____________________________________ #5. Given the function:
f ( x, y ) = xe xy Evaluate fxy(1,0). a) 0
b) e
c) 1
d) 2
e)
3
_____________________________________ #6. Find the average value on [-1,2] of the function:
p(x) = 4 ‒ 3x³ a)
l
b)
1/2
c) 1/4
d) − 11.75
e)
4 1 x − x4 3 4
_____________________________________
Continued on page #5
Page #5 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #7. Find the area enclosed between the graphs:
y = e2 x , a) 9/8
y = e− 3x ,
x = ln2
1 2 1 5 e + − e) 29 / 24 3 2 6 3e _____________________________________ b) 12
c) 3+2ln2
d)
#8. Which of the given points is in the domain of both functions.
Q ( x, y ) =
1 25 − 4 x 2 − 3 y 2
a) ( 0, 0 ) b) ( 0 ,−2 )
P( x, y ) = ln( x 2 + y 2 − 4)
c) ( 1, 5/2 ) d) (−2, 2 ) e) ( 1, 1 )
_____________________________________
#9.
Solve the equation for x:
ln(4 − 3 x) = 7 a) x = 1
b) x = 3ln(4 / 7)
4 − e7 c) x = 3
d) x = log 7 (4 / 3)
4 − 7e e) x = 3
_____________________________________
Continued on page #6
Page #6 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #10. Find the (approximate) mean, μ, of the random variable x with the probability density function given below defined on the given interval.
1 f ( x) = 4 x 0 a) 7
b) 12
4 ≤ x ≤ 16 otherwise
c) 9.3
d) 4
e) 9
_____________________________________ #11. Given the function:
f ( x, y ) = x 2 + 4 xy 2 + y 2 and the points:
i ) ( 0, 0 )
1 1 ii ) − , 4 8
1 1 iii ) − , − 8 4
1 iv ) − ,0 4
v ) ( 1, 1 )
which are critical points?
a) i ) and ii ) b) ii) and iii )
c) i) and iv)
d) iv) only e) i ), ii ) and iii)
_____________________________________ #12. The population of Mogwai in a wilderness preserve follows the equation:
P(t ) =
B 20 + 10e− 0.1t
Here, P is the population in thousands of Mogwai, and t is time in months. If, after 10 months the population is found to be at 20 thousand Mogwai, what is (approximately) the carrying capacity of the preserve? a) 47.4 thousand
b) 30.0 thousand
d) 50.3 thousand
c) 23.7 thousand
e) 41.5 thousand
_____________________________________ Continued on page #7
Page #7 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #13. Find the derivative of:
g ( x) = 2 x ln x at x = e.
a) 2e + 1(ln2)
b) 2e
c) 2e
d) e 2 x
e) 2
_____________________________________ #14. Evaluate: Gu ( 1, 3 )
G (u , v) =
a) 0
b) 22.45
u+ v v + eu
c) 1.052
d) − 0.743
e) − 0.123
_____________________________________ #15. Given the probability density function:
evaluate P(1 < X < 2).
a) 0.09
x 2e− 2 x , x ≥ 0 f ( x) = otherwise 0 , b) 0.11
c) 1.23
d) 0.53
e) 0.09
_____________________________________
Continued on page #8
Page #8 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #16. Find the constant, k, such that:
k , x≥ 2 2 f ( x) = x ( ln x ) otherwise 0, is a probability density function.
a) 1 + e3
b) e
c) 2ln2
d) ln 2
e) ln 5
_____________________________________ #17. Find the (approximate) value of σ (the standard deviation) for the probability distribution:
5 , f ( x) = 4 x 2 0, a) 5.13
b) 3.00
1≤ x ≤ 5 otherwise
c) 2.01
d) 0.95
e) 5.00
_____________________________________ #18. Use the fundamental theorem of calculus to find the approximate value of the value of the definite integral:
9
∫
2
a) 96.2
b) 62.5
x 2 ( x − 1) c) 137.3
−1 3
dx
d) 203.6
e) Undefined
_____________________________________ Continued on page #9
Page #9 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #19. Given that:
z = x 2 y,
with
y = f (t ), x = g (t )
and that:
f (1) = 2, f '(1) = 3, g (1) = 1, compute
g '(1) = 4
dz at t = 1. dt a) 19
b) 48
c) 5
d) 12
e) − 7
_____________________________________ #20. Solve for the value of a :
ln(a) + 2ln(a1/2 ) + 5 − ln(a − 1) − ln(2a ) = 0 a) 0
b)
2
c) 1 + e5
2 − e5
d)
2 − e5
e) No Solution
_____________________________________ #21. It is found that a chemical reaction proceeds at a rate given by the equation:
C ' ( t ) = 90 + 9t 2 + (6 − t )5 Here, the rate is measured in grams per second, and time is measured in seconds. If at the start of the reaction (t=0) there are 100 units of the chemical, approximately how much do we have when t = 2?
a) 7,397g
b) 479g
c) 215g
d) 8,912g
e) 158g
_____________________________________ Continued on page #10
Page #10 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #22. Solve the differential equation:
y' = ln x
a)
x3
+Cx
b)
2 ln x
x3
−
2y x
ln x − 1 + Cx
c)
x3
2
C ln x d) + 2 x x
( ln x ) 2 + C
ln x e) x
x2
2
_____________________________________ #23. Find the solution of:
y ' + 2 y2 x = 4 y2 passing through the point (0,1).
a) y =
1
b)
2
x − 4x + C d)
y=
y=
1 2
x − 4x + 1
1 2
x − 4x
+1
1
c) y =
(
e) y = ln x 2 − 4 x + e
2
x − 4x
+C
)
_____________________________________
Continued on page #11
Page #11 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded.
#24. An open-topped triangular prism (see right) is constructed from 300cm2 of cardboard. Find the approximate maximum volume of this open prism. Hint: If the side panels are both c long and a, and b wide respectively, then the area of each end is given by: End Area = ab/2 And the volume is (length) x (end area). a) 100cm3
b) 500cm3
c) 360cm3
d) 60cm3
e) 96cm3
_____________________________________
#25. Consider two Riemann sums for the function:
y = 3x
1/2
over the interval 0 ≤ x ≤ 4. The first approximation uses has 12 sub-intervals, and left endpoints. The second has 10 sub-intervals and uses right endpoints. Which of the following statements are true: a) b) c) d) e)
Both approximations are less than the area under the curve. Both approximations are greater than the area under the curve. The first is less, and the second is more than the area under the curve. The first is greater and the second is less than the area under the curve. The relationship between the values cannot be determined. _____________________________________
Continued on page #12
Page #12 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #26. A roller coaster travels on it's track following the equation:
H (t ) =
10 2 + e − 0.2t
where t is measured in seconds, and H(t) is the height above the ground. At t = 4 the track breaks and the coaster, with the passengers still inside, initially flies off, following a tangent line. What is (approximately) the equation of this line?
a) y = 4.083 x − 16.125 b) y = 0.206 x + 3.259 c) y = 0.150 x + 3.484 d) y = 0.206 x + 4.083 e) y = 0.150 x + 4.083 _____________________________________ #27. A continuous random variable has a mean of 2. If it's exponentially distributed, what is the equation of it's probability density?
e2 x , x ≥ 0 a) f ( x) = 0 , otherwise
e− 0.5 x , x ≥ 0 b) f ( x ) = otherwise 0 ,
0.5e− 0.5 x , x ≥ 0 c) f ( x) = 0 , otherwise
2e− 2 x , x ≥ 0 d) f ( x) = 0 , otherwise
0.69e0.69 x , x ≥ 0 e) f ( x) = 0 , otherwise _____________________________________
THE END
Student Number:____________________
MATHEMATICS 1M03 Winter Session, 2011
Final Examination - Practice Version Please Read the Instructions: They will be the same as on the actual examination!
FINAL EXAMINATION DAY CLASS DURATION OF EXAMINATION: 3 hours MAXIMUM GRADE: 27 MCMASTER UNIVERSITY EXAMINATION Thursday, April 8, 2010 THIS TEST INCLUDES 12 PAGES AND 25 QUESTIONS. YOU ARE RESPONSIBLE FOR ENSURING YOUR COPY OF THE TEST IS COMPLETE. BRING ANY DISCREPANCY TO THE ATTENTION OF YOUR INVIGILATOR. Instructions: 1. Only the Casio FX-991 calculator is allowed to be used on this test. 2. Make sure your name and student number is at the top of each page. 3. Each question is worth one mark. 4. A blank answer is an automatic zero for any question, even if the correct solution is circled on the question itself. 5. Unless specifically stated otherwise in the question, incorrect or multiple answers are also worth zero marks. No negative marks or part marks will be assigned. 6. In the event of a discrepancy between instructions provided in a question and these instructions, the instructions in the question supersede those written here. 7. Rough work materials will be provided. All rough work must be handed in with the test, but any solutions written on the rough paper will NOT be graded. 8. Good Luck! _________________________________________________ PLEASE READ THE OMR INSTRUCTIONS ON PAGE #2
Page #2 of 12 Name: _______________________________
Student Number:____________________
OMR EXAMINATION INSTRUCTIONS NOTE: IT IS YOUR RESPONSIBILITY TO ENSURE THAT THE ANSWER SHEET IS PROPERLY COMPLETED: YOUR EXAMINATION RESULT DEPENDS ON PROPER ATTENTION TO THESE INSTRUCTIONS. The scanner which reads the sheets senses shaded areas by their non-reflection of light. A heavy mark must be made, completely filling the circular bubble, with an HB pencil. Marks made with a pen or a felt–tip marker will not be sensed. Erasures must be thorough or the scanner may still sense a mark. Do not use correction fluid on the sheets. Do not put any unnecessary marks or writing on the sheet. 1. Print your name, student number, course name, section number, instructor name, and the date in the spaces provided at the top of Side 1 (red side) of the sheet. Then you must sign in the space marked SIGNATURE. 2. Write your student number in the space provided and fill in the corresponding bubble numbers underneath. 3. Mark only one choice from the alternatives (1, 2, 3, 4, 5, or A, B, C, D, E) provided for each question. The question number is to the left of the bubbles. Make sure that the number of the question on the scan sheet is the same as the question number on the test paper. 4. Pay particular attention to the Marking Directions on the form. 5. Begin answering the questions using the first set of bubbles, marked “1”.
134
Continued on page #2
Page #3 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #1. Evaluate:
e
∫
x ln x dx
1
a) (2e2 + 1) / 4
b)
1 2
x 2 ln e −
1 4
c) (e2 + 1) 4
e) (e2 + 1)
d) 1/4
_____________________________________ #2. Evaluate:
∞
∫
2e x
x 2 0 (1 + e )
b) − 2/e
a) 1
dx
c) 0
d) 1/e
e) 2
_____________________________________ #3. A scientist discovers a new radioactive element, "Zorkium" with a half-life of 4 days. If he has 10g of pure Zorkium when he announces his discovery to the world, how much will he have left in 8 days when he receives his Nobel Prize?
a) 1.25g
b ) 2.5g
c) 3.33g
d ) 6.66g
e) 8g
_____________________________________
Continued on page #4
Page #4 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #4. At what point(s) does the function:
f ( x, y ) = x 4/3 y 2/3 attain it's maximum value when restricted to the curve:
a) (1,‒1),(‒1,1)
x 2 + y 2 = 12
b) ( 2√2, ‒2 ),( ‒2√2, 2 ) d) ( 2√2, 2 ) only
c) ( ‒2√2, ‒2 ),( 2√2, 2 )
e) ( 1, 1 ) only
_____________________________________ #5. Given the function:
f ( x, y ) = xe xy Evaluate fxy(1,0). a) 0
b) e
c) 1
d) 2
e)
3
_____________________________________ #6. Find the average value on [-1,2] of the function:
p(x) = 4 ‒ 3x³ a)
l
b)
1/2
c) 1/4
d) − 11.75
e)
4 1 x − x4 3 4
_____________________________________
Continued on page #5
Page #5 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #7. Find the area enclosed between the graphs:
y = e2 x , a) 9/8
y = e− 3x ,
x = ln2
1 2 1 5 e + − e) 29 / 24 3 2 6 3e _____________________________________ b) 12
c) 3+2ln2
d)
#8. Which of the given points is in the domain of both functions.
Q ( x, y ) =
1 25 − 4 x 2 − 3 y 2
a) ( 0, 0 ) b) ( 0 ,−2 )
P( x, y ) = ln( x 2 + y 2 − 4)
c) ( 1, 5/2 ) d) (−2, 2 ) e) ( 1, 1 )
_____________________________________
#9.
Solve the equation for x:
ln(4 − 3 x) = 7 a) x = 1
b) x = 3ln(4 / 7)
4 − e7 c) x = 3
d) x = log 7 (4 / 3)
4 − 7e e) x = 3
_____________________________________
Continued on page #6
Page #6 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #10. Find the (approximate) mean, μ, of the random variable x with the probability density function given below defined on the given interval.
1 f ( x) = 4 x 0 a) 7
b) 12
4 ≤ x ≤ 16 otherwise
c) 9.3
d) 4
e) 9
_____________________________________ #11. Given the function:
f ( x, y ) = x 2 + 4 xy 2 + y 2 and the points:
i ) ( 0, 0 )
1 1 ii ) − , 4 8
1 1 iii ) − , − 8 4
1 iv ) − ,0 4
v ) ( 1, 1 )
which are critical points?
a) i ) and ii ) b) ii) and iii )
c) i) and iv)
d) iv) only e) i ), ii ) and iii)
_____________________________________ #12. The population of Mogwai in a wilderness preserve follows the equation:
P(t ) =
B 20 + 10e− 0.1t
Here, P is the population in thousands of Mogwai, and t is time in months. If, after 10 months the population is found to be at 20 thousand Mogwai, what is (approximately) the carrying capacity of the preserve? a) 47.4 thousand
b) 30.0 thousand
d) 50.3 thousand
c) 23.7 thousand
e) 41.5 thousand
_____________________________________ Continued on page #7
Page #7 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #13. Find the derivative of:
g ( x) = 2 x ln x at x = e.
a) 2e + 1(ln2)
b) 2e
c) 2e
d) e 2 x
e) 2
_____________________________________ #14. Evaluate: Gu ( 1, 3 )
G (u , v) =
a) 0
b) 22.45
u+ v v + eu
c) 1.052
d) − 0.743
e) − 0.123
_____________________________________ #15. Given the probability density function:
evaluate P(1 < X < 2).
a) 0.09
x 2e− 2 x , x ≥ 0 f ( x) = otherwise 0 , b) 0.11
c) 1.23
d) 0.53
e) 0.09
_____________________________________
Continued on page #8
Page #8 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #16. Find the constant, k, such that:
k , x≥ 2 2 f ( x) = x ( ln x ) otherwise 0, is a probability density function.
a) 1 + e3
b) e
c) 2ln2
d) ln 2
e) ln 5
_____________________________________ #17. Find the (approximate) value of σ (the standard deviation) for the probability distribution:
5 , f ( x) = 4 x 2 0, a) 5.13
b) 3.00
1≤ x ≤ 5 otherwise
c) 2.01
d) 0.95
e) 5.00
_____________________________________ #18. Use the fundamental theorem of calculus to find the approximate value of the value of the definite integral:
9
∫
2
a) 96.2
b) 62.5
x 2 ( x − 1) c) 137.3
−1 3
dx
d) 203.6
e) Undefined
_____________________________________ Continued on page #9
Page #9 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #19. Given that:
z = x 2 y,
with
y = f (t ), x = g (t )
and that:
f (1) = 2, f '(1) = 3, g (1) = 1, compute
g '(1) = 4
dz at t = 1. dt a) 19
b) 48
c) 5
d) 12
e) − 7
_____________________________________ #20. Solve for the value of a :
ln(a) + 2ln(a1/2 ) + 5 − ln(a − 1) − ln(2a ) = 0 a) 0
b)
2
c) 1 + e5
2 − e5
d)
2 − e5
e) No Solution
_____________________________________ #21. It is found that a chemical reaction proceeds at a rate given by the equation:
C ' ( t ) = 90 + 9t 2 + (6 − t )5 Here, the rate is measured in grams per second, and time is measured in seconds. If at the start of the reaction (t=0) there are 100 units of the chemical, approximately how much do we have when t = 2?
a) 7,397g
b) 479g
c) 215g
d) 8,912g
e) 158g
_____________________________________ Continued on page #10
Page #10 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #22. Solve the differential equation:
y' = ln x
a)
x3
+Cx
b)
2 ln x
x3
−
2y x
ln x − 1 + Cx
c)
x3
2
C ln x d) + 2 x x
( ln x ) 2 + C
ln x e) x
x2
2
_____________________________________ #23. Find the solution of:
y ' + 2 y2 x = 4 y2 passing through the point (0,1).
a) y =
1
b)
2
x − 4x + C d)
y=
y=
1 2
x − 4x + 1
1 2
x − 4x
+1
1
c) y =
(
e) y = ln x 2 − 4 x + e
2
x − 4x
+C
)
_____________________________________
Continued on page #11
Page #11 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded.
#24. An open-topped triangular prism (see right) is constructed from 300cm2 of cardboard. Find the approximate maximum volume of this open prism. Hint: If the side panels are both c long and a, and b wide respectively, then the area of each end is given by: End Area = ab/2 And the volume is (length) x (end area). a) 100cm3
b) 500cm3
c) 360cm3
d) 60cm3
e) 96cm3
_____________________________________
#25. Consider two Riemann sums for the function:
y = 3x
1/2
over the interval 0 ≤ x ≤ 4. The first approximation uses has 12 sub-intervals, and left endpoints. The second has 10 sub-intervals and uses right endpoints. Which of the following statements are true: a) b) c) d) e)
Both approximations are less than the area under the curve. Both approximations are greater than the area under the curve. The first is less, and the second is more than the area under the curve. The first is greater and the second is less than the area under the curve. The relationship between the values cannot be determined. _____________________________________
Continued on page #12
Page #12 of 12 Name: _______________________________
Student Number:____________________
Remember: All answers must be entered on the OMR card - This paper will NOT be graded. #26. A roller coaster travels on it's track following the equation:
H (t ) =
10 2 + e − 0.2t
where t is measured in seconds, and H(t) is the height above the ground. At t = 4 the track breaks and the coaster, with the passengers still inside, initially flies off, following a tangent line. What is (approximately) the equation of this line?
a) y = 4.083 x − 16.125 b) y = 0.206 x + 3.259 c) y = 0.150 x + 3.484 d) y = 0.206 x + 4.083 e) y = 0.150 x + 4.083 _____________________________________ #27. A continuous random variable has a mean of 2. If it's exponentially distributed, what is the equation of it's probability density?
e2 x , x ≥ 0 a) f ( x) = 0 , otherwise
e− 0.5 x , x ≥ 0 b) f ( x ) = otherwise 0 ,
0.5e− 0.5 x , x ≥ 0 c) f ( x) = 0 , otherwise
2e− 2 x , x ≥ 0 d) f ( x) = 0 , otherwise
0.69e0.69 x , x ≥ 0 e) f ( x) = 0 , otherwise _____________________________________
THE END
