course Mth 174
Can you go over the following test with me So that I can see how to work them:Time and Date Stamps (logged): 18:52:13 07-28-2007 ?????????
With 100+ students I can't take time to work out tests on request, unless a student makes a good attempt to work the test out and includes the details of the attempted solutions.
However you are working hard this Summer and I'll take a few minutes to insert some hints. Then work the problems out as best you can, and I can respond.
Calculus II Test 2
--------------------------------------------------------------------------------
Completely document your work and your reasoning.
You will be graded on your documentation, your reasoning, and the correctness of your conclusions.
Except where the need for more precision dictates otherwise (e.g., in nuclear physics) all quantities may be rounded to three significant figures. The generating program works in binary and often generates extraneous digits (e.g., 1.5001 for 1.5, 3.6999 for 3.7).
** Write clearly in dark pencil or ink, on one side of the paper only. **
--------------------------------------------------------------------------------
function
general antiderivative
sin(ax)cos(bx)
1/(b^2-a^2) [ a cos(ax)sin(ax) - b sin(ax)cos(bx) ] + c
cos(ax)cos(bx) 1/(b^2-a^2) [ b cos(ax)sin(bx) - a sin(ax)cos(bx) ] + c
sin(ax)cos(bx) 1/(b^2-a^2) [ b sin(ax)sin(bx) + a cos(ax)cos(bx)] + c
p(x) e^(ax) 1/a p(x)e^(ax) - 1/a INTEGRAL(p'(x)e^(ax),x) + c
p(x) sin(ax) 1/a p(x)cos(ax) + 1/a INTEGRAL(p'(x)cos(ax),x) + c
p(x) cos(ax) 1/a p(x)sin(ax) - 1/a INTEGRAL(p'(x)sin(ax),x) + c
1/(sin(x))^m -1/(m-1) cos(x) / (sin(x))^(m-1) + (m-2)/(m-1) INTEGRAL(1/(sin(x))^(m-2), x) + c
1/sin(x) 1/2 ln | (cos(x)-1) / (cos(x) + 1) | + c
1/(cos(x))^m 1/(m-1) sin(x) / (cos(x))^(m-1) + (m-2)/(m-1) INTEGRAL(1/(cos(x))^(m-2), x) + c
1/cos(x) 1/2 ln | (sin(x)-1) / (sin(x) + 1) | + c
(bx+c)/(x^2+x^2) b/s ln | x^2+x^2 | + c/a arctan(x/a) + c
(cx + d) / [ (x-a)(x-b) ] 1/(a-b) [ (ac + d) ln | x-a | - (bc+d) ln | x-b | ] + c
1 / `sqrt( x^2 +- a^2 ) ln | x + `sqr(x^2 +- a^2 | + c
`sqrt(a^2 +- x^2 ) 1/2 ( x `sqrt(a^2 +- x^2) + a^2 INTEGRAL(1/`sqrt(a^2 +- x^2 ) + c
`sqrt(x^2 - a^2) 1/2 ( x `sqrt(a^2 +- x^2) + a^2 INTEGRAL(1/`sqrt(a^2 +- x^2 ) + c
10-05-2001 16:44:20
Test should be printed using Internet Explorer. If printed from different browser check to be sure test items have not been cut off. If items are cut off then print in Landscape Mode (choose File, Print, click on Properties and check the box next to Landscape, etc.).
Write on ONE SIDE of paper only
If a distance student be sure to email instructor after taking the test in order to request results.
Signed by Attendant, with Current Date and Time: ______________________
If picture ID has been matched with student and name as given above, Attendant please sign here: _________
Instructions:
Test is to be taken without reference to text or outside notes.
Graphing Calculator is allowed, as is blank paper or testing center paper.
No time limit but test is to be taken in one sitting.
Please place completed test in Dave Smith's folder, OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va., 24212-0828 OR email copy of document to dsmith@vhcc.edu, OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken. Student may, if proctor deems it feasible, make and retain a copy of the test..
Directions for Student:
Completely document your work.
Numerical answers should be correct to 3 significant figures. You may round off given numerical information to a precision consistent with this standard.
Undocumented and unjustified answers may be counted wrong, and in the case of two-choice or limited-choice answers (e.g., true-false or yes-no) will be counted wrong. Undocumented and unjustified answers, if wrong, never get partial credit. So show your work and explain your reasoning.
Due to a scanner malfunction and other errors some test items may be hard to read, incomplete or even illegible. If this is judged by the instructor to be the case you will not be penalized for these items, but if you complete them and if they help your grade they will be counted. Therefore it is to your advantage to attempt to complete them, if necessary sensibly filling in any questionable parts.
Please write on one side of paper only, and staple test pages together.
Test Problems:
. . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
Problem Number 1
Antidifferentiate ( 3 x + 10) / [ (x - 8) (x + 2) ] without the use of tables.
.
.
Use partial fractions.
.
.
.
.
.
.
.
Problem Number 2
If p(x) = k x (1-x), 0 <= x <= 1, represents a probability distribution, then
What is the value of k?
What is the probability that .53 < x < .56?
.
.
If it's a probability distribution then the integral of the function over its domain must be 1. What does k have to be equal to in order for that to be so?
The probability of an occurrence between x = a and x = b is the integral of the probability distribution function p(x) between x = a and x = b.
.
.
.
.
.
.
.
Problem Number 3
Prove whether the integral of 1 / ( .9 + e^( 2.7 x)), from x = 1 to infinity, converges or diverges.
.
Use a comparison test. Find a function with known to be convergent which is greater than the given function, or find a function known to be divergent which is less than the given function.
Known convergent integrals include, for example,
int(1 / x^p with p > 1 from x = any positive number to infinity)
int(e^-(kx) for positive k, from x = any number to infinity)
Divergent integrals include, for example,
int(ln(x) from x = any positive number to infinity)
int(1 / x^p with 0 < p < 1 from x = any positive number to infinity)
.
.
.
.
.
.
.
.
Problem Number 4
The probability distribution for molecular speeds in a gas is given at temperature T by p(v) = A v^2 e^(-c v^2 / T), where c = m / (1.76 * 10^-23) and T is temperature in degrees Kelvin. For hydrogen molecules, which have mass approximately 3 * 10^-27 kg, find the value of A when the temperature is 270 Kelvin. Then find the average molecular velocity.
See note on #1 to evaluate A.
The average velocity is equal to the integral of v * p(v) over the domain of the function.
.
.
.
.
.
.
.
.
.
Problem Number 5
Find the volume of the solid obtained by rotating the region bounded by the curve y = x^- 1 between x = 0 and x = .36 and the y axis about the line y = -.6001.
.
Describe the graph of the region.
For a given value of x, what is the y interval of the region above and below that value?
How far is that interval from the line y = -.6 at its closest and at its furthest?
Describe in detail the shape that will result if that y interval is rotated about the line y = -.6.
Use this desription as a basis for a 'washer-method' construction of the volume integral.
.
.
.
.
.
.
.
.
Problem Number 6
A right triangle has its right angle at the origin. Its leg along the x axis has length 46 cm and its leg along the y axis has length 5 cm. At distance x from the y axis the density of the triangle is 1 / ( 1.8 + x) grams / cm^2. Use a Riemann sum to represent the approximate mass of the triangle. What is the precise mass?
Describe the region corresponding to an x-interval of length `dx which contains the position x.
What is the area of that region?
What is the density of that region?
What therefore is the mass of that region?
What Riemann sum do you get, what integral does it approach, and what do you get when you do the integration?
.
.
.
.
.
.
.
.
.
.
Problem Number 7
Find the integral of the function f(x) = sin(x + `pi), evaluated between x = 0 and x = `pi/2. Then calculate the error obtained by TRAP(3), MID(3) and SIMP(3).
Find the ratios of the errors in MID(3) and SIMP(3) to the error in TRAP(3).
Each estimate will either overestimate or underestimate the integral. For each specify which, and tell why this happens.
.
Integrate the function, find TRAP(3), MID(3), SIMP(3), and based on these results answer the questions.
.
.
.
.
.
.
.
.
Problem Number 8
For the function f(x) = sin(x-`pi/2), evaluated between x = 0 and x = `pi/2, place the following in order:
LEFT(100)
RIGHT(100)
TRAP(100)
MID(100)
the actual value of the integral
.
Describe in detail the graph of the function.
Which will be greater, LEFT(100) or RIGHT(100)?
TRAP(100) will be between these and MID(100) will also be between these.
Why is this so and which of the two will be greater?
What two of the approximations will the correct integral be between?
.
.
.
.
.
.
.
.
Problem Number 9
Find the present value after 6 years of income stream function $ 160,000 * e^( .061 t) per year, where t is in years from the present, provided we expect money to grow at a constant annual rate of 3%, compounded continuously.
How much income will accrue during time interval `dt containing time instant t years?
How much money would you have to have now to grow into that amount in t years?
This gives you a Riemann sum, which approaches an integral.
You might also want to review Mth 173 qa 10.
.
.
.
.
.
.
.
.
Problem Number 10
Antidifferentiate sin^ 10( 4 x) cos^ 8( 4 x) with or without the use of tables.
What substitutions or integrations by parts are possible here?
Is there a formula on the given table that applies?
course Mth 174
Can you go over the following test with me So that I can see how to work them:Time and Date Stamps (logged): 18:52:13 07-28-2007 ?????????
With 100+ students I can't take time to work out tests on request, unless a student makes a good attempt to work the test out and includes the details of the attempted solutions.
However you are working hard this Summer and I'll take a few minutes to insert some hints. Then work the problems out as best you can, and I can respond.
Calculus II Test 2
--------------------------------------------------------------------------------
Completely document your work and your reasoning.
You will be graded on your documentation, your reasoning, and the correctness of your conclusions.
Except where the need for more precision dictates otherwise (e.g., in nuclear physics) all quantities may be rounded to three significant figures. The generating program works in binary and often generates extraneous digits (e.g., 1.5001 for 1.5, 3.6999 for 3.7).
** Write clearly in dark pencil or ink, on one side of the paper only. **
--------------------------------------------------------------------------------
function
general antiderivative
sin(ax)cos(bx)
1/(b^2-a^2) [ a cos(ax)sin(ax) - b sin(ax)cos(bx) ] + c
cos(ax)cos(bx) 1/(b^2-a^2) [ b cos(ax)sin(bx) - a sin(ax)cos(bx) ] + c
sin(ax)cos(bx) 1/(b^2-a^2) [ b sin(ax)sin(bx) + a cos(ax)cos(bx)] + c
p(x) e^(ax) 1/a p(x)e^(ax) - 1/a INTEGRAL(p'(x)e^(ax),x) + c
p(x) sin(ax) 1/a p(x)cos(ax) + 1/a INTEGRAL(p'(x)cos(ax),x) + c
p(x) cos(ax) 1/a p(x)sin(ax) - 1/a INTEGRAL(p'(x)sin(ax),x) + c
1/(sin(x))^m -1/(m-1) cos(x) / (sin(x))^(m-1) + (m-2)/(m-1) INTEGRAL(1/(sin(x))^(m-2), x) + c
1/sin(x) 1/2 ln | (cos(x)-1) / (cos(x) + 1) | + c
1/(cos(x))^m 1/(m-1) sin(x) / (cos(x))^(m-1) + (m-2)/(m-1) INTEGRAL(1/(cos(x))^(m-2), x) + c
1/cos(x) 1/2 ln | (sin(x)-1) / (sin(x) + 1) | + c
(bx+c)/(x^2+x^2) b/s ln | x^2+x^2 | + c/a arctan(x/a) + c
(cx + d) / [ (x-a)(x-b) ] 1/(a-b) [ (ac + d) ln | x-a | - (bc+d) ln | x-b | ] + c
1 / `sqrt( x^2 +- a^2 ) ln | x + `sqr(x^2 +- a^2 | + c
`sqrt(a^2 +- x^2 ) 1/2 ( x `sqrt(a^2 +- x^2) + a^2 INTEGRAL(1/`sqrt(a^2 +- x^2 ) + c
`sqrt(x^2 - a^2) 1/2 ( x `sqrt(a^2 +- x^2) + a^2 INTEGRAL(1/`sqrt(a^2 +- x^2 ) + c
10-05-2001 16:44:20
Test should be printed using Internet Explorer. If printed from different browser check to be sure test items have not been cut off. If items are cut off then print in Landscape Mode (choose File, Print, click on Properties and check the box next to Landscape, etc.).
Write on ONE SIDE of paper only
If a distance student be sure to email instructor after taking the test in order to request results.
Signed by Attendant, with Current Date and Time: ______________________
If picture ID has been matched with student and name as given above, Attendant please sign here: _________
Instructions:
Test is to be taken without reference to text or outside notes.
Graphing Calculator is allowed, as is blank paper or testing center paper.
No time limit but test is to be taken in one sitting.
Please place completed test in Dave Smith's folder, OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va., 24212-0828 OR email copy of document to dsmith@vhcc.edu, OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken. Student may, if proctor deems it feasible, make and retain a copy of the test..
Directions for Student:
Completely document your work.
Numerical answers should be correct to 3 significant figures. You may round off given numerical information to a precision consistent with this standard.
Undocumented and unjustified answers may be counted wrong, and in the case of two-choice or limited-choice answers (e.g., true-false or yes-no) will be counted wrong. Undocumented and unjustified answers, if wrong, never get partial credit. So show your work and explain your reasoning.
Due to a scanner malfunction and other errors some test items may be hard to read, incomplete or even illegible. If this is judged by the instructor to be the case you will not be penalized for these items, but if you complete them and if they help your grade they will be counted. Therefore it is to your advantage to attempt to complete them, if necessary sensibly filling in any questionable parts.
Please write on one side of paper only, and staple test pages together.
Test Problems:
. . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
Problem Number 1
Antidifferentiate ( 3 x + 10) / [ (x - 8) (x + 2) ] without the use of tables.
.
.
Use partial fractions.
.
.
.
.
.
.
.
Problem Number 2
If p(x) = k x (1-x), 0 <= x <= 1, represents a probability distribution, then
What is the value of k?
What is the probability that .53 < x < .56?
.
.
If it's a probability distribution then the integral of the function over its domain must be 1. What does k have to be equal to in order for that to be so?
The probability of an occurrence between x = a and x = b is the integral of the probability distribution function p(x) between x = a and x = b.
.
.
.
.
.
.
.
Problem Number 3
Prove whether the integral of 1 / ( .9 + e^( 2.7 x)), from x = 1 to infinity, converges or diverges.
.
Use a comparison test. Find a function with known to be convergent which is greater than the given function, or find a function known to be divergent which is less than the given function.
Known convergent integrals include, for example,
int(1 / x^p with p > 1 from x = any positive number to infinity)
int(e^-(kx) for positive k, from x = any number to infinity)
Divergent integrals include, for example,
int(ln(x) from x = any positive number to infinity)
int(1 / x^p with 0 < p < 1 from x = any positive number to infinity)
.
.
.
.
.
.
.
.
Problem Number 4
The probability distribution for molecular speeds in a gas is given at temperature T by p(v) = A v^2 e^(-c v^2 / T), where c = m / (1.76 * 10^-23) and T is temperature in degrees Kelvin. For hydrogen molecules, which have mass approximately 3 * 10^-27 kg, find the value of A when the temperature is 270 Kelvin. Then find the average molecular velocity.
See note on #1 to evaluate A.
The average velocity is equal to the integral of v * p(v) over the domain of the function.
.
.
.
.
.
.
.
.
.
Problem Number 5
Find the volume of the solid obtained by rotating the region bounded by the curve y = x^- 1 between x = 0 and x = .36 and the y axis about the line y = -.6001.
.
Describe the graph of the region.
For a given value of x, what is the y interval of the region above and below that value?
How far is that interval from the line y = -.6 at its closest and at its furthest?
Describe in detail the shape that will result if that y interval is rotated about the line y = -.6.
Use this desription as a basis for a 'washer-method' construction of the volume integral.
.
.
.
.
.
.
.
.
Problem Number 6
A right triangle has its right angle at the origin. Its leg along the x axis has length 46 cm and its leg along the y axis has length 5 cm. At distance x from the y axis the density of the triangle is 1 / ( 1.8 + x) grams / cm^2. Use a Riemann sum to represent the approximate mass of the triangle. What is the precise mass?
Describe the region corresponding to an x-interval of length `dx which contains the position x.
What is the area of that region?
What is the density of that region?
What therefore is the mass of that region?
What Riemann sum do you get, what integral does it approach, and what do you get when you do the integration?
.
.
.
.
.
.
.
.
.
.
Problem Number 7
Find the integral of the function f(x) = sin(x + `pi), evaluated between x = 0 and x = `pi/2. Then calculate the error obtained by TRAP(3), MID(3) and SIMP(3).
Find the ratios of the errors in MID(3) and SIMP(3) to the error in TRAP(3).
Each estimate will either overestimate or underestimate the integral. For each specify which, and tell why this happens.
.
Integrate the function, find TRAP(3), MID(3), SIMP(3), and based on these results answer the questions.
.
.
.
.
.
.
.
.
Problem Number 8
For the function f(x) = sin(x-`pi/2), evaluated between x = 0 and x = `pi/2, place the following in order:
LEFT(100)
RIGHT(100)
TRAP(100)
MID(100)
the actual value of the integral
.
Describe in detail the graph of the function.
Which will be greater, LEFT(100) or RIGHT(100)?
TRAP(100) will be between these and MID(100) will also be between these.
Why is this so and which of the two will be greater?
What two of the approximations will the correct integral be between?
.
.
.
.
.
.
.
.
Problem Number 9
Find the present value after 6 years of income stream function $ 160,000 * e^( .061 t) per year, where t is in years from the present, provided we expect money to grow at a constant annual rate of 3%, compounded continuously.
How much income will accrue during time interval `dt containing time instant t years?
How much money would you have to have now to grow into that amount in t years?
This gives you a Riemann sum, which approaches an integral.
You might also want to review Mth 173 qa 10.
.
.
.
.
.
.
.
.
Problem Number 10
Antidifferentiate sin^ 10( 4 x) cos^ 8( 4 x) with or without the use of tables.
What substitutions or integrations by parts are possible here?
Is there a formula on the given table that applies?