Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Transcription

1 Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking mathematics, statistics and computer science courses. There are also numerous majors and minors at CC which require some calculus or statistics (see the bottom of this document for a list of such majors). This worksheet and packet are designed to help ou determine what might be a good first mathematics, statistics or computer science course for ou. Work our wa through the decision tree on the net page. Consider the questions carefull and ask for advice or assistance if ou are unsure. Use the following questions to help ou as well. ote that the decision tree refers to three attached documents titled, Sample Pre-Calc, Sample Calc I and Sample Calc II. Please look at and read these documents carefull, when appropriate.. Make a list of majors ou are considering. This is not binding, just some majors ou ve been thinking about.. Are an of the majors on the list below? If es, be sure ou followed es for do ou intend to stud calculus at CC? 3. As ou work through the decision tree if ou are stuck between two possible paths, consider the following questions: (a) When was the last time ou took a math class? (b) How independent are ou in our learning? Would ou feel comfortable reviewing some topics on our own? 4. Write down the courses the tree suggested for ou. Look these up on attached document, Choosing our First Math Class. After reading the corresponding document, do these suggestions seem reasonable to ou? If not, think about wh not and talk to one of the mentors. To help answer the question, do ou intend to stud calculus at CC?, hereisalist of majors that require Calculus (MA5 or MA6) or higher (some of these majors require significantl more mathematics). Biolog, Chemistr, Computer Science, Economics, Environmental Science, Geolog, International Political Econom, Mathematics, Mathematical Economics, Phsics, Pscholog. The following is a list of majors that require some statistics or modeling course from the list MA7, B, EC, EV8, MA56, MA7. The specific course required varies across these majors, but MA7 satisfies most of these requirements (as will MA7). Biolog, Economics, Environmental Science, Geolog, International Political Econom, Mathematical Economics.

2 Consider MA 5 Do ou intend to stud calculus at CC or do ou have previous eperience with calculus? ou still need to satisf the QR and SI requirements. Consider MA, 7, CP. Do ou have a good understanding of functions and graphs and algebraic manipulation? To check our knowledge of these topics look at the document Sample Pre-Calc. Do ou have eperience with calculus (AP, IB, or other)? Consider MA 6, 7 or CP Do ou have a good understanding of derivatives and introductor integration? To check our knowledge of these topics look at the document Sample Calc I Did ou receive a score on an AP or IB eam? Consider taking MA9, MA 7, CP Did ou receive a score of at least 5 on the IB Math SL eam, or of 4 or 5 on the AP Calculus AB eam? Did ou take the AP BC or IB Math HL eam? Consider taking MA4, 7,, CP Do ou have a good understanding of integration, Talor polnomials, and infinite series and feel comfortable learning some new material independentl? To check our knowledge of these topics look at the document Sample Calc II. Did ou receive a score of at least 5 on the IB Math HL eam, or of 4 or 5 on the AP Calculus BC eam? Success in quantitative disciplines at CC requires knowing our resources. Be sure to talk to our professor, the departmental paraprofessional, and the Quantitative Reasoning Center (in Tutt Librar).

3 Sample Pre-Calculus Questions A student need not get % on the questions below in order to take Math 6 rather than Math 5. An initial score of at least 5% is good enough as long as ou feel that a little review would bring our score up to at least 8%. If ou feel that ou would need more than a little review, then ou ought to enroll in Math 5.. Suppose =. What can be concluded? a. = b. = 3 c. = d. = or = 3. Which of the following is the equation of the line passing through (, -) and parallel to = 4 + 5? a. = b. = 4 9 c. + 4 = d. + 4 = 7 3. Which of the following is the graph of = 3 ( )? a b c d

4 4. What is the equation of a circle of radius, centered at the point (3, 4)? a. ( 3) + ( 4) = 4 b. ( + 3) + ( + 4) = 4 c. ( 3) + ( 4) = d. + = 9 5. If /3 is three less than half of 4/5 of some number, what is that number? a. 3/5 b. 55/6 c. 5/3 d. one of the above 6. If f () = 3 5, which of the following is equivalent to f ( +)? a. 3 4 b. 4 5 c d Which of the following equations are valid? (I) + = + (II) a + b =+ a b b a. Onl I b. II and III c. Onl II d. Onl III (III) + 3 = In the following diagram, find the value of cosθ : a. b. 5 q c. 3 d. 5

5 9. If sinθ = 3, what is tanθ? a. 3 b. 3 or 3 c. or d. 3. Which of the following is the graph of = sin(3)? 4 a 4 b c d Solve for : = a. log b. log c. d. log. Solve for : 3log log 4 = a. b. c. 8 3 d. (log 3) /3

6 Sample Calculus I Questions A student need not get % on the questions below in order to skip Calculus I (Math ) and go straight into Calculus II (Math 6). An initial score of at least 5% is good enough as long as ou feel that a little review would bring our score up to at least 8%. If ou feel that ou would need more than a little review, then ou ought to enroll in Math. Click here for solutions.. Derivatives Smbolicall (a) Find the derivatives of the following functions. i. = 3 + iii. =( 4 +7) /3 v. = sin() ii. = cos(7)+ +6 iv. = e vi. =ln( + 7) (b) Let f() = i. Find all intervals on which f is increasing. ii. Find all intervals on which f is concave up. iii. Find all local maima for f.. Derivatives Graphicall Let f() be a function. The graph of its derivative, f () is shown below. Give approimate answers to the questions below (a) On what interval(s) is f increasing? (b) On what interval(s) is f concave up? (c) At what point(s) does f reach a local maimum? (d) At what point(s) does f reach a local maimum? (e) Is f (.5) greater than or less than? 3. Derivatives umericall (a) Some numerical data is given for a function below f() i. What is the average rate of change of f on the interval [, 4.]?

7 ii. Estimate the instantaneous rate of change of f at =3.5 (there is more than one wa to do this). (b) The function g() satisfies g(3) = and g (3) =. Estimate the value of g(3.). 4. Derivatives Theoreticall Which of the following is a correct definition of f ()? (i) lim f( + h) f() h f()+h (iii) lim h h (ii) lim h f( + h) f() h f()+f(h) (iv) lim + h 5. Applications (a) Find the equation of the line tangent to f() =3 3 +ln() at =. (b) A commercial nurser has ards of fencing which the owners plan to use to enclose as large a rectangular garden as possible. The garden will be bounded on one side b a barn, so no fencing is needed on that side. How large will the garden be (in square ards)? 6. A Little Bit of Integration (a) Find the following definite and indefinite integrals. i. ( 3 +) d ii. (3 +) d (b) Use a left hand sum with 4 equal subintervals to estimate the value of the integral eactl with the Fundamental Theorem of Calculus.) d. (Do not evaluate

8 Sample Calculus II Questions A student need not get % on the questions below in order to skip Calculus II (Math 6) and go straight into Linear Algebra (Math ). An initial score of at least 5% is good enough as long as ou feel that a little review would bring our score up to at least 8%. If ou feel that ou would need more than a little review, then ou ought to enroll in Math 6. Click here for solutions.. Integration Techniques and the Fundamental Theorem of Calculus (a) Use either substitution or integration-b-parts to find the following indefinite integrals. i. ( ) 6 d iii. e d ii. 3 ln( 7 ) d iv. e d (b) Suppose that the function f() isdefinedbf() = e t t dt. Findf ().. Anti-derivatives graphicall The graph of the function f() on the interval [, 4] is shown below. Define F (t) = t f() d. (a) What is the value of F (.5)? (b) On what interval(s) is F increasing? (c) On what interval(s) is F concave up? (d) At what point(s) does F reach a local minimum? 3. Anti-derivatives numericall The Dundas cit engineers have collected some data concerning the rate r(t) of energ consumption per hour in their town. The data below represents a tpical da in Dundas where the rate is measured in megawatts per hour, t hours after midnight. t r(t) Make a good estimate of the total 4 hour consumption of electricit in Dundas.

9 4. Applications of Integration (a) Compute the area bounded b the curves =6 and =3. (b) Rotate the region bounded b the -ais, the curve =, and the line = 3 about the -ais and find the volume of the resulting solid. 5. Improper Integrals Which of the following improper integrals converge? (a) d (c) d (b) 6. Series d (d) e d (a) Which of the following series converge? i. (3 k ) iii. k= ii. ( 3 )k iv. k= (b) To what value do the following series converge? k= k k k= i. k= 5( 3 )k ii. k= 5( 3 )k (c) Which of the following limits equals the sum of the series n= a n? i. lim n n= a n iii. lim a n k n=k ii. lim k 7. Talor Series k n= a n iv. lim n a n (a) The Talor series for f() =e about =is: i. What is the interval of convergence of the Talor series for e? ii. Use the first 4 terms of the Talor series for e to estimate the value of e to three decimal places. (b) The Talor series for f() = about =is with an interval of convergence of (, ) i. Find the first 4 non-zero terms of the Talor series for f() =. ii. What is the interval of convergence for the Talor series in the problem above?

10 Choosing our First Math Class If ou took an AP or IB eam or a course at another universit, ou should receive official notification of credit and ou are encouraged to take the net appropriate class. However, our calculus sequence is not traditional and ou ma find the following information helpful. Regardless, here are some rough guidelines on where to start, including some options for more advanced students or those wondering what might be a good net course. These descriptions are intended to enhance the Decision tree, which is a quick and eas place to start. In calculus choose: MA 5 (Precalc/Calc) if ou need Calculus, but our algebra and pre-calculus background is weak. In particular, choose this course if ou have trouble simplifing complicated algebraic epressions involving eponents and fractions, if ou have trouble seeing relationships between formulas and their graphs, or if ou have not become acquainted with trig and log functions. Check the document Sample Pre-Calculus Questions to help ou with this decision. MA 6 (Calc ) if ou have a solid algebra and pre-calculus background, but have either never seen calculus, or do not feel comfortable with what ou have seen. MA 9 (Calc ) if ou successfull completed (and do not need help reviewing) a high school calculus course that covered techniques and applications of differentiation, and included a little introduction to integration, or if ou completed calculus (AP BC, for eample), but did not score well on the eam, or do not feel comfortable with the work ou have done. MA 4 (Calc 3) if ou successfull completed (and do not need help reviewing) a high school calculus course (AP BC, for eample), feel comfortable with that material and feel comfortable learning some new material independentl. MA (Linear Algebra) if ou successfull completed (and do not need help reviewing) a high school (or other) calculus 3 course that covered multivariable derivatives and integrals, alternate coordinate sstems, and introductor vector calculus. MA 56 (Biological Modeling) if ou successfull completed (and do not need help reviewing) a high school (or other) calculus 3 course that covered multivariable derivatives and integrals, alternate coordinate sstems, and introductor vector calculus. We recommend against taking MA 5 in the first ear, but ver strong students might consider it, especiall if the take another mathematics course at CC prior to this one. In statistics choose: MA 7 (Prob&Stat) if ou have not had a calculus or statistics course, and/or are not ver interested in seeing the reasoning that lies behind statistical tests. MA 7 (Prob&Stat Modeling) if ou have successfull completed a calculus or statistics course, and like to see a little theor with our formulas. If ou epect to be in a statisticall need major, ou are strongl encouraged to consider this statistics course. If ou have taken calculus (or higher) and feel comfortable working with mathematics at that level ou are strongl encouraged to consider this statistics course.

11 In Computer Science: CP (CS) This is our first course in computer science and it is appropriate for students with little or no programming eperience. This should be the first computer science class ou take unless ou have a ver strong programming background in a language like Java/C/C++. CP (CS ) - This course is tpicall the second or third computer science class taken b students. This course should be our first computer science class onl if ou alread have a ver strong programming background in a language like Java/C/C++. To take this class as a first class ou should be ver comfortable with basic programming techniques, conditional statements, loops, functions, objectedoriented programming, recursion, and arras/arralists. CP 5 (Application Design) - This course is tpicall the second or third computer science class taken b students (it can be taken before or after CP). This course should be our first computer science class onl if ou alread have a ver strong programming background in a language like Java/C/C++. To take this class as a first class ou should be ver comfortable with basic programming techniques, conditional statements, loops, functions, objected-oriented programming, recursion, and arras/arralists. Selecting a QR course: For those who think that the will not continue in computer science, mathematics, the sciences (biolog, chemistr, environmental science, geolog, phsics, or pscholog), or economics (including mathematical economics and international political econom) should consider Math or CP (there is at least one each ear and the cover a variet of topics, Please check the course schedule to see what is offered this ear). These courses tpicall have no prerequisites and are aimed at a general audience or an of the man other QR courses offered on campus. A listing of QR courses currentl offered can be found using the online course scheduling sstem.