Math Summer Peter Alfeld. WeBWorK assignment number 1. due 9/6/06 at 11:59 PM.

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1 Math Summer 2004 Peter Alfeld. WeBWorK assignment number. due 9/6/06 at :59 PM. This first home work serves as an introduction to WeBWorK and as a review of some relevant precalculus topics, including: The language of the number system, Solution of linear, quadratic, and polynomial equations, Language of Polynomials Cartesian Coordinates Equations and properties of straight lines logic, converse, contrapositive, and negation of statements functions, even, odd, graphs, composition The questions include a few word problems. Here are some hints on how to use WeBWorK effectively: After first logging into WeBWorK change your password. Find out how to print a hard copy on the computer system that you are going to use. Contact me if you have any problems. Print a hard copy of this assignment. Note, however, that the online versions of these problems may have links to the web pages of this course that are absent on the hard copy. Get to work on each set right away and answer these questions well before the deadline. Not only will this give you the chance to figure out what s wrong if an answer is not accepted, you also will avoid the likely rush and congestion prior to the deadline. The primary purpose of the WeBWorK assignments in this class is to give you the opportunity to learn by having instant feedback on your active solution of relevant problems. Make the best of it! Peter Alfeld, JWB 27, (0 pts) set/20s0p.pg This first question is just an exercise in entering answers into WeBWorK. It also gives you an opportunity to experiment with entering different arithmetic and algebraic expressions into WeBWorK and seeing what WeBWorK really thinks you are doing (as opposed to what you believe it should think). Notice the buttons on this page and try them out before moving to the next problem. Use the Back Button on your browser to get back here when needed. Prob. List gets you back to the list of all problems in this set. Next gets you to the next question in this set. Submit Answer submits your answer as you might expect, but there may be other ways to do so. Specifically, in this problem, there is only one question. In that case you can submit your answer by typing it into the answer window and then pressing Return (or Enter ) on your keyboard. But even in this case, you can also type the answer and click on the submit button. There is no harm in submitting an answer even if you are not quite sure that it s correct, since if it is not you have an unlimited number of additional tries. On the other hand, it is usually more efficient to print your own private problems set, work out the answers in a quiet environment like your home, and then sit down in front of a computer and enter your answers. If some are wrong you can try to fix them right at the computer, or you may want to go back and work on them quietly elsewhere before returning to the computer. Pressing on the Preview Answer Button makes WeBWorK display what it thinks you entered in the answer window. After using Preview you can modify your answer and use a Preview Again button. typeset denotes the ordinary display mode on your workstation. images is a bit faster but looks less fancy. Very occasionally the output from typeset gets corrupted. In that

2 case you can send me an and I will fix it. You can also see the intended output, if not quite as nicely, using images. The options of plaintext and jsmath are not useful. Logout terminates this WeBWorK session for you. You can of course log back in and continue. Feedback enables you to send a message to me. If you use this way of sending I receive information about your WeBWorK state, in addition to your actual message. The Help Button transports you to an official WeBWorK help page that has more information than this first problem. Problem Sets transports you back to the page where you can select a certain problem set. When you do this particular problem in this first set, there is only one set, but eventually there will be 3 of them. For all problems in this course you will be able to see the Answers to the problems after the due date. Go to a problem, click on show correct answers, and then click on submit answer. You can also download and print a hard copy with the answers showing. These answers are the precise strings against which WeBWorK compares your answer. If the answer is an algebraic expression your answer needs to be equivalent to the WeBWorK answer, but it may be in a different form. For example if WeBWorK thinks the answer is 2 a, it is OK for you to type a + a instead. If WeBWorK expects a numerical answer then you can usually enter it as an arithmetic expression (like /7 instead of.42857), and usually WeBWorK will expect your answer to be within one tenth of one percent of what it thinks the answer is. Most of the problems (including this one) in this course will also have solutions attached that you can see after the due date by clicking on show solutions followed by submit answers. The solutions are text typed by your instructor that gives more information than the answers, and in particular often explains how the answers can be obtained. 2 Now for the meat of this problem. Notice that the answer window is extra large so you can try the things suggested above and below. Type the number 3 here:. Try entering other expressions and use the preview button to see what WeBWorK thinks you entered. Return to this problem to try out things when you get stuck somewhere else. Here are some good examples to try. Check them all out using the Preview button. (In later questions on this set you will get to use what you learn here.) Never mind that you may have already answered the correct answer 3. Once you get credit for an answer it won t be taken away by trying other answers. a/2b versus a/2/b versus a/(2b) a/b+c versus a/(b+c) a+b**2 versus (a+b)**2 sqrt a+b versus sqrt(a+b) 4/3 pi r**2 versus (4/3) pi r**2 (In other words, if you are not sure use parentheses freely.) Note: WeBWorK will not usually let you enter algebraic expressions when the answer is a number, and it will only let you use certain variables when the answer is in fact an algebraic expression. So the above window, and the opportunity for experimentation that it offers is unique. Make good use of it! Presumably this has been your first encounter with WeBWorK. Come back here to try things out and to refresh your memory if you get stuck somewhere down the line. 2.(0 pts) set/20s0p2.pg This problem demonstrates how you enter numerical answers into WeBWorK. Evaluate the expression 3( 8)( 2(0)): In the case above you need to enter a number, since the question is testing whether you can multiply out these numbers. (Of course you can, this is just a ww intro. You can use a calculator if you want.) For most problems, you will be able to get WeB- WorK to do some of the work for you. For example Calculate (-8) * (): The asterisk is what most computers use to denote multiplication and you can use this with WeBWorK. But WeBWorK will also allow use to use a space to

3 denote multiplication. You can enter 8 or -88 or even 8. All will work. Try them. Now try calculating the sine of 45 degrees ( that s sine of pi over 4 in radians and numerically sin(pi/4) equals or, more precisely, / 2 ). You can enter this as sin(pi/4), as sin( /4), as /sqrt(2), as 2**(-.5), etc. This is because WeBWorK knows about functions like sin and sqrt (square root). (Note: exponents can be indicated by either a caret or **). Try it. sin(π/4) = Note that WeBWorK always uses radian mode for trig functions. It simply does not know what a degree is! You can also use juxtaposition to denote multiplication. E.g. enter 2sin(3π/2). You can enter this as 2*sin(3*pi/2) or more simply as 2sin(3pi/2). Try it: Sometimes you need to use ( ) s to make your meaning clear. E.g. /2+3 is 3.5, but /(2+3) is.2 Why? Try entering both and use the Preview button below to see the difference. In addition to ( ) s, you can also use [ ] s and { } s. You can always try to enter answers and let WeB- WorK do the calculating. WeBWorK will tell you if the problem requires a strictly numerical answer. There is no penalty for getting an answer wrong. What counts is that you get the answer right eventually (before the due date). For complicated answers, you should use the Preview button to check for syntax errors and also to check that the answer you enter is really what you think it is. 3.(0 pts) set/20s0p3.pg This problem demonstrates how you enter function answers into WeBWorK. First enter the function sin x. When entering the function, you should enter sin(x), but WeBWorK will also except sin x or even sinx. If you remember your trig identities, sin(x) = -cos(x+pi/2) and WeBWorK will accept this or any other function equal to sin(x), e.g. sin(x) +sin(x)**2+cos(x)**2- We said you should enter sin(x) even though WeB- WorK will also except sin x or even sinx because you are less likely to make a mistake. Try entering sin(2x) 3 without the parentheses and you may be surprised at what you get. Use the Preview button to see what you get. WeBWorK will evaluate functions (such as sin) before doing anything else, so sin 2x means first apply sin which gives sin(2) and then multiple by x. Try it. Now enter the function 2cost. Note this is a function of t and not x. Try entering 2cos x and see what happens. 4.(0 pts) set/20s0p4.pg This problem will help you learn the rules of precedence, i.e. the order in which mathematical operations are performed. You can use parentheses (and also square brackets [ ] and/or curly braces { }) if you want to change the normal way operations work. So first let us review the normal way operations are performed. The rules are simple. Exponentiation is always done before multiplication and division and multiplication and division are always done before addition and subtraction. (Mathematically we say exponentiation takes precedence over multiplication and division, etc.). For example what is +2*3? and what is 2 3 2? Now, sometime you want to force things to be done in a different way. This is what parentheses are used for. The rule is: whatever is enclosed in parentheses is done before anything else (and things in the innermost parentheses are done first). For example how do you enter + sin(3) 2 + tan(4)? Hint: this is a good place to use [ ] s and also to use the Preview button. Here are some more examples: (+3)9 =36, (2*3)**2 = 6**2 = 36, 3**(2*2) = 3**4 = 8, (2+3)**2 = 5**2 = 25, 3**(2+2) = 3**4 = 8 (Here we have used ** to denote exponentiation and you can also use this instead of a caret if you want).

4 Try entering some of these and use the Preview button to see the result. The correct result for this answer blank is 36, but by using the Preview button, you can enter whatever you want and use WeBWorK as a hand calculator. There is one other thing to be careful of. Multiplication and division have the same precedence and there are no universal rules as to which should be done first. For example, what does 2/3*4 mean? (Note that / is the division symbol, which is usually written as a line with two dots, but unfortunately, this line with two dots symbol is not on computer keyboards. Don t think of / as the horizontal line in a fraction. Ask yourself what /2/2 should mean.) WeBWorK and most other computers read things from left to right, i.e. 2/3*4 means (2/3)*4 or 8/3, IT DOES NOT MEAN 2/2. Some computers may do operations from right to left. If you want 2/(3*4) = 2/2, you have to use parentheses. The same thing happens with addition and subtraction = 0 but -(3+2) = -4. This is one case where using parentheses even if they are not needed might be a good idea, e.g. write (2/3)*4 even though you could write 2/3*4. This is also a case where previewing your answer can save you a lot a grief since you will be able to see what you entered. Enter 2/3*4 and use the Preview button to see what you get. 5.(0 pts) set/20s0p5.pg This problem demonstrates a WeBWorK True/False question. Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false.. π < Notice that if one of your answers is wrong then, in this problem, WeBWorK will tell you which parts are wrong and which parts are right. This is the behavior for most problems, but for true/false or multiple choice questions WeBWorK will usually only tell you whether or not all the answers are correct. It 4 won t tell you which ones are wrong. In those cases you also will need to get all the answers correct before getting credit. The idea is encourage you think rather than to just try guessing. 6.(0 pts) set/20s0p6.pg This problem demonstrates a WeBWorK Matching question. Match the statements defined below with the letters labeling their equivalent expressions.. x is any real number 2. The distance from x to 8 is less than or equal to 2 3. x is greater than 8 4. x is less than or equal to 8 5. x is greater than or equal to 8 A. x 8 B. 8 < x C. x 8 2 D. x 8 E. < x < For matching questions, as for true/false questions (but not for this demonstration problem) you will usually need to get all answers correct to obtain credit, and if some answers are false ww will not tell you which or how many. 7.(0 pts) set/20s0p7.pg This problem demonstrates a WeBWorK question that requires you to enter a number or a fraction. Evaluate the expression Give you answer in decimal notation correct to three decimal places or give your answer as a fraction. Now that you have finished you can use the Prob. List button at the top of the page to return to the problem list page. You ll see that the problems you have done have been labeled as correct or incorrect, so you can go back and do problems you skipped or couldn t get right the first time. Once you have done a problem correctly it is ALWAYS listed as correct even if you go back and do it incorrectly later. This means you can use WeBWorK to review course material without any danger of changing your score.

5 8.(0 pts) set/20sp.pg Recall that the natural numbers are the integers are,2,3,...,..., 3, 2,,0,,2,3,..., rational numbers are ratios of integers (with the denominator being non-zero), and real numbers are all numbers corresponding to points on the number line. You can also think of real numbers as repeating or non-repeating decimals. There are more technical definitions which you will learn in real analysis. Indicate whether the following statements are True (T) or False (F).. The difference of two natural numbers is always a natural number. 2. The sum of two natural numbers is always a natural number. 3. The quotient of two natural numbers is always a rational number 4. The product of two natural numbers is always a natural number. 5. The difference of two natural numbers is always an integer. 6. The ratio of two natural numbers is always positive 7. The quotient of two natural numbers is always a natural number. 9.(0 pts) set/20sp2.pg Indicate whether the following statements are True (T) or False (F).. The quotient of two integers is always an integer (provided the denominator is non-zero). 2. The ratio of two integers is always positive 3. The difference of two integers is always a natural number. 4. The product of two integers is always an integer. 5. The difference of two integers is always an integer. 6. The quotient of two integers is always a rational number (provided the denominator is non-zero). 7. The sum of two integers is always an integer. 5 0.(0 pts) set/20sp3.pg Indicate whether the following statements are True (T) or False (F).. The difference of two rational numbers is always a rational number. 2. The difference of two rational numbers is always a natural number. 3. The quotient of two rational numbers is always a rational number (provided the denominator is non-zero). 4. The ratio of two rational numbers is always positive 5. The product of two rational numbers is always a rational number. 6. The quotient of two rational numbers is always a real number (provided the denominator is non-zero). 7. The sum of two rational numbers is always a rational number..(0 pts) set/20sp4.pg Solve the equation for x =. 3(x + 2) + 4 = 5(x ) 2 2.(0 pts) set/20sp5.pg Solve the equation 2 3 t + 3 +t + 9 t 2 = 0 for t = 3.(0 pts) set/20sp6.pg The equation 5x 4 6x 3 3x 2 = 0 has three real solutions A, B, and C where A < B < C and A is: and B is: and C is: 4.(0 pts) set/20sp7.pg Evaluate the expression 64 4/3. (You may enter a fraction as your answer.) 5.(0 pts) set/20sp8.pg The expression (3a 2 b 5 c 5 ) 2 (2a 2 b 2 c 5 ) 3 equals na r b s c t where n, the leading coefficient, is: and r, the exponent of a, is: and s, the exponent of b, is:

6 and finally t, the exponent of c, is: 6.(0 pts) set/20sp9.pg Consider the two points (2, 4) and (5,7). The distance between them is: The x co-ordinate of the midpoint of the line segment that joins them is: The y co-ordinate of the midpoint of the line segment that joins them is: 7.(0 pts) set/20sp0.pg Find the distance between (5, 6) and (-5, -). 8.(0 pts) set/20sp.pg Find the perimeter of the triangle with the vertices at (4, 2), (-3, 5), and (-2, -2). 25.(0 pts) set/20sp8.pg An equation of a line through (, 4) which is parallel to the line y = 2x + 2 has slope: and y intercept at: 26.(0 pts) set/20sp9.pg Find the slope of the line through (0, -3) and (-3, 4). 27.(0 pts) set/20sp20.pg The distance of the point (5,5) from the line y = 2x is: 9.(0 pts) set/20sp2.pg The equation of the line with slope 3 that goes through the point (7,3) can be written in the form y = mx + b where m is: and where b is: 20.(0 pts) set/20sp3.pg This problem is like the preceding problem. The equation of the line with slope 2.3 that goes through the point ( 6.3,4) can be written in the form y = mx + b where m is: and where b is: 2.(0 pts) set/20sp4.pg The equation of the line that goes through the point (9,43) and is parallel to the x-axis can be written in the form y = mx + b where m is: and where b is: 22.(0 pts) set/20sp5.pg The equation of the line that goes through the point (6,9) and is parallel to the line 4x + 3y = 4 can be written in the form y = mx + b where m is: and where b is: 23.(0 pts) set/20sp6.pg The equation of the line that goes through the point (9,5) and is perpendicular to the line 4x + 4y = 5 can be written in the form y = mx + b where m is: and where b is: 24.(0 pts) set/20sp7.pg A line through (5, 4) with a slope of -7 has a y- intercept at 6 28.(0 pts) set/20sp2.pg This is like the preceding problem. The distance of the point ( 4, 4) from the line y = x 3 is: 29.(0 pts) set/20s2p2.pg In this question you will derive a general formula for the distance of a point from a line. Let P be the point (p,q) and L the line y = mx + b. The slope of L is. The slope of a line perpendicular to L is. The line through P perpendicular to L can be written as y = sx + c where s is: and c is:. That line intersects L in the point Q = (u,v), where u is: and v is:. The distance of P and Q is. (Note, all answers must be in terms of m, b, p and q.) 30.(0 pts) set/20sp22.pg You drop a rock into a deep well. You can t see the rock s impact at the bottom, but you hear it after 4 seconds. The depth of the well is feet. Ignore air resistance. The time that passes after you drop the rock has two components: the time it takes the rock to reach the bottom of the well, and the time that it takes the sound of the impact to travel back to you. Assume the speed of sound is 00 feet per second.

7 Note: After t seconds the rock has reached a depth of d feet where d = 6t 2. 3.(0 pts) set/20sp23.pg This is the general version of the preceding problem. Suppose the speed of sound is c, and gravity is g. Thus if you throw the rock from an initial height h 0 and with an initial velocity v 0 the height h(t) of the rock after time t is h(t) = g 2 t2 + v 0 t + h 0. (The height is negative when the rock is below ground level. Thus you can think of depth as negative height.) Suppose you hear the impact after t seconds. Then the depth of this well is d =. (Your answer will be a mathematical expression involving t, g, and c.) 32.(0 pts) set/20sp24.pg Suppose you obtain 00 percent credit on all WeB- WorK assignments in this class. Then the minimum average percentage on the exams that will still get you an A in this class is. Your answer should be a number between 0 and 00. You may enter a fraction. 33.(0 pts) set/20sp25.pg Jamestown is 0 miles downstream from Aliceville and on the opposite side of a river half a mile wide. Mary will run from Aliceville along the river for 6 miles, and then swim diagonally (i.e., along a straight line) across the river to Jamestown. If she runs at 8 miles per hour and swims at three miles per hour she will reach Jamestown after hours. Assume that the current is negligible. Note that unless otherwise specified WeBWorK expects your answer to be within one tenth of one percent of the true answer which is the answer it has been given by the author of the problem. Practically speaking this means you should specify at least 4 digits total (including any before the decimal point). To be safe, you want to compute your answer with your calculator much more accurately. 34.(0 pts) set/20sp27.pg The next few problems are simple exercises in logic. Consider the statement A implies B. The converse 7 of this statement is B implies A, its contrapositive is not B implies not A, and its negation is A and not B. For example, consider the statement all natural numbers are real numbers. This is a true statement, but this is actually not important for this discussion. This statement can be put as an implication: if x is a natural number, then x is a real number. The converse of this statement is all real numbers are natural numbers, or if x is a real number then it is a natural number, (which is a false statement), the contrapositive is if x is not a real number then it isn t a natural number (which is a true statement), and the negation of the statement is some natural numbers are not real numbers (which is a false statement). The purpose this particular problem is to illustrate the pattern of the next couple of problems. You already know the answers, so it is just a matter of entering them. In this problem, WeBWorK will tell you separately for each answer whether it is correct or not, but in the next two you will have to enter everything correctly to get credit. Let S be the statement All natural numbers are real numbers Complete the sentences below, filling in A-G from the following list, and T or F for true or false, as appropriate. A. Numbers aren t natural. B. All real numbers are natural. C. Some natural numbers aren t real numbers. D. Some real numbers aren t natural numbers. E. No real number is natural. F. A number can t be natural if it isn t real. G. A number can t be real if it isn t natural. S is (true or false). The converse of S is (enter a letter from A-G) and that statement is. The contrapositive of S is and that statement is. The negation of S is and that statement is (true or false). 35.(0 pts) set/20sp28.pg Let S be the statement All women are humans.

8 Complete the sentences below, filling in A-G from the following list, and T or F for true or false, as appropriate. A. Some women aren t humans. B. Some women are humans. C. Some humans aren t women. D. No women are humans. E. If you aren t a human you aren t a woman. F. Men aren t human. G. All humans are women. S is (true or false). The converse of S is (enter a letter from A-G) and that statement is (true or false). The contrapositive of S is and that statement is. The negation of S is and that statement is. There are two versions of this problem with the roles of men versus women randomly interchanged. 36.(0 pts) set/20sp29.pg Throughout this problem, suppose that x stands for a real number. Let S be the statement if x > 0 then x 3 > 0. Complete the sentences below, filling in A-G from the following list, and T or F for true or false, as appropriate. A. Real numbers are all zero. B. If x 3 > 0, then x > 0. C. There is some x > 0 such that x 3 0 D. If x 0 then x 3 0. E. If x > 0 then x 3 > x. F. x 3 > x. G. If x 3 0 then x 0. S is (true or false). The converse of S is (enter a letter from A-G) and that statement is (true or false). The contrapositive of S is and that statement is. The negation of S is and that statement is. 37.(0 pts) set/20sp30.pg This problem addresses some common algebraic errors. For the equalities stated below assume that x and y stand for real numbers. Assume that any denominators are non-zero. Mark the equalities with T 8 (true) if they are true for all values of x and y, and F (false) otherwise. (x + y) 2 = x 2 + y 2. (x + y) 2 = x 2 + 2xy + y 2. x x+y = y. x (x + y) = y. x 2 = x. x 2 = x. x = x + 2. x+y = x + y. 38.(0 pts) set/20s2p.pg Let f (x) = mx + b (where m 0). The graph of f is a straight line with slope. Any line perpendicular to that graph has a slope. The graph of the inverse function of f is a straight line with slope. 39.(0 pts) set/20s2p3.pg This problem is about polynomial degrees. Recall that the degree of a polynomial p is n if p can be written as p(x) = a n x n + a n x n a x + a 0 where the leading coefficient a n 0. The degree of p(x) = is. The degree of p(x) = x 2 + 2x + 3 is. The degree of p(x) = ( (x + 2) 2 (x ) 3) 2 is. Let p and q be polynomials of degree m and n, respectively. The degree of the product of p and q is. and the degree of the composition of p and q is. 40.(0 pts) set/20s2p4.pg The next few problems concern even and odd functions. Recall that a function f is even if f (x) = f ( x) for all x in its domain, and it is odd if f (x) = f ( x) for all x in its domain. The graph of an even functions is symmetric with respect to the y-axis, and an

9 odd function is symmetric with respect to the origin. This is an example of one of our major themes: the interplay between algebra and geometry. For each of the following functions enter E to indicate that the function is even, O to indicate it is odd, and N to indicate that is neither even nor odd.. f (x) = x 4 2. f (x) = x 2 6x 2 + 3x 6 3. f (x) = x 3 + x 5 + x 3 4. f (x) = 5x 2 3x (0 pts) set/20s2p5.pg Oddly, there is a function that is both even and odd. It is f (x) =. 42.(0 pts) set/20s2p6.pg For each of the following functions enter E to indicate that the function is even, O to indicate it is odd, and N to indicate that is neither even nor odd. f (x) = sinx. f (x) = cosx. f (x) = tanx. 43.(0 pts) set/20s2p7.pg For each of the following functions enter E to indicate that the function is even, O to indicate it is odd, and N to indicate that is neither even nor odd. f (x) = sin 2 x. f (x) = sinx 2. f (x) = sin(cosx). f (x) = sin(sinx). f (x) = sinx + cosx. 44.(0 pts) set/20s2p8.pg Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. You must get all of the answers correct to receive credit.. A function cannot be both even and odd. 2. The composition of an odd function and an odd function is even 3. The product of two even function is even 4. The composition of an even and an odd function is even 5. The ratio of two odd functions is odd 6. The product of two odd function is odd 9 7. The sum of an even and an odd function is usually neither even or odd, but it may be even. 8. The sum of two even functions is even All of the answers must be correct before you get credit for the problem. 45.(0 pts) set/20s2p9.pg The next few problems concern the composition of functions, and the fact that this is different from the multiplication of functions. If f and g are two functions then their product f g is defined by ( f g)(x) = f (x)g(x). As usual, the absence of an arithmetic operator denotes multiplication. The composition f g of these two functions is quite different. We evaluate first one, and then the other. Thus ( f g)(x) = f ( g(x) ). Let f (x) = 3x + 3 and g(x) = 7x 2 + 3x. ( f g)(4) = 46.(0 pts) set/20s2p0.pg Let f (x) = 2x + 6 and g(x) = 6x 2 + 2x. Then ( f g)(x) = 47.(0 pts) set/20s2p.pg Let f (x) = 4x + 4 and g(x) = 6x 2 + 6x. ( f g)(8) = 48.(0 pts) set/20s2p2.pg Let f (x) = 3x + 3 and g(x) = 3x 2 + 3x. Then, ( f g)(x) = 49.(0 pts) set/20s2p3.pg Let f (x) = x + and g(x) = x+. Then ( f f )(x) =, ( f g)(x) =, (g f )(x) =, (g g)(x) =. 50.(0 pts) set/20s2p4.pg Let g(x) = x + and h(x) = ( f g)(x) = x 2 + 2x +. Then f (x) =, and (g f )(x) =.

10 5.(0 pts) set/20s2p5.pg Let g(x) = x 2 and h(x) = ( f g)(x) = sinx 2. Then f (x) =, and (g f )(x) =. 52.(0 pts) set/20s2p6.pg The next few questions provide another variation of the interplay between algebra and geometry. Simple algebraic modifications have simple effects on the graph. Adding to x shifts the graph left or right, adding to y shifts it up or down, multiplying x rescales it horizontally, and multiplying y rescales it vertically. These effects can of course be combined. Relative to the graph of y = sin(x) the graphs of the following equations have been changed in what way?. y = sin(x)/8 2. y = sin(x + 8) 3. y = sin(x) y = 8sin(x) A. shifted 8 units left B. stretched vertically by the factor 8 C. compressed vertically by the factor 8 D. shifted 8 units up 53.(0 pts) set/20s2p7.pg Relative to the graph of y = x 2 the graphs of the following equations have been changed in what way?. y = (x/4) 2 2. y = x y = 4x 2 4. y = (x 2 )/4 A. shifted 4 units up B. compressed vertically by the factor 4 C. stretched vertically by the factor 4 D. stretched horizontally by the factor 4 54.(0 pts) set/20s2p8.pg Let f (x) = x 2 + sinx and let g(x) = where g is the function whose graph has been obtained from that of f by shifting it 2 to the right and 6 up. 55.(0 pts) set/20s2p9.pg The function f (x) = x 2 +0x +32 can be obtained from an even function g by shifting its graph horizontally and vertically. That even function is g(x) =. Its graph has been shifted by to the left and up. 56.(0 pts) set/20s2p20.pg Relative to the graph of y = x 3 the graphs of the following equations have been changed in what way?. y = (x) 3 / y = 7x 3 3. y = x y = (x 3 )/7 A. stretched vertically by the factor 7 B. shifted 7 units down C. compressed vertically by the factor 7 D. stretched horizontally by the factor 7 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 0

11 Math Summer 2004 lim x 4 x 4 x 2 =. Peter Alfeld. WeBWorK assignment number 2. due 9/3/06 at :59 PM. This problem set is on limits and derivatives, and some applications. Peter Alfeld, JWB 27, (0 pts) set2/20s2p2.pg Evaluate the limit x 3 lim x 3 x 2 + 4x 2 =. 2.(0 pts) set2/20s2p22.pg Evaluate the limit s 3 s lim s s 2 =. 3.(0 pts) set2/20s2p23.pg Evaluate the limit 8 a lim a 8 9 a =. 4.(0 pts) set2/20s2p24.pg y 3 Evaluate the limit lim y 3 y 3 =. 5.(0 pts) set2/20s2p25.pg 7x 2 3x + 5 Evaluate the limit lim = x. x 7 6.(0 pts) set2/20s3p2.pg x + Evaluate lim x 3 x + 2 =. 7.(0 pts) set2/20s3p3.pg x 2 x + 30 lim =. x 5 x 5 8.(0 pts) set2/20s3p4.pg x 2 7x + 0 lim =. x 2 x 2 9.(0 pts) set2/20s3p5.pg 0.(0 pts) set2/20s3p6.pg x 4 lim x lim x x + 4 =. x 4 x =..(0 pts) set2/20s2p26.pg (x + h) 3 x 3 lim =. h 0 h (Your answer will be a mathematical expression in x.) 2.(0 pts) set2/20s2p27.pg Let Then lim h 0 f (x + h) f (x) h f (x) = x 2 2x (0 pts) set2/20s2p28.pg Let Then lim h 0 f (x + h) f (x) h =. f (x) = 4x 2 9x + 5. =. 4.(0 pts) set2/20s2p29.pg Recall that lim x c f (x) = L means: For all ε > 0 there is a δ > 0 such that for all x satisfying 0 < x c < δ we have that f (x) L < ε. What if the limit does not equal L? Think about what the means in ε,δ language. Consider the following phrases:. ε > 0 2. δ > < x c < δ 4. f (x) L > ε 5. but 6. such that for all 7. there is some 8. there is some x such that

12 Order these statements so that they form a rigorous assertion that lim x c f (x) L and enter their reference numbers in the appropriate sequence in these boxes: 5.(0 pts) set2/20s2p30.pg This exercise will help you review some basic trigonometry on a right triangle. Recall that, by the Pythagorean Theorem, a triangle with sides a. b and c has a right angle opposite the (longest) side c if and only if a 2 + b 2 = c 2. This is certainly true for the triangle in this problem. If necessary, review the definitions of the trigonometric functions and their inverses to solve this problem. Consider the familiar right triangle with sides of lengths 3, 4, and 5 feet. Let A be the angle opposite the side of length 3, and B the angle opposite the side of length 4 feet. A = radians = degrees, and B = radians = degrees. 6.(0 pts) set2/20s3p.pg Many times in this class we will solve quadratic equations. You may be used to applying the quadratic formula, and that s fine, but of course the variables a, b, and c may occur in your quadratic equations in different ways than they are used in the quadratic formula. This problem illustrates the translation from formula to problem, that may be necessary. There are two solutions of the equation bx 2 cx + a 2 = 0 (where a, b, and c are constants, and x is the unknown). They differ by the sign of the square root. Enter the one with the plus sign here. 7.(0 pts) set2/20s3p0.pg If you toss a rock at an initial height H with an initial velocity V then its height h(t) after t seconds is given by the formula h(t) = 2 gt2 +Vt + H 2 where g = 32 feet second 2 on Earth. (On other celestial bodies g would be different. The minus sign is due to the convention that the positive direction is up and gravity pulls down.) You have probably seen this formula before, and chances are you were simply told that this is the way it is. With Calculus, we can make perfect sense of this formula. The underlying observation is the experimentally observed fact that, ignoring air resistance, on Earth a free falling object increases its speed by 32 feet per second every second. The velocity of the object is the derivative of height, and the acceleration is the derivative of velocity. So we have to work out that the formula given above has the right properties. If h is given by the above expression, then h(0) =. The velocity v(t) is v(t) = h (t) = and v(0) =. Moreover, the acceleration is v (t) =. 8.(0 pts) set2/20s3p.pg The point P(5,36) lies on the curve y = x 2 + x + 6. If Q is the point (x,x 2 + x + 6), find the slope of the secant line PQ for the following values of x. If x = 5., the slope of PQ is: and if x = 5.0, the slope of PQ is: and if x = 4.9, the slope of PQ is: and if x = 4.99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P(5,36). 9.(0 pts) set2/20s3p2.pg If a ball is thrown straight up into the air with an initial velocity of 45 ft/s, its height in feet after t second is given by y = 45t 6t 2. Find the average velocity for the time period beginning when t = and lasting (i) 0.5 seconds (ii) 0. seconds (iii) 0.0 seconds Finally based on the above results, guess what the instantaneous velocity of the ball is when t =.

13 20.(0 pts) set2/20s3p3.pg This a simple exercise in computing derivatives of polynomials. The derivative of is p (x) =. The derivative of p(x) = 9x 2 + 3x + 8 q(x) = 5x 5 9x 4 + 8x 3 6x 2 + 8x + 3 is q (x) =. 2.(0 pts) set2/20s3p4.pg More derivatives: The derivative of is p (x) =. The derivative of p(x) = (2x ) 2 q(x) = (2x ) 3 is q (x) =. 22.(0 pts) set2/20s3p5.pg If f (x) = 7x + 7, then f ( 7)=. 23.(0 pts) set2/20s3p6.pg If f (x) = 4 + 7x 5x 2 then f ( ) =. 24.(0 pts) set2/20s3p7.pg If f (x) = 2x 2 4x 36, then f (x) =, and f (3)=. 25.(0 pts) set2/20s3p8.pg Let f (x) = x Use the limit definition of the derivative on page 07 to find (i) f ( 3) = (ii) f ( ) = (iii) f (3) = (iv) f (4) = To avoid calculating four separate limits, I suggest that you evaluate the derivative at the point when x = a. Once you have the derivative, you can just plug in those four values for a to get the answers (0 pts) set2/20s3p9.pg This problem will help you practice computing tangents in the next problem. Let f (x) = x 2. Then f (x) =. The tangent to the graph of f through the point (, ) has the slope and the y-intercept. It intercepts the x-axis at x =. 27.(0 pts) set2/20s3p20.pg The slope of the tangent line to the parabola y = 4x 2 + 3x + 4 at the point (2,26) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 28.(0 pts) set2/20s3p2.pg The slope of the tangent line to the curve y = 3x 3 at the point ( 2, 24) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 29.(0 pts) set2/20s3p22.pg This problem is our first introduction to Newton s Method for the solution of nonlinear equations. The positive solution of f (x) = x 2 2 = 0 is obviously x = 2. How might one calculate a numerical value of 2? The idea of Newton s method is to pick a guess x 0, compute the tangent to the graph of f at ( x 0, f (x 0 ) ), and then replace the guess x 0 with x which is the x intercept of the tangent. (You should draw a picture of this idea.) Suppose x 0 =.5. The tangent to the graph of f at the point (x 0, f (x 0 )) can be written in slope intercept form as y = x+. The tangent intercepts the x-axis at x =. Now let s repeat the process. The tangent to the graph of f at the point (x, f (x )) can be written in slope intercept form as y = x+. The tangent intercepts the x-axis at x 2 =.

14 Note the accuracy of this approximation: 2 x2 =. Note: To obtain the same result in your last answer as ww you need to compute all intermediate answers to the full accuracy of your calculator. To accomplish this store all intermediate results in your calculator. Do not copy them on paper and then reenter them by hand later. Doing so introduces rounding errors and compromises the accuracy of your calculations and solutions. In fact, you should make a habit of this every time you use your calculator: store intermediate results and avoid having to reenter them. Most calculators keep more digits internally than they can display, so losing accuracy by copying and reentering is inevitable. 30.(0 pts) set2/20s3p23.pg A particle moves along a straight line and its position at time t is given by s(t) = 2t 3 27t t where s is measured in feet and t in seconds. Find the velocity (in ft/sec) of the particle at time t = 0: The particle stops moving (i.e. is in a rest) twice, once when t = A and again when t = B where A < B. A is and B is What is the position of the particle at time 8? Finally, what is the TOTAL distance the particle travels between time 0 and time 8? 3.(0 pts) set2/20s3p24.pg You toss a rock up vertically at an initial speed of 36 feet per second and release it at an initial height of 6 feet. The rock will remain in the air for seconds. It will reach a maximum height of feet after seconds. Note: Ignore air resistance. 32.(0 pts) set2/20s3p25.pg This is the general version of the previous problem. Again, ignore air resistance. Assume a free falling object accelerates at g feet per second. (On Earth, of course, g = 32.) Your answers will be mathematical expressions involving g, V, and H. You toss a rock up vertically at an initial speed of V feet per second and release it at an initial height of H feet. The rock will remain in the air for seconds. 4 It will reach a maximum height of after seconds. feet 33.(0 pts) set2/20s3p26.pg Let s now combine our rock tossing with a horizontal motion. This kind of problem can be handled by considering the vertical and horizontal motions separately. The horizontal velocity is constant. (Again, we ignore air resistance.) Recall from your study of trigonometry that if you release a rock at a speed v in a direction that makes an angle α with the horizontal, then the initial vertical velocity v v and the horizontal velocity v h are given by v v = vsinα and v h = vcosα. You shoot a rifle at an angle of 24 degrees. The bullet leaves your rifle at a height of 6 feet and a speed of 808 feet per second. It hits the ground after seconds at a distance of feet. Note: Again, ignore air resistance. When shooting a rifle this is of course a gross simplification. Even so, that s some rifle! 34.(0 pts) set2/20s3p27.pg The ideas we are practicing in these exercises can be put to practical use. My son likes to shoot arrows. He was wondering at what speed the arrow leaves the bow when it is released. So we went to the Great Salt Lake and shot the bow at an angle of 45 degrees. The arrow hit the ground at a distance of 600 feet. All of this was measured very roughly, but lets work with these figures, and, again, let s ignore air resistance. For simplicity let s also assume that the arrow is released at an initial height of 0 feet. The arrow leaves the bow at a speed of miles per hour. Note: My son and I did worry about the effect of air resistance. The arrows were sticking out of the sand where they had hit the beach. So my son shot another arrow downwards at an angle of 45 degrees straight into the sand. It penetrated to about the same depth as the arrows that had flown the distance. So they must have hit at about the same speed. We concluded that ignoring air resistance in this case was reasonable. 35.(0 pts) set2/20s3p28.pg Actually, once you get the hang of it, you ll prefer using variable rather than numbers, and putting in the

15 numbers only at the end of the calculation. So let s do the previous problem in general. However, since we are not likely to leave Earth, let s continue with g = 32 feet per second squared. (To write the solution of this problem I did very little beyond taking the solution of the previous problem and replacing 600 with d throughout.) You shoot an arrow at an angle of 45 degrees. It hits the ground at a distance of d feet. The arrow left the bow at a speed of miles per hour. 36.(0 pts) set2/20s3p29.pg If a ball is thrown vertically upward from the roof of a 64 foot tall building with a velocity of 80 ft/sec, its height after t seconds is s(t) = 64+80t 6t 2. What is the maximum height the ball reaches? What is the velocity of the ball when it hits the ground (height 0)? 37.(0 pts) set2/ind.pg This fun little question was motivated by a discussion I had with students after class. Consider the absurd statement: All numbers are equal. Figure out what is wrong with the proof below and me your explanation. I will forward the best explanations to the class, with your name attached. Proof: Let n be a natural number. Let S n be the statement that in a set of n numbers all numbers are equal. I shall prove by induction that S n is true for all natural numbers n =,2,3,... Clearly, S is true. If a set contains just one number then all numbers in that set are equal. I now need to show that the truth of S k implies the truth of S k+. (Note: In class I used n instead of k. I m hoping that k is clearer. Using k instead of n is not the problem.) Let M = {a,a 2,...,a k+ } be a set of k + numbers. Then the subset M = {a,a 2,...,a k } of M contains k numbers, and by the induction hypothesis these are all equal. Similarly, the subset M 2 = {a 2,a 3,...,a k+ } of M contains k numbers all of which are equal. In other words, we have a = a 2 =... = a k a 2 =... = a k = a k+ Together these equations imply that a = a 2 =... = a k = a k+ Thus all the numbers in M are equal. QED To get credit for this ww question, type the sentence I read this, without the quotation marks, here:. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 5

16 Math Summer 2004 Peter Alfeld. WeBWorK assignment number 3. due 9/20/06 at :59 PM. This problem set is on limits and derivatives, and some applications. Peter Alfeld, JWB 27, (0 pts) set3/20s4p5.pg If f (x) = (x 2 + 5x + 4) 4 then f (x) = 2.(0 pts) set3/20s4p7.pg If f (x) = (x 3 + 5x + 6) 2 then f (x) = 3.(0 pts) set3/20s4p8.pg If f (x) = (4x + 4) 2 then f (x) = and f (5) =. 4.(0 pts) set3/20s4p.pg If f (x) = 4x + 6 then f (x) =. 5.(0 pts) set3/20s4p3.pg If f (x) = 3x 2 + 3x + 7 then f (x) = and f (5) =. 7.(0 pts) set3/20s4p20.pg A space traveler is moving from left to right along the curve y = x 2. When she shuts off the engines, she will continue traveling along the tangent line at the point where she is at that time. At what point (x,y) = (, ) should she shut off the engines in order to reach the point (4,5)? If she was traveling from right to left she would have to shut off the engines at the point (x,y) = (, ). 8.(0 pts) set3/20s4p22.pg As discussed in class (and in the textbook in section.4), to solve the equation f (x) = 0 by Newton s Method we start with a good initial guess x 0 and then run the iteration x n+ = x n f (x n) f (x n ), n = 0,,2,... until we get an approximation x n+ that is good enough for our purposes. Suppose you want to compute the cube root of 4 by solving the equation x 3 4 = 0. Since 3 = and 2 3 = 8 Let s start with Then x =, x 2 =, x 0 =.5 x 3 =, and To check your answer compute x 3 3 =. Enter x, x 2 and x 3 with at least 6 correct digits beyond the decimal point. 6.(0 pts) set3/20s4p9.pg If 9.(0 pts) set3/20s4p23.pg f (x) = 8x4 The equation x 0(x )(x 2)(x 3) = then f (4) (x) =. has three real solutions Note: There is a way of doing this problem without using the quotient rule 4 times. a < b < c

17 where a =, b =, and c =. Enter your answers with at least six correct digits beyond the decimal point. 0.(0 pts) set3/20s4p24.pg Newton s Method will converge to a true solution if you have a good initial approximation. If you don t it may not converge at all. Consider, for example, the equation f (x) = x(x )(x + ) = x 3 x = 0. Obviously, the solutions are x =, 0,. If we start Newton s Method with x 0 being close to one of these solutions we will get convergence to that solution. On the other hand, note that is zero when f (x) = 3x 2 x = ± 3. Thus the tangent will be horizontal in those two cases, and Newton s can t even be carried out. In this problem we ll investigate what happens in the contrived case that x 0 = 5 5. You can enter x 0 into WeBWorK as sqrt(5)/5. Try it: x 0 =, Now do your computations using exact arithmetic, and you ll recognize a pattern: x =, x 2 =, x 3 =, x 4 =, Draw a picture to see what s going on. Note, however, that Newton s method may fail in many different ways. A detailed analysis of Newton s method and related methods is a huge subject and well beyond the scope of our class..(0 pts) set3/20s4p25.pg As discussed in class, there may be more variables floating around than the one with respect to which we differentiate. In that situation it is important to be aware with respect to which variable we differentiate. For example, the volume of a square box width s and height h is given Thus D s V = and D h V =. V = s 2 h. 2.(0 pts) set3/20s4p26.pg The total resistance R of two parallel resistors S and T is given by R = ST S + T. Thus D S R = and D T R =. 3.(0 pts) set3/20s4p27.pg This problem is in preparation for the next two problems. Suppose you have a cube of length s. The volume of that cube is V = s 3. Now let s suppose the dimensions of that cube (and hence its volume) depend on time. We are wondering about the relationship between the growth of the length versus the growth of the volume. Suppose s(t) = t. Then s (t) = and V (t) =. Next, suppose V(t) = t. Then s (t) = and V (t) =. 4.(0 pts) set3/20s4p28.pg The radius of a spherical watermelon is growing at a constant rate of 2 centimeters per week. The thickness of the rind is always one tenth of the radius. The volume of the rind is growing at the rate cubic centimeters per week at the end of the fifth week. Assume that the radius is initially zero. 5.(0 pts) set3/20s4p29.pg Your neighbor is growing a slightly different watermelon. It also has a rind whose thickness is one tenth of the radius of that watermelon. However, 2

18 the rind of your neighbor s water melons grows at a constant rate of 20 cubic centimeters a week. The radius of your neighbor s watermelon after 5 weeks is and at that time it is growing at centimeters per week. 6.(0 pts) set3/20s5p.pg Let f (x) = x(x 2 + 4) 5. f (x) =. f (x) =. 7.(0 pts) set3/20s5p6.pg Compute D x x =, D 2 xx 2 =, D 3 x x3 =, D 4 x x4 =, D 5 x x5 =, D 6 x x6 =. Do you recognize a pattern? Compute D 6 x(2x 6 + 3x 5 + 4x 4 + 5x 3 + 7x 2 5x + π) = and D 5 x (2x + )5 =. 8.(0 pts) set3/20s5p7.pg Consider these statements written in ordinary language: A The speed of the car is proportional to the distance it has traveled. B The car is speeding up. C The car is slowing down. D The car always travels the same distance in the same time interval. E We are driving backwards. F Our acceleration is decreasing. Denoting by s(t) the distance covered by the car at time t, and letting k denote a constant, match these statements with the following mathematical statements by entering the letters A through E on the appropriate boxes: s < 0. s is constant s < 0 s < 0 s > 0 s = ks 9.(0 pts) set3/20s5p8.pg Find the slope of the tangent line to the curve at the point (7,6). Your answer: 4x 2 5xy 2y 3 = (0 pts) set3/20s5p9.pg suppose 5x 2 +4x+xy = 4 and y(4) = 23. Find y (4) by implicit differentiation. Your answer: 2.(0 pts) set3/20s5p.pg Let A be the area of a circle with radius r. If dr dt = 2, find da dt when r = 4. Your answer: 22.(0 pts) set3/20s5p2.pg Use implicit differentiation to find the equation of the tangent line to the curve xy 3 + xy = 0 at the point (5,). The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 23.(0 pts) set3/20s5p3.pg Suppose 64 x2 + y2 25 = and y(3) = Find y (3) by implicit differentiation. 24.(0 pts) set3/20s5p4.pg Find the slope of the tangent line to the curve (a lemniscate) 2(x 2 + y 2 ) 2 = 25(x 2 y 2 ) at the point ( 3, ). m = 25.(0 pts) set3/20s5p5.pg Suppose x + y = 0 and y() = 8. Find y () by implicit differentiation. 26.(0 pts) set3/20s5p6.pg Let xy = and let Find dx dt when x =. dy dt = 4 3

19 27.(0 pts) set3/20s5p7.pg The graph of the equation x 2 + xy + y 2 = 9 is a slanted ellipse. Think of y as a function of x. Differentiating implicitly and solving for y gives: y =. (Your answer will depend on x and y.) The ellipse has two horizontal tangents. The upper one has the equation y =. The right most vertical tangent has the equation x =. That tangent touches the ellipse where y =. 28.(0 pts) set3/20s5p8.pg A street light is at the top of a 3.5 ft. tall pole. A man 5.4 ft tall walks away from the pole with a speed of 4.5 feet/sec along a straight path. How fast is the tip of his shadow moving when he is 35 feet from the pole? Your answer: 29.(0 pts) set3/20s5p9.pg The altitude of a triangle is increasing at a rate of.5 centimeters/minute while the area of the triangle is increasing at a rate of 4.5 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 7.5 centimeters and the area is 82.0 square centimeters? Your answer: 30.(0 pts) set3/20s5p20.pg Gravel is being dumped from a conveyor belt at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other. How fast is the height of the pile increasing when the pile is 2 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V = π 3 r2 h. Your answer: feet per minute. 3.(0 pts) set3/20s5p2.pg Water is leaking out of an inverted conical tank at a rate of 6200 cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate. The tank has height 5 meters and the diameter at the top is 4.5 meters. If the water level 4 is rising at a rate of 6 centimeters per minute when the height of the water is.5 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. Your answer: cubic centimeters per minute. 32.(0 pts) set3/20s5p23.pg A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 7 cm. Your answer (cubic centimeters per minute) should be a positive number. 33.(0 pts) set3/20s5p24.pg You are blowing air into a spherical balloon at a rate of 4π 3 cubic inches per second. (The reason for this strange looking rate is that it will simplify your algebra a little.) Assume the radius of your balloon is zero at time zero. Let r(t), A(t), and V (t) denote the radius, the surface area, and the volume of your balloon at time t, respectively. (Assume the thickness of the skin is zero.) All of your answers below are expressions in t: r (t) = inches per second, A (t) = square inches per second, and V (t) = cubic inches per second. 34.(0 pts) set3/20s5p25.pg You may want to solve this problem only after you solve the next one, which is the general version of this one. On the other hand, you also may want to solve this one first, and when solving the general one compare your evolving solution against your results in this problem. A child is flying a kite. If the kite is 85 feet above the child s hand level and the wind is blowing it on a horizontal course at 5 feet per second, the child is paying out cord at feet per second when 95 feet of cord are out. Assume that the cord remains straight from hand to kite. (If you have ever flown a kite you know that this is an unrealistic assumption.) For a solution of this problem (after the set closes) see the next problem. 35.(0 pts) set3/20s5p26.pg A child is flying a kite. If the kite is h feet above the child s hand level and the wind is blowing it on

20 a horizontal course at v feet per second, the child is paying out cord at feet per second when s feet of cord are out. Assume that the cord remains straight from hand to kite. 36.(0 pts) set3/20s5p29.pg You say goodbye to your friend at the intersection of two perpendicular roads. At time t = 0 you drive off North at a (constant) speed v and your friend drives West at a (constant) speed w. You badly want to know: how fast is the distance between you and your friend increasing at time t? Enter here the derivative of the distance from your friend with respect to t: Being scientifically minded you ask yourself how does the speed of separation change with time. In other words, what is the second derivative of the distance between you and your friend? Suppose that after your friend takes off (at time t = 0) you linger for an hour to contemplate the spot on which he or she was standing. After that hour you drive off too (to the North). How fast is the distance between you and your friend increasing at time t (greater than one hour)? Again, you ask what is the second derivative of your separation: If you wish me your comments on how lingering makes things harder, mathematically speaking. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 5

21 Math Summer 2004 Peter Alfeld. WeBWorK assignment number 4. due 9/27/06 at :59 PM. This problem set is on antiderivatives and minimization and maximization problems. Peter Alfeld, JWB 27, (0 pts) set4/20s6p3.pg d ( dx ax 2 + bx + c ) =. 2.(0 pts) set4/20s6p4.pg Suppose f (t) = t 2 +. Then f (t) =. f (t) =. f (t) =. 3.(0 pts) set4/20s6p5.pg Suppose you want to compute the fifth root of 6 by solving the equation f (x) = x 5 6 = 0 ( ) using Newton s method. Newton s method starts with an initial approximation x 0 and then computes a sequence of approximations x, x 2, x 3,... via the formula x k+ = g(x k ), k = 0,,2,... where g(x) = x f (x) f (x). For the function defined above in ( ), g(x) =. Letting x 0 = you obtain x =, x 2 =, and x 3 =. 4.(0 pts) set4/20s6p7.pg For what values of x does the graph of f (x) = 0x 3 30x 2 90x + 0 have a horizontal tangent? Enter the x values in order, smallest first, to 4 places of accuracy: x = x 2 = 5.(0 pts) set4/20s6p8.pg The function f (x) = 2x 3 + 2x 2 36x + 7 is increasing on the interval (, ). It is decreasing on the interval (, ) and the interval (, ). The function has a local maximum at. 6.(0 pts) set4/20s6p9.pg Find the point on the line 3x + 3y 6 = 0 which is closest to the point ( 4, 7). (, ) 7.(0 pts) set4/20s6p0.pg A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 0 x 2. What are the dimensions of such a rectangle with the greatest possible area? Width = Height = 8.(0 pts) set4/20s6p.pg A cylinder is inscribed in a right circular cone of height 2 and radius (at the base) equal to 6. What are the dimensions of such a cylinder which has maximum volume? Radius = Height = 9.(0 pts) set4/20s6p2.pg A fence 6 feet tall runs parallel to a tall building at a distance of 2 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? 0.(0 pts) set4/20s6p3.pg If 2300 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = cubic centimeters..(0 pts) set4/20s6p4.pg The function f (x) = 4x + 2x has one local minimum and one local maximum.

22 It is helpful to make a rough sketch of the graph to see what is happening. This function has a local minimum at x equals with value and a local maximum at x equals with value 2.(0 pts) set4/20s6p6.pg A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 25 feet? 3.(0 pts) set4/20s6p7.pg A rancher wants to fence in an area of square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use? 4.(0 pts) set4/20s6p8.pg A University of Rochester student decided to depart from Earth after his graduation to find work on Mars. Before building a shuttle, he conducted careful calculations. A model for the velocity of the shuttle, from liftoff at t = 0 s until the solid rocket boosters were jettisoned at t = 00 s, is given by v(t) = t t t (in feet per second). Using this model, estimate the absolute maximum value and absolute minimum value of the acceleration of the shuttle between liftoff and the jettisoning of the boosters. 5.(0 pts) set4/20s6p9.pg You are going to make many cylindrical cans. The cans will hold different volumes. But you d like them all to be such that the amount of sheet metal used for the cans is as small as possible, subject to the can holding the specific volume. How do you choose the ratio of diameter to height of the can? Assume that the thickness of the wall, top, and bottom of the can is everywhere the same, and that you can ignore the material needed for example to join the top to the wall. Put differently, you ask what ratio of diameter to height will minimize the area of a cylinder with a given volume? That ratio equals. 2 6.(0 pts) set4/20s6p20.pg It takes a certain power P to keep a plane moving along at a speed v. The power needs to overcome air drag which increases as the speed increases, and it needs to keep the plane in the air which gets harder as the speed decreases. So assume the power required is given by P = cv 2 + d v 2 where c and d are positive constants. (They depend on your plane, your altitude, and the weather, among other things.) Enter here the choice of v that will minimize the power required to keep moving at speed v. Suppose you have a certain amount of fuel and the fuel flow required to deliver a certain power is proportional to to that power. What is the speed v that will maximize your range (i.e., the distance you can travel at that speed before your fuel runs out)? Enter your speed here Finally, enter here the ratio of the speed that maximizes the distance and the speed that minimizes the required power. 7.(0 pts) set4/20s6p2.pg This is a related rates problem with a twist. Suppose you have a street light at a height H. You drop a rock vertically so that it hits the ground at a distance d from the street light. Denote the height of the rock by h. The shadow of the rock moves along the ground. Let s denote the distance of the shadow from the point where the rock impacts the ground. Of

23 course, s and h are both functions of time. To enter your answer into WeBWorK use the notation v to denote h : v = h. Then the speed of the shadow at any time while the rock is in the air is given by s = (where s is an expression depending on h, s, H, and v (You will find that d drops out of your calculation.) Now consider the time at which the rock hits the ground. At that time h = s = 0. The speed of the shadow at that time is s = where your answer is an expression depending on H, v, and d. 8.(0 pts) set4/20s7p3.pg Consider the function f (x) = 2x x 4 00x f (x) has inflection points at (reading from left to right) x = D, E, and F where D is and E is and F is For each of the following intervals, tell whether f (x) is concave up (type in CU) or concave down (type in CD). (,D]: [D,E]: [E,F]: [F, ): 9.(0 pts) set4/20s7p4.pg A right circular cone is to be inscribed in another right circular cone of given volume, with the same axis and with the vertex of the inner cone touching the base of the outer cone. Draw a picture of the cones. What must be the ratio of their altitudes for the inscribed cone to have maximum volume? 20.(0 pts) set4/20s7p5.pg I have enough pure silver to coat 3 square meter of surface area. I plan to coat a sphere and a cube. Allowing for the possibility of all the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a maximum? The radius of the sphere is, 3 and the length of the sides of the cube is. Again, allowing for the possibility of all the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a minimum? The radius of the sphere is, and the length of the sides of the cube is. 2.(0 pts) set4/20s7p6.pg One end of a ladder of length L rests on the ground and the other end rests on the top of a wall of height h, as illustrated in the Figure on this page. As the bottom end is pushed along the ground towards the wall, the top end extends beyond the wall. The value of x that maximizes the horizontal overhang s is x =. (Your answer will depend on L and h.) In the particular case that L = 6 and h = 7 this value is x =. The corresponding numerical value of s =. 22.(0 pts) set4/20s7p7.pg The illumination at a point is inversely proportional to the square of the distance of the point from the light source and directly proportional to the intensity of the light source. Suppose two light sources are s feet apart and their intensities are I and J, respectively. Let P be the point between them where the sum of their illuminations is a minimum. The distance of P from light source I will be feet. (Your answer will depend on I, J, and s.)