Math Summer Peter Alfeld. WeBWorK assignment number 0. due 5/21/03 at 11:00 PM. Notice the buttons on this page and try them out before

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1 Math 0- Summer 003 Peter Alfeld. WeBWorK assignment number 0. due 5//03 at :00 PM. The main purpose of this first WeBWorK set is to to help you familiarize yourself with WeBWorK. You don t have to do the set for credit so don t worry if you already know WeBWorK or you have a late start in this class. 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 7 or 36, ( pt) set0/q0.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, but formatted text is a little faster. Very occasionally the output from typeset gets corrupted. IN that case you can send me an and I will fix it. You can also see the intended output, if not quite as nicely, using formated text. The options of plain text and typeset are not useful. Logout terminates this WeBWorK session for you. You can of course log back in and continue.

2 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 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 4857), 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. Now for the meat of this problem. Notice that the answer window is extra large so you can try the things suggested above. 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/b versus a//b versus a/(b) a/b+c versus a/(b+c) a+b** versus (a+b)** sqrt a+b versus sqrt(a+b) 4/3 pi r** versus (4/3) pi r** (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..( pt) set0/prob.pg This problem demonstrates how you enter numerical answers into WeBWorK. Evaluate the expression : 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 (-9) * (): 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 denote multiplication. You can enter 9 or -99 or even 9. 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, ). You can enter this as sin(pi/4), as

3 sin(3.4596/4), as /sqrt(), as **(-.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 sin 3π. You can enter this as *sin(3*pi/) or more simply as sin(3pi/). Try it: Sometimes you need to use ( ) s to make your meaning clear. E.g. /+3 is 3.5, but /(+3) is. 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 strict 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.( pt) set0/proba.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/) and WeBWorK will accept this or any other function equal to sin(x), e.g. sin(x) +sin(x)**+cos(x)**- 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(x) 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 x means first apply sin which gives sin() and then multiple by x. Try it. 3 Now enter the function cost. Note this is a function of t and not x. Try entering cos x and see what happens. 4.( pt) set0/probb.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 +*3? and what is 3? 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 inner most parentheses are done first). For example how do you enter sin 3 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, (*3)** = 6** = 36, 3**(*) = 3**4 = 8, (+3)** = 5** = 5, 3**(+) = 3**4 = 8 (Here we have used ** to denote exponentiation and you can also use this instead of a caret if you want). 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.

4 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 /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 // should mean.) WeBWorK and most other computers read things from left to right, i.e. /3*4 means (/3)*4 or 8/3, IT DOES NOT MEAN /. Some computers may do operations from right to left. If you want /(3*4) = /, you have to use parentheses. The same thing happens with addition and subtraction. -3+ = 0 but -(3+) = -4. This is one case where using parentheses even if they are not needed might be a good idea, e.g. write (/3)*4 even though you could write /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 /3*4 and use the Preview button to see what you get. 5.( pt) set0/prob.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. You must get all of the answers correct to receive credit 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 won t tell you which ones are wrong. The idea is encourage you think rather than to just try guessing. In every case all of the answers must be correct before you get credit for the problem. 4 6.( pt) set0/prob3.pg This problem demonstrates a WeBWorK Matching question. Match the statements defined below with the letters labeling their equivalent expressions. You must get all of the answers correct to receive credit.. x is greater than -8. x is any real number 3. x is less than The distance from x to -8 is less than or equal to 5. The distance from x to -8 is more than A. x 8 B. x 8 C. x D. 8 x E. x 8 For this problem WeBWorK only tells you that all your answers are correct or that at least one is wrong. This makes the problem harder and is usually used only for T/F and matching questions. The idea is encourage you to think rather than just try guessing. 7.( pt) set0/prob4/prob4.pg This problem demonstrates a WeBWorK problem involving graphics. The simplest functions are linear (or affine) functions the functions whose graphs are a straight line. They are important because many functions (the so-called differentiable functions) locally look like straight lines. ( locally means that if we zoom in and look at the function at very powerful magnification it will look like a straight line.) Enter the letter of the graph of the function which corresponds to each statement.. The graph of the line is increasing. The graph of the line is decreasing 3. The graph of the line is constant 4. The graph of the line is not the graph of a function A B C D

5 This is another problem where you aren t told if some of your answers are right. (With matching questions and true false questions, this is the standard behavior otherwise it is too easy to guess your way to the answer without learning anything.) If you are having a hard time seeing the picture clearly, click on the picture. It will expand to a larger picture on its own page so that you can inspect it more closely. 8.( pt) set0/prob5.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. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 5

6 Math 0- Summer 003 Peter Alfeld. WeBWorK assignment number. due 5/8/03 at :00 PM. This is the first of 0 regular home works. It covers some of the prerequisites of this class, specifically: Solving linear and quadratic equations. Manipulating powers. The geometry and algebra of straight lines. The distance between two points. A few word problems. Enabling you to solve word problems is the reason why the system requires you to take Calculus! Note that some of the problems come with hints or solutions. A hint becomes available after you submit your first answer. To see it click on the new button Show Hint and submit an answer again. Solutions become available after the set closes. To see it go back to the problem, click on the new button Show Solution and, once again, click on Submit Answer. Peter Alfeld, JWB 7, (0 pts) set/prob3.pg Solve the equation 3 x 4 5 x for x..(0 pts) set/prob4.pg Solve the equation for t 3 t 3.(0 pts) set/prob5.pg 3 t 9 t 0 The equation x 4 4x 3 x 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/prob6.pg Evaluate the expression 5 4! 3. (You may enter a fraction as your answer.) 5.(0 pts) set/prob7.pg The expression 3a b 4 c 5 a 5 b 4 c 4 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: and finally t, the exponent of c, is: 6.(0 pts) set/prob8.pg Consider the two points 5" and 8" 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/prob9.pg Find the distance between (5, 4) and (0, -). 8.(0 pts) set/prob0.pg Find the perimeter of the triangle with the vertices at (5, ), (-, ), and (-4, -6). 9.(0 pts) set/prob.pg The equation of the line with slope that goes through the point 6" 6 can be written in the form y mx b where m is: and where b is: 0.(0 pts) set/prob.pg This problem is like the preceding problem. The equation of the line with slope that goes through the point 3 3" 7 can be written in the form y mx b where m is: and where b is:.(0 pts) set/prob3.pg The equation of the line that goes through the point " 4 and is parallel to the x-axis can be written in the form y mx b where m is: and where b is:.(0 pts) set/prob4.pg The equation of the line that goes through the point 6" 0 and is parallel to the line 5x y 3 can be written in the form y mx b where m is: and where b is: 3.(0 pts) set/prob5.pg The equation of the line that goes through the point

7 9" 6 and is perpendicular to the line x 4y 4 can be written in the form y mx b where m is: and where b is: 4.(0 pts) set/prob6.pg A line through (-, -9) with a slope of 4 has a y- intercept at 5.(0 pts) set/prob7.pg An equation of a line through (0, 3) which is parallel to the line y x has slope: and y intercept at: 6.(0 pts) set/prob8.pg Find the slope of the line through (0, -5) and (-7, ). 7.(0 pts) set/prob9.pg The distance of the point " 4 from the line y x is: 8.(0 pts) set/prob0.pg This is like the preceding problem. The distance of the point 5" from the line y x 4 is: 9.(0 pts) set/prob.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. 0.(0 pts) set/prob.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 g h t # t v 0 t h 0 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.).(0 pts) set/prob3.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. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

8 Math 0- Summer f x & 5x 4 3x 0 4. f x & x 4 3x 0 x 5 Peter Alfeld. WeBWorK assignment number. due 6/5/03 at :00 PM. This home work covers some additional prerequisites, in addition to our first Calculus concept, limits. Peter Alfeld, JWB 7, (0 pts) set/pi.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 inverse function of f is a straight line with slope..(0 pts) set/p.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 where the leading coefficient a n % 0 ' a x a 0 The degree of p x ( is. The degree of p x ( x x 3 is. The degree of p x (*)+ x x 3, 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 is. 3.(0 pts) set/prob.pg For each of the following functions, decide whether it is even, odd, or neither. Enter E for an EVEN function, O for an ODD function and N for a function which is NEITHER even nor odd.. f x ( x 3 x 3 x 5. f x ( x 4 6x 0 3x 4.(0 pts) set/prob0.pg Oddly, there is a function that is both even and odd. It is f x (. 5.(0 pts) set/prob.pg Let f x ( 5x 4 and g x ( 3x 5x. f g - Recall that by convention the absence of an arithmetic operator denotes multiplication. 6.(0 pts) set/prob3.pg Let f x ( 5x and g x ( 5x Then f g x 7.(0 pts) set/prob4.pg. Let f x ( x 4 and g x ( 3x f g 4 8.(0 pts) set/prob5.pg. Let f x ( 5x 4 and g x ( 4x Then, f g x 9.(0 pts) set/prob6.pg Relative to the graph of y sin x 4x. x. 4x. the graphs of the following equations have been changed in what way?. y sin x 6. y sin x 7 3. y sin x 6 4. y 7sin x A. stretched vertically by the factor 7 B. compressed vertically by the factor 7 C. shifted 6 units up D. shifted 6 units left 0.(0 pts) set/prob7.pg Relative to the graph of y x 3

9 0 0 the graphs of the following equations have been changed in what way?. y x 3 3. y x y / x y / x 3 3 A. compressed vertically by the factor 3 B. shifted 3 units up C. shifted 3 units down D. shifted 3 units left.(0 pts) set/prob3.pg Let f x ( x sinx and let g x ( where g is the function whose graph has been obtained from that of f by shifting it 4 to the right and 9 up..(0 pts) set/prob4.pg The function f x $ x 4x 0 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. 3.(0 pts) set/prob8.pg Relative to the graph of y x the graphs of the following equations have been changed in what way?. y x 9. y x 9 3. y 9x 4. y 8x A. shifted 9 units left B. compressed vertically by the factor 9 C. compressed horizontally by the factor 9 D. shifted 9 units right 4.(0 pts) set/prob9.pg This question is modeled after problems 7 and 8 of the first home work set. 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 3 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.) 5.(0 pts) set/prob0.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.. The ratio of two odd functions is odd. The sum of an even and an odd function is usually neither even or odd, but it may be even. 3. The sum of two even functions is even 4. The product of two even function is even 5. The product of two odd function is odd 6. A function cannot be both even and odd. 7. The composition of an even and an odd function is even 8. The composition of an odd function and an odd function is even All of the answers must be correct before you get credit for the problem. 6.(0 pts) set/prob.pg x h 3 x 3 lim h54 0 h is. 7.(0 pts) set/prob.pg Recall that lim x 4 f x & c L means: For all ε 0 there is a δ 0 such that for all x satisfying 0 # x c5 δ we have that f x 6 L7 ε.

10 What if the limit does not equal L? Think about what the means in ε" δ language. Consider the following phrases:. ε 0. 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 Order these statements so that they form a rigorous assertion that lim x54 f x 8% L c and enter their reference numbers in the appropriate sequence in these boxes: 8.(0 pts) set/s 3 8.pg Evaluate the limit x lim x4 x 7x 8 9.(0 pts) set/s 3 9.pg Evaluate the limit s 3 s lim s4 s 0.(0 pts) set/s 3 7.pg Evaluate the limit 64 b lim b4 64.(0 pts) set/s 3 36.pg Evaluate the limit.(0 pts) set/s 3 6.pg Evaluate the limit lim x49 8 b lim a 6 a4 6 a 6 7x 8x 7 x 8 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 3

11 = Math 0- Summer 003 Peter Alfeld. WeBWorK assignment number 3. due 6//03 at :00 PM. Peter Alfeld, JWB 7, (0 pts) set3/asymp.pg The function x f x & 5x 3 x has a vertical asymptote at x asymptote with the equation y mx b and an oblique where m and b. (You can use synthetic or long division to compute the equation of that asymptote.).(0 pts) set3/bis.pg The function f x & x 3 x has a root between 0.6 and 0.8. It is:. Enter your answer with at least three digits. You can compute it using the bisection method described in class. (You might also find a more efficient way, but the bisection method is sure to work!) 3.(0 pts) set3/eps.pg In this problem we consider three functions f. Each of them is continuous at x 0, i.e., lim x54 f x & f 0 : 0 In order to show by the ε δ definition that this is true one has to give a definition of δ in terms of ε such that x 0 δ ; f x f 0 <7 ε Match these choices of δ. δ ε. δ ε 3. δ ε with the functions so that that choice of δ establishes continuity of the function (at x 0. You can use each choice only once. Enter the reference numbers of the given functions in the appropriate answer boxes. f x & x: f x & x : f x & x: 4.(0 pts) set3/s.pg The point P 5" 38 lies on the curve y x x 8. If Q is the point x" x x 8, 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" (0 pts) set3/s 5.pg If a ball is thrown straight up into the air with an initial velocity of 45 ft/s, it height in feet after t second is given by y 45t 6t. Find the average velocity for the time period begining 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. 6.(0 pts) set3/s 5 37.pg For what value of the constant c is the function f continuous on " where f x ( cx 9 if x >? " 3@ cx 9 if x >? 3" 7.(0 pts) set3/s 6.pg The slope of the tangent line to the parabola y 4x 3x 6 at the point 3" 5 is: The equation of this tangent line can be written in the form y mx b where m is: and where b is: 8.(0 pts) set3/s 6.pg The slope of the tangent line to the curve y x 3 at the point 4" 8 is: The equation of this tangent line can be written in the form y mx b where m is: and where b is:

12 9.(0 pts) set3/s 9.pg If f x ( 5x 3, find f AB. 0.(0 pts) set3/s 7.pg If f x ( 3 5x 3x, find f AB 0..(0 pts) set3/s.pg If f x ( x x 6, find f AC x. Find f AB..(0 pts) set3/s 4.pg If f x ( 7x 8 4x 5 5x 3 Find f A. x, find f AB x. 3.(0 pts) set3/s 3.pg A particle moves along a straight line and its position at time t is given by s t & t 3 4t 90t 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 6? Finally, what is the TOTAL distance the particle travels between time 0 and time 6? 4.(0 pts) set3/s 3 8.pg If a ball is thrown vertically upward from the roof of a 3 foot tall building with a velocity of ft/sec, its height after t seconds is s t $ 3 t 6t. What is the maximum height the ball reaches? What is the velocity of the ball when it hits the ground (height 0)? 5.(0 pts) set3/ur dr 0.pg Let f x $ x 3 Use the limit definition of the derivative on page 07 to find (i) f A D 7 ( (ii) f AB 4 (iii) f AB D ( (iv) f AB 0 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. 6.(0 pts) set3/well.pg This question is in preparation for a quantitative word problem that will appear on a later set. Suppose you drop a rock into the center of a very deep well with a diameter of feet, located on the equator. Suppose there is no air resistance. (Think of the well as being sealed or located at the moon.) Enter T or F for the following statements. The top of the well rotates with the Earth towards the East. The top of the well rotates with the Earth towards the West. The top of the well rotates with the Earth towards the South. The top of the well rotates with the Earth towards the North. The rock will fall straight down until it hits the center of the bottom of the well. The rock will hit the East wall of the well. The rock will hit the West wall of the well. The rock will hit the North wall of the well. The rock will hit the South wall of the well. None of these statements are true. 7.(0 pts) set3/formula.pg There are two solutions of the equation bx cx a 0 (where a, b, and c are constants, and x is the unknow). They differ by the sign of the suare root. Enter the one with the plus sign here Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

13 G G A H A 0 Math 0- Summer 003 Peter Alfeld. WeBWorK assignment number 4. due 6/8/03 at :00 PM. The exercises in this set mostly concern differentiation rules, particularly the product, quotient, and chain rules. The first problem is meant to help you prepare for Exam. The last problem is a word problem that I hope you will find intriguing. Remember that in WeBWorK (and in general unless stated otherwise) angles are measured in radians. Peter Alfeld, JWB 7 or 36, (0 pts) set4/e.pg The purpose of this problem is to help you prepare for exam. The exam will cover the definition and computation of derivatives. Remember that we define the derivative f A of a function f by f x hë f x f A x ( lim h54 0 h In class we derived a bunch of differentiation rules, three of which are non-intuitive: The Product Rule: ) f x g x, A f A x g x The Quotient Rule: f x g xïh The Chain Rule: f ) g x, f x ga x F f AB x g xë f x gac x g x f A ) g x, ga x F The exam will contain simple applications of each of these rules and one example where all three have to be applied. To prepare you should apply these rules to so many examples that you can remember and apply them with ease. The exam will be closed notes and books, and no calculators. Below are some examples of the kind of problems that will be on the exam. I recommend you do this problem a few days before the exam just you like would do the exam itself, without notes, books, or calculators. In WeBWorK you can enter the answer in any form, in the exam you should do obvious simplifications like canceling common factors in numerator and denominator, or combining powers of x. If f x & 3x 4x then f AB x (. If f x & sinx 4cosx then f A x (. The limit lim 30 hj 4 3K 4 h54 0 h defines the derivative of the function f x ( at the point x. If f x & x sinx then f A x (. cosx If f x & then f AB x (. x If f x & sin ) x, then f A x &. xtan If f x & 0 x then f AB x (. x.(0 pts) set4/p.pg Let f x ( 5cosx 5tanx f AB x 3.(0 pts) set4/s 4.pg If f x & sinx 6cosx, then f AB x & 4.(0 pts) set4/s 4 4.pg If find f AB x. f x ( sinx cosx 5.(0 pts) set4/s 5.pg If f x &/ x 4x 8 4, find f AB x. 6.(0 pts) set4/s 5.pg If f x & sin sin x, find f A x. 7.(0 pts) set4/s 5.pg If f x & cos sin x, find f A x. 8.(0 pts) set4/s 5.pg Let f x & x 3 f AB x 5x 6 3

14 9.(0 pts) set4/s 5 3.pg If f x (/ 4x 8 3, find f AB x. Find f AB. 0.(0 pts) set4/s 5 4.pg If f x ( sin x 5, find f A x..(0 pts) set4/s 5 5.pg If f x ( sin 4 x, find f AB x. Find f A 4..(0 pts) set4/s 5 6.pg If f x ( 3x 5, find f AB x. 3.(0 pts) set4/s 5 7.pg If f x ( tan3x, find f A x. 4.(0 pts) set4/s 5 8.pg Let f x &ML 3x 3x 7 f AB x f A 4 5.(0 pts) set4/s 5 9.pg If f x ( cos 4x 6, find f AB x. 6.(0 pts) set4/s 5 9a.pg Let f x $ cos 5x 4 f AB x 7.(0 pts) set4/ur dr 5 3.pg Let f x ( 8x 4 x f 0 4 x ( Note: There is a way of doing this problem without using the quotient rule 4 times. 8.(0 pts) set4/z.pg The purpose of this problem is to show pretty much all of our rules at work at once. If xsin f x ( x x find f AB x. 9.(0 pts) set4/z.pg Let f x $ Then f A x = f ANAC x = f ANANAC x = f ANANANAC x = tanx 0.(0 pts) set4/z3.pg This problem is a bit on the difficult side, but it s OK to have a challenge among routine problems. Here are a couple of hints: Think of motion as composed of a horizontal and a vertical component, and remember that velocity is the derivative of location. So if you know velocity and want to know location you have to do the opposite of differentiation: find a function of which velocity is the derivative. Suppose again you drop a rock into a deep cylindrical well that is located on the equator. Assume gravity is g, the radius of the well is r, and the radius of the earth is R. Again, for simplicity, assume that we can ignore air drag. Due to the earth s rotation, points on the wall of the well move in circular arcs. However, for the purpose of this exercise suppose that each point on the wall is moving east along a straight line at a speed that is proportional to the distance of the point from the center of earth. (This is a reasonable assumption for the kind of time frame we anticipate.) Also suppose that gravity is constant throughout the earth. (In reality it isn t, in fact it s 0 at the center since an object there gets pulled equally in all directions.) Suppose that the earth takes T seconds for one rotation. Finally, suppose you release the rock at the center of the circle that forms the cross section of the well. At what depth d and what time t will the rock strike the east wall of the well? Your answer should of course be two formulas that depend on g, r, R, and T. t and d Once you have that formula compute d and t for the particular values R 3" 963 miles" T 86" 64 seconds"

15 g 3 ft/sec " and r ft For these values you obtain t and d seconds feet. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 3

16 Math 0- Summer 003 Peter Alfeld. WeBWorK assignment number 5. due 6/5/03 at :00 PM. This home work set is focused on related rates and implicit differentiation. Peter Alfeld, , JWB 7..(0 pts) set5/language.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: sana 0 sa is constant sao 0 sananap 0 sanap 0 sa ks.(0 pts) set5/cs6p.pg Find the slope of the tangent line to the curve at the point 5" 4. x 4xy 4y (0 pts) set5/s 6.pg suppose 4x 3x xy by implicit differentiation. and y 6Q 5. Find yab 4.(0 pts) set5/s 6 3.pg Find ya by implicit differentiation. Match the expressions defining y implicitly with the letters labeling the expressions for ya.. 6sin x y & 6ycosx. 6sin x y & 6ysinx 3. 6cos x y ( 6ysinx 4. 6cos x y ( 6ycosx 0 6cos x yj 6ysin x A. 0 6cos x yj 6cos x 0 6sin x yj 6ysin x B. 6cos x 0 6sin x y C. 0 6sin x yc 6ycos x 6sinx 0 0 6sin x y 6cos D. x 0 yc 6ycos x 6cos x yj 6sin x 5.(0 pts) set5/s 6 9.pg Use implicit differentiation to find the equation of the tangent line to the curve xy 3 xy 6 at the point 3". The equation of this tangent line can be written in the form y mx b where m is: and where b is: 6.(0 pts) set5/s 6.pg Suppose 64 x y 49 by implicit differentiation. and y ( Find ya 7.(0 pts) set5/s 6 5a.pg Find the slope of the tangent line to the curve (a lemniscate) x y 5 x y at the point 3". m 8.(0 pts) set5/s 6 4.pg Suppose x y and y 6 ( 64. Find ya 6 by implicit differentiation. 9.(0 pts) set5/cs8p.pg A street light is at the top of a ft. tall pole. A man ft tall walks away from the pole with a speed of feet/sec along a straight path. How fast is the tip of his shadow moving when he is feet from the pole? 0.(0 pts) set5/cs8p.pg The altitude of a triangle is increasing at a rate of

17 500 centimeters/minute while the area of the triangle is increasing at a rate of square centimeters/minute. At what rate is the base of the triangle changing when the altitude is centimeters and the area is square centimeters?.(0 pts) set5/cs8p3.pg Water is leaking out of an inverted conical tank at a rate of cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate. The tank has height meters and the diameter at the top is meters. If the water level is rising at a rate of centimeters per minute when the height of the water is meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute..(0 pts) set5/cs8p5.pg A plane flying with a constant speed of 34 km/min passes over a ground radar station at an altitude of 8 km and climbs at an angle of 40 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing minutes later? 3.(0 pts) set5/s 8.pg Let A be the area of a circle with radius r. If dr dt 3, find da dt when r. 4.(0 pts) set5/s 8.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 the same. How fast is the height of the pile increasing when the pile is 8 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V 3 πr h. 6.(0 pts) set5/s 8 5.pg A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0. cm/min. At what rate is the volume of the snowball decreasing when the diameter is 6 cm. (Note the answer is a positive number). 7.(0 pts) set5/bye.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. 5.(0 pts) set5/s 8 3.pg Let xy 3 and let Find dx dt when x 5. dy dt 8.(0 pts) set5/ellipse.pg The graph of the equation x xy y 9 is a slanted ellipse illustrated in this figure:

18 Think of y as a function of x. Differentiating implicitly and solving for ya gives: ya. (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. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 3

19 Math 0- Summer 003 Peter Alfeld. WeBWorK assignment number 6. due 7//03 at :00 PM. This homework covers Newton s method and using derivatives to find maxima and minima. The first problem is meant to help you prepare for exam. Peter Alfeld, JWB 7, (0 pts) set6/e.pg The purpose of this problem is to help you prepare for exam. There will be 7 problems on the exam, covering the following areas: Finding critical points and extreme values. For example, the function f x & x x has two stationary points. The smaller is and the larger is. The maximum value of f in the interval R is. Drawing graphs. For example, draw the graph of the function in the preceding problem. Use of derivatives to determine where a function is increasing and where it is decreasing. (Simple) Word problems involving minimization and maximization. The Mean Value Theorem..(0 pts) set6/newton.pg Suppose you want to compute the fifth root of 6 by solving the equation f x ( x S T using Newton s method. Newton s method starts with an initial approximation x 0 and then computes a sequence of approximations x, x, x 3, ' via the formula x k g x k F" k 0" " " where f x g x & x f A x For the function defined above in S T, g x -. Letting x 0 you obtain x =, x =, and x 3 =. 3.(0 pts) set6/newton.pg Repeat the preceding exercise for f x ( x cosx 0 and x 0 For this function g x. You obtain x =, x =, and x 3 =. 4.(0 pts) set6/csp.pg For what values of x does the graph of f x ( 4x 3 30x 48x 48 have a horizontal tangent? Enter the x values in order, smallest first, to 4 places of accuracy: x x 5.(0 pts) set6/c3s3p.pg The function f x ( x 3 4 x 96 6x 7 57 is increasing on the interval (, ). It is decreasing on the interval (, ) and the interval (, ). The function has a local maximum at. 6.(0 pts) set6/c3s8p.pg Find the point on the line x 8y 0 which is closest to the point " 5. (, ) 7.(0 pts) set6/c3s8p.pg A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y x. What are the dimensions of such a rectangle with the greatest possible area? Width = Height = 8.(0 pts) set6/c3s8p3.pg A cylinder is inscribed in a right circular cone of height 7 and radius (at the base) equal to 3. What are

20 the dimensions of such a cylinder which has maximum volume? Radius = Height = 9.(0 pts) set6/nsc4 6 6.pg A fence feet tall runs parallel to a tall building at a distance of 4 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) set6/nsc4 6 3.pg If 000 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) set6/s3 43a.pg The function f xü 6x 9x has one local minimum and one local maximum. 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.(0 pts) set6/s3 4 6a.pg Consider the function f x V 7 x 3! 3. For this function there are two important intervals: " A and A" where A is a critical point. Find A For each of the following intervals, tell whether f x is increasing (type in INC) or decreasing (type in DEC). D " A : A" : For each of the following intervals, tell whether f x is concave up (type in CU) or concave down (type in CD). D " A : A" : 3.(0 pts) set6/s3 8 6.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 3 feet? 4.(0 pts) set6/s3 8 6.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? 5.(0 pts) set6/sc4 53.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 = 75.9 s, is given by v t t t 6 4t 5 35 (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. 6.(0 pts) set6/can.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. 7.(0 pts) set6/fly.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 d v 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.

21 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. 8.(0 pts) set6/relate.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 course, s and h are both functions of time. To enter your answer into WeBWorK use the notation v to denote ha : v ha Then the speed of the shadow at any time while the rock is in the air is given by saw (where sa 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 saw where your answer is an expression depending on H, v, and d. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 3

22 Math 0- Summer 003 Peter Alfeld. WeBWorK assignment number 7. due 7/9/03 at :00 PM..(0 pts) set7/csp8.pg At what point does the normal to y 3 4 3x 4x at " 3 intersect the parabola a second time? (, ) The normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals i.e. if the slope of the first line is m then the slope of the second line is m.(0 pts) set7/c3sp.pg Consider the function f x & x 3 3x 3x Find the average slope of this function on the interval D "'. By the Mean Value Theorem, we know there exists a c in the open interval " such that f A c is equal to this mean slope. Find the value of c in the interval which works 3.(0 pts) set7/c3s4p.pg Answer the following questions for the function f x ( xl x 6 defined on the interval RX 6" 4@. A. f x is concave down on the region to B. f x is concave up on the region to C. The inflection point for this function is at D. The minimum for this function occurs at E. The maximum for this function occurs at 4.(0 pts) set7/s3 4.pg The function f xÿ x 3 36x 0x has one local minimum and one local maximum. 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 5.(0 pts) set7/s3 43.pg The function f x Z x 3 36x 6x 4 has one local minimum and one local maximum. 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 6.(0 pts) set7/s3.pg Consider the function f x 5x 3 4x on the interval R[ 5" 5@. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there exists at least one c in the open interval 5" 5 such that f AC c is equal to this mean slope. For this problem, there are two values of c that work. The smaller one is and the larger one is 7.(0 pts) set7/s3.pg Consider the function f x & x 3 6x 48x 3 on the interval R[ 6" 7@. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there exists a c in the open interval 6" 7 such that f A c is equal to this mean slope. For this problem, there are two values of c that work. The smaller one is and the larger one is 8.(0 pts) set7/s3 3 6.pg Consider the function f x # x 3 4x 4x 6. For this function there are three important intervals: " A@, R A" B@, and R B" where A and B are the critical points Find A and B For each of the following intervals, tell whether f x is increasing (type in INC) or decreasing (type in DEC). " A@ : R A" B@ : R B" : 9.(0 pts) set7/s3 3 6a.pg Consider the function f x \ 4x 5x. For this function there are four important intervals: " A@,

23 R A" B, B" C, and R C" where A, and C are the critical points and the function is not defined at B. Find A and B and C For each of the following intervals, tell whether f x is increasing (type in INC) or decreasing (type in DEC). D " A@ : R A" B : B" C@ : R C" : 0.(0 pts) set7/s3 4 0.pg Consider the function f x & x 5 30x 4 60x 3 7. 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@ : R D" E@ : R E " F@ : R F" :.(0 pts) set7/vpr4 4.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?.(0 pts) set7/vpr4 4 3.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, 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. 3.(0 pts) set7/ladder.pg (This is essentially problem 3 on page 87 of the textbook.) 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 8 and h 5 this value is x. The corresponding numerical value of s. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR