1.(24 pts) This problem demonstrates how you enter numerical

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1 Tom Robbins MATH 0- Summer 00 Homework Set 0 due 5//0 at 6:00 AM This first set (set 0) is designed to acquaint you with using WeBWorK. Your score on this set will not be counted toward your final grade You may need to give 4 or 5 significant digits for some (floating point) numerical answers in order to have them accepted by the computer..(4 pts) This problem demonstrates how you enter numerical answers into WeBWorK. Evaluate the expression : In the case above you need to enter a number, since we re testing whether you can multiply out these numbers. (You can use a calculator if you want.) For most problems, you will be able to get WeBWorK to do some of the work for you. For example Calculate (-4) * (7): 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 either 4 7 or -8 or even 4 7. 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 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. 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: 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 WeBWorK 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 mutiple by x. Try it. Now enter the function cost. Note this is a function of t and not x. Try entering cos x and see what happens. 3.(4 pts) This problem will help you learn the rules of precedence, i.e. the order in which mathematical operations are preformed. 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? 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 WeBWorK do the calculating. WeBWorK will tell you if the problem requires a strict numerical answer. The way we use WeBWorK in this class 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..(4 pts) 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 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.

2 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 /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. 4.(4 pts) 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 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. 5.(4 pts) 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.. The distance from x to 3 is less than or equal to. x is greater than or equal to 3 3. x is greater than 3 4. x is any real number 5. The distance from x to 3 is more than A. x 3 B. x 3 C. x 3 D. 3 x E. x 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 think rather than to just try guessing. If you are having trouble reading the mathematics on the screen, this means that you are using text mode. If you are using Netscape or MSIE then you can get an easier to read version of the equations by returning to the problem list page (use the button at the top of this page) and choosing formatted-text or typeset instead of text. Sometimes there is a 5-0 second delay in viewing a problem in typeset mode the first time. 6.(4 pts) This problem demonstrates a WeBWorK problem involving graphics. The simplest functions are the 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 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. Some problems display a link to a web page where you can get additional information or a hint:hint 7.(4 pts) 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

3 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 grade. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 3

4 Tom Robbins MATH 0- Summer 00 Homework Set due 5/30/0 at 7:00 AM This is the first of 0 WeBWorK based home work sets. All questions have equal weight. Note that problems and, 9 and 0, and and are identical pairs with different numbers. The purpose of this is to encourage you to think in terms of a general approach to the problem. Future assignments will ask for answers in terms of general parameters. Due to Memorial Day, this assignment is due Wednesday, May 30, 7:00am. The maximum point count per problem is 4 points (send me a note with your guess of how I came up with that funny number). Most of the questions are routine but some are a little more challenging. You should print the problems, do them at home or in another private place, and then return to a computer and enter your answers. Then think about those problems for which your answers were incorrect. Talk to each other, to the tutors, and don t hesitate to contact me if you have any questions. When you think you have the correct answers return to a computer and enter them. Getting the answers right may take you several rounds of rethinking and entering them. You have an arbitrary number of attempts for each questions. However, particularly the more complicated problems may take you quite a while to figure out correctly, so remember: Procrastination is hazardous Significant effort has been made to make sure these problems make sense and the answers provided to the computer system are correct. Please let us know (using the feedback button) if there are any discrepancies or your are sure your answers are correct and the computer keeps refusing them. If you are right the class will be informed and you will receive credit. Peter Alfeld, JWB 7 or 36, (4 pts) Evaluate the expression (Your answer cannot be an algebraic expression. ).(4 pts) Evaluate the expression Give you answer in decimal notation correct to three decimal places or give your answer as a fraction. [NOTE: Your answer can be an algebraic expression. Make sure to include all necessary (, ). ] 3.(4 pts) Solve the equation for x 6 x 0 x 0 x 4.(4 pts) Solve the equation for t t 9 t 8 t 0 t 5.(4 pts) The equation 4x 4 8x 3 x 0 has three real solutions A, B, and C where A B C and A is: and B is: and C is: 6.(4 pts) Evaluate the expression [NOTE: Your answer cannot be an algebraic expression. ] 7.(4 pts) The expression 3a b c 3 a 5 b 3 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: and finally t, the exponent of c, is: [NOTE: Your answers cannot be algebraic expressions.] 5 8.(4 pts) Consider the two points 5 5 and 7 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: 9.(4 pts) Find the distance between (6, 4) and (0, 0). 0.(4 pts) Find the perimeter of the triangle with the vertices at (4, 0), (-5, ), and (-3, -3)..(4 pts) The equation of the line with slope 5 that goes through the point 5 7 can be written in the form y mx b where m is: and where b is:.(4 pts) This problem is like the preceding problem. The equation of the line with slope 9 that goes through the point can be written in the form y mx b where m is: and where b is: 3.(4 pts) The equation of the line that goes through the point 8 and is parallel to the x-axis can be written in the form y mx b where m is: and where b is: 4.(4 pts) The equation of the line that goes through the point 7 5 and is parallel to the line 4x 3y can be written in the form y mx b where m is: and where b is: 5.(4 pts) The equation of the line that goes through the point 6 and is perpendicular to the line x y 4 can be written in the form y mx b where m is: and where b is: 6.(4 pts) A line through (8, 0) with a slope of 0

5 has a y-intercept at 7.(4 pts) An equation of a line through (-, 4) which is perpendicular to the line y x has slope: and y intercept at: 8.(4 pts) Find the slope of the line through (-9, 0) and (-4, 7). 9.(4 pts) The distance of the point from the line y 4x 4 is: 0.(4 pts) This is like the preceding problem. The distance of the point 5 from the line y x 3 is:.(4 pts) You drop a rock into a well. You can t see the impact at the bottom but you hear it after 5 seconds. You wonder how deep the well is. Assume the speed of sound is 00 feet per second and the acceleration of the rock is 3 feet per second squared. Ignore air drag (this is a major simplifying assumption). (Hint: the time till you hear the impact is composed of the time the rock takes to reach the bottom and the time the sound of the impact takes to travel back to your ear.) The depth of the well (in feet) is:.(4 pts) This is the same as the preceding problem but suppose you hear the impact after 4 6 seconds. The depth of this well (in feet) is: 3.(4 pts) Suppose you obtain 00 percent credit on all WeBWorK 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

6 Tom Robbins MATH 0- Summer 00 Homework Set due 6/5/0 at 7:00 AM This is the second of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. Remember our central guideline: Procrastination is hazardous Again, significant effort has been made to make sure these problems make sense and the answers provided to the computer system are correct. Please let us know (using the feedback button) if there are any discrepancies or you are sure your answers are correct and the computer keeps refusing them. If you are right the class will be informed and you will receive credit. Peter Alfeld, JWB 7 or 36, (4 pts) 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 NEI- THER even nor odd.. f x 5x 6 3x 0. f x x 3 x 9 x 5 3. f x x 6 3x 0 x 5 4. f x x 6 6x 0 3x.(4 pts) Let f x 5x 4 and g x x 4x. f g Recall that by convention the absence of an arithmetic operator denotes multiplication. 3.(4 pts) Let f x 5x 3 and g x 3x 3x. Then f g x 4.(4 pts) Let f x 3x 3 and g x 4x x. f g 7 5.(4 pts) Let f x x 4 and g x 4x 3x. Then, f g x 6.(4 pts) Relative to the graph of y sin x the graphs of the following equations have been changed in what way?. y sin x 8. y sin x 5 3. y sin x 8 4. y sin 5x A. compressed vertically by the factor 5 B. compressed horizontally by the factor 5 C. shifted 8 units down D. shifted 8 units up 7.(4 pts) Relative to the graph of y sin x the graphs of the following equations have been changed in what way?. y 0sin x. y sin x 0 3. y sin x 0 4. y sin 0x A. stretched vertically by the factor 0 B. stretched horizontally by the factor 0 C. compressed vertically by the factor 0 D. compressed horizontally by the factor 0 8.(4 pts) 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 3 to the right and 9 up. 9.(4 pts) This question is modeled after problems 0 and 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 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.) 0.(4 pts) 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.. The composition of an odd function and an odd function is even 3. The composition of an even and an odd function is even 4. The sum of an even and an odd function is usually neither even or odd, but it may be even. 5. The product of two even function is even

7 6. The sum of two even functions is even 7. The ratio of two odd functions is odd 8. The product of two odd function is odd All of the answers must be correct before you get credit for the problem..(4 pts) 3 x h x 3 lim h 0 h is..(4 pts) Recall that lim x f x c 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. 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 x f x c L and enter their reference numbers in the appropriate sequence in these boxes: 3.(4 pts) Evaluate the limit x 3 lim x 3 x 3x 8 4.(4 pts) Evaluate the limit x 3 x lim x x 5.(4 pts) Evaluate the limit 5 t lim t 5 5 t 6.(4 pts) Evaluate the limit y 8 lim y 8 y 8 7.(4 pts) Evaluate the limit 4x 3x 5 lim x 3 x 4 8.(4 pts) The function f x x 4x can be obtained from an even function g by shifting its graph horizontally and vertically. That even function is g x and it has been shifted by to the left and up. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

8 Tom Robbins MATH 0- Summer 00 Homework Set 3 due 6//0 at 7:00 AM This is the third of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. Remember our central guideline: Procrastination is hazardous Again, significant effort has been made to make sure these problems make sense and the answers provided to the computer system are correct. Please let us know (using the feedback button) if there are any discrepancies or you are sure your answers are correct and the computer keeps refusing them. If you are right the class will be informed and you will receive credit. Peter Alfeld, JWB 7 or 36, (4 pts) For each of the following angles, find the degree measure of the angle with the given radian measure: π 6 5π 4 4π 3 7π π.(4 pts) For each of the following angles, find the radian measure of the angle with the given degree measure (you can enter π as pi in your answers): (4 pts) For each of the followings angles (in radian measure), find the sin of the angle (your answer cannot contain trig functions, it must be an arithmetic expression or number): π 6 π 4 π 3 π π π 4.(4 pts) A plane is flying at an elevation of 000 feet. It is within sight of the airport and the pilot finds that the angle of depression to the airport is 7. (The angle of depression is the angle that the line from the airport to the plane makes with a horizontal line.) Find the distance between the plane and the airport. Find the distance between a point on the ground directly below the plane and the airport. 5.(4 pts) A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 4 and 6. How high (in feet) is the ballon? 6.(4 pts) The point P 4 7 lies on the curve y x x 7. If Q is the point x x x 7, find the slope of the secant line PQ for the following values of x. If x 4, the slope of PQ is: and if x 4 0, the slope of PQ is: and if x 3 9, the slope of PQ is: and if x 3 99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P (4 pts) If a ball is thrown straight up into the air with an initial velocity of 65 ft/s, it height in feet after t second is given by y 65t 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. ca if a 8.(4 pts) For what value of the constant c is the function f continuous on where 3 f a ca if a 3 9.(4 pts) The slope of the tangent line to the parabola y x 7x at the point 4 is: The equation of this tangent line can be written in the form y mx b where m is: and where b is: 0.(4 pts) The slope of the tangent line to the curve y 4x 3 at the point is: The equation of this tangent line can be written in the form y mx b where m is: and where b is:.(4 pts) If f x 4x 0, find f 3..(4 pts) If f x 5 x x, find f. 3.(4 pts) If f x x x 34, find f x. Find f 5.

9 4.(4 pts) If f x 6x 8 4x 5 x 3 6x, find f x. Find f. 5.(4 pts) A particle moves along a straight line and its position at time t is given by s t t 3 t 36t 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 4? Finally, what is the TOTAL distance the particle travels between time 0 and time 4? 6.(4 pts) If a ball is thrown vertically upward from the roof of 48 foot building with a velocity of ft/sec, its height after t seconds is s t 48 t 6t. What is the maximum height the ball reaches? What is the velocity of the ball when it hits the ground (height 0)? 7.(4 pts) Let f x x 3 Use the limit definition of the derivative on page 07 to find (i) f 6 (ii) f 5 (iii) f (iv) f 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. 8.(4 pts) 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, loacted 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 statmenets.. The top of the well rotates with the Earth towards the South.. The rock will hit the North wall of the well. 3. The rock will hit the West wall of the well. 4. The rock will hit the East wall of the well. 5. The top of the well rotates with the Earth towards the North. 6. The rock will fall straight down until it hits the center of the bottom of the well. 7. None of these statements are true. 8. The top of the well rotates with the Earth towards the East. 9. The top of the well rotates with the Earth towards the West. 0. The rock will hit the South wall of the well. 9.(4 pts) 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

10 Tom Robbins MATH 0- Summer 00 Homework Set 4 due 6/9/0 at 7:00 AM This is the fourth of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. By now you remember our central guideline and so I will mention it for the last time: Procrastination is hazardous As usual, please let us know (using the feedback button) if there are any discrepancies or you are sure your answers are correct and the computer keeps refusing them. Have fun! Peter Alfeld, JWB 7 or 36, (4 pts) Let 4.(4 pts) If f x cos x 3, find f x. f x 6cosx 7tanx 5.(4 pts) Let f x f x cos 4x 5.(4 pts) If f x 5 sin x cosx, then x x f 3.(4 pts) If 3sinx f x cosx find f x. 4.(4 pts) If f x x x 7, find f x. 5.(4 pts) If f x sin sin x, find f x. 6.(4 pts) If f x cos sin x, find f x. 7.(4 pts) Let f f x x 3 3x 8 x 8.(4 pts) If f x 3x 6, find f x. Find f 3. 9.(4 pts) If f x sin x 4, find f x. 0.(4 pts) If f x sin 4 x, find f x. Find f..(4 pts) If f x x, find f x..(4 pts) If f x tan5x, find f x. 3.(4 pts) Let f f x 4 f x 3x 4x 3 f 6.(4 pts) Let 3x 4 f x x f 4 x Note: There is a way of doing this problem without using the quotient rule 4 times. 7.(4 pts) The purpose of this problem is to show pretty much all of our rules at work at once. If xsin x f x x find f x. f f f 8.(4 pts) Let Then x = x = x = x = f x tan x 9.(4 pts) 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

11 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 miles T seconds 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

12 Tom Robbins MATH 0- Summer 00 Homework Set 5 due 6/6/0 at 7:00 AM This is the fifth of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. As usual, please let us know (using the feedback button) if there are any discrepancies or you are sure your answers are correct and the computer keeps refusing them. Have fun! Peter Alfeld, JWB 7 or 36, (4 pts) Find the slope of the tangent line to the curve at the point 3 8. x 3xy y (4 pts) If 5x 5x xy 30 and y 5 by implicit differentiation. 4, find y 3.(4 pts) Find y by implicit differentiation. Match the expressions defining y implicitly with the letters labeling the expressions for y.. 3sin x y 3ysinx. 3cos x y 3ycosx 3. 3cos x y 3ysinx 4. 3sin x y 3ycosx A. B. C. D. 3sin x y 3ycosx 3sinx 3sin x y 3cos x y 3ycosx 3cos x y 3sinx 3sin x y 3ysinx 3cosx 3sin x y 3cos x y 3cos x y 3ysinx 3cosx 4.(4 pts) 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: 5.(4 pts) If x 64 y 49 and y , find y by implicit differentiation. 6.(4 pts) Find the slope of the tangent line to the curve (a lemniscate) x y 5 x y at the point 3. m 7.(4 pts) If x y 8 and y 9 5, find y 9 by implicit differentiation. 8.(4 pts) 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? 5 9.(4 pts) The altitude of a triangle is increasing at a rate of 500 centimeters/minute while the area of the triangle is increasing at a rate of 000 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.(4 pts) 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 500 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute..(4 pts) A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of km and climbs at an angle of 5 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing minutes later? dr dt.(4 pts) Let A be the area of a circle with radius r. If 3, find da dt when r 3. 3.(4 pts) Gravel is being dumped from a conveyor belt at a rate of 50 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. 4.(4 pts) Let and let Find dx dt when x. xy 5 dy dt 3 5.(4 pts) 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 9 cm. (Note the answer is a positive number). 6.(4 pts) 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)

13 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

14 Tom Robbins MATH 0- Summer 00 Homework Set 6 due 7/3/0 at 7:00 AM This is the sixth of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. As usual, please let us know if there are any problems with these questions. Have fun! Peter Alfeld, JWB 7 or 36, (4 pts) For what values of x does the graph of f x 4x 3 8x 4x 0 have a horizontal tangent? Enter the x values in order, smallest first, to 4 places of accuracy: x x.(4 pts) The function f x 6x x x 0 is increasing on the interval (, ). It is decreasing on the interval (, ) and the interval (, ). The function has a local maximum at. 3.(4 pts) Find the point on the line 8x 8y 4 0 which is closest to the point. (, ) 4.(4 pts) A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y 0 x. What are the dimensions of such a rectangle with the greatest possible area? Width = Height = 5.(4 pts) A cylinder is inscribed in a right circular cone of height 7.5 and radius (at the base) equal to 3. What are the dimensions of such a cylinder which has maximum volume? Radius = Height = 6.(4 pts) A fence 3 feet tall runs parallel to a tall building at a distance of 5 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? 7.(4 pts) If 500 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. 8.(4 pts) The function f x 6x 4x 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 9.(4 pts) Consider the function f x 9 x 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). A : A : For each of the following intervals, tell whether f x is concave up (type in CU) or concave down (type in CD). A : A : 0.(4 pts) 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 36 feet?.(4 pts) 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 pts) 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 = 6.8 s, is given by v t t t 3 4t 0 4 (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. 3.(4 pts) 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. ratio =. 4.(4 pts) 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

15 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. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

16 Tom Robbins MATH 0- Summer 00 Homework Set 7 due 7/0/0 at 7:00 AM This is the seventh of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. As usual, please let us know if there are any problems with these questions. Have fun! Peter Alfeld, JWB 7 or 36, (4 pts) At what point does the normal to y 4x x at 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.(4 pts) Note: despite rumors to the contrary this is a fine problem. However, rather than going back and forth, please ignore this problem. The circumference of a sphere was measured to be cm with a possible error of cm. Use the approximation techniques based on differentials, as described in section 3.0 of the text, to estimate the maximum error in the calculated surface area. Estimate the relative error in the calculated surface area. 3.(4 pts) Consider the function f x x 3 x 3x 4 Find the average slope of this function on the interval 3 0. By the Mean Value Theorem, we know there exists a c in the open interval 3 0 such that f c is equal to this mean slope. Find the value of c in the interval which works 4.(4 pts) Answer the following questions for the function f x x x defined on the interval 4 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 5.(4 pts) The function f x x 3 4x 7x 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.(4 pts) The function f x x 3 36x 6x 5 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 7.(4 pts) Consider the function f x 4x 3 5x on the interval 4 4. 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 4 4 such that f 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.(4 pts) Consider the function f x x 3 x 7x 0 on the interval 6 0. 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 0 such that f 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 9.(4 pts) Consider the function f x x 3 30x 6x. For this function there are three important intervals: A, A B, and 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 : A B : B : 0.(4 pts) Consider the function f x 3x x. For this function there are four important intervals: A, A B, B C, and 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). A : A B : B C : C :

17 .(4 pts) Consider the function f x x 5 45x 4 360x 3 4. 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 :.(4 pts) 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? 3.(4 pts) I have enough pure silver to coat 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. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

18 Tom Robbins MATH 0- Summer 00 Homework Set 8 due 7/7/0 at 7:00 AM This is the eighth of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. As usual, please let us know if there are any problems with these questions. Have fun! Peter Alfeld, JWB 7 or 36, f.(4 pts) Given x x 6 and f 6 and f 4. Find f x and find f 4.(4 pts) A car traveling at 50 ft/sec decelerates at a constant 8 feet per second squared. How many feet does the car travel before coming to a complete stop? 3.(4 pts) A ball is shot straight up into the air with initial velocity of 50 ft/sec. Assuming that the air resistance can be ignored, how high does it go? Hint: The acceleration due to gravity is 3 ft per second squared. 4.(4 pts) A ball is shot at an angle of 45 degrees into the air with initial velocity of 45 ft/sec. Assuming no air resistance, how high does it go? How far away does it land? Hint: The acceleration due to gravity is 3 ft per second squared. 5.(4 pts) Consider the function f x 3x 3 6x 3x 4. An antiderivative of f x is F x Ax 4 Bx 3 Cx Dx where A is and B is and C is and D is 6.(4 pts) Consider the function f x 3x 3 4x 6x 0. Enter an antiderivative of f x 7.(4 pts) Consider the function f x whose second derivative is f x x 8sin x. If f 0 and f 0 3, what is f 5? 8.(4 pts) Consider the function f x whose second derivative is f x 7x 6sin x. If f 0 3 and f 0 3, what is f x? 9.(4 pts) Given that the graph of f x passes through the point 8 and that the slope of its tangent line at x f x is 6x 3, what is f 4? 0.(4 pts) Consider the function f x 0x 9 0x 6 x 3. Enter an antiderivative of f x.(4 pts) Consider the function f x 0 x 5 x 5. Let F x be the antiderivative of f x with F 0. Then F equals 7 6.(4 pts) Consider the function f x. x 3 x 7 Let F x be the antiderivative of f x with F 0. Then F x 3.(4 pts) The ubiquitous oz aluminum cans used to distribute drinks in this country have a diameter of approximately.75 inches and a height of 5.0 inches. Estimate the amount of aluminum that could be saved if the cans where designed with the ratio d h that you computed in problem set 6 (while having the same vlume as the standard cans). Amount of aluminum saved, as a percentage of the amount used to make the optimal cans, equals. What would be the diameter and height (in inches) of the new and improved cans?. 4.(4 pts) A can manufacturer located in Oklahoma City produces aluminum cans for one major cola company. The distribution area for these cans includes about 3% of the US population located in New Mexico, Texas, Oklahoma, Kansas, Arkansas, Louisiana and southwestern Missouri. Inside the plant, there are eight stamping presses that manufacture can lids. Each press stamps two lids per stroke and on average runs at 500 strokes per minute. The machines run 4 hours a day for an average of 300 days per year. (The only time that the machines stop running is for two two week maintenance breaks, 0 holidays, and an additional days due to miscellaneous break downs.) If the total volume of aluminum for one can is V c cubic meters and the cost of processed aluminum is dollars per cubic meter, how many dollars would be saved by changing the design of the can? Savings equals dollars. 5.(4 pts) In problems 3 and 4 we define a can to be optimal if it has the least surface area for a given volume. In this problem we assume the real issue in can manufacturing is cost, and the industry has figured out which shape minimizes the cost of making the cans. (Of course there are other issues, like how the can fits into one s hand and whatever marketing appeal a given shape might have. Also consider that once the country is saturated with vending machines it would be very expensive to make substantial changes in the shape of the can.) The familiar oz aluminum can is made of two parts, a shell and a lid. Assume that the total manufacturing cost for the lid and the shell is C 6 5 cents.

19 The manufacturing cost of the lid consists of a fixed cost of M L 4 4 cents per lid, and an additional cost that is proportional to the area of the lid. Thus the total cost is C L M L κ L π d 4 cents per lid, for some constant κ L. Similarly, the manufacturing cost of the shell consists of a fixed cost of M S 85 cents per shell, and an additional cost that is proportional to the surface area of the shell. Thus the total cost is C S M S κ S πhd π 4 d cents per shell, for some constant κ S. Assuming that the total manufacturing cost of the can is minimized for a diameter of.75 inches and a height of 5.0 inches, what are the variable costs involved with the manufacture of the cans? κ L κ S cents per square inch, cents per square inch. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

20 sin Tom Robbins MATH 0- Summer 00 Homework Set 9 due 7/5/0 at 7:00 AM This is the 9-th of 0 WeBWorK based home work sets. All questions have equal weight. The maximum point count per problem is 4 points. As usual, please let us know if there are any problems with these questions. Have fun! Peter Alfeld, JWB 7 or 36, f f x.(4 pts) If f x t8 dt then x.(4 pts) The value of 0 x 3 dx is 3.(4 pts) The value of 3 x dx is 7 4.(4 pts) Evaluate the definite integral 9 4 6x 8 dx 5.(4 pts) Evaluate the definite integral 3 x 0x 7 dx 6 6.(4 pts) Evaluate the definite integral x dx 7.(4 pts) Evaluate the definite integral π 0x 5 dx x 8.(4 pts) Evaluate the definite integral 0 sin x dx 9.(4 pts) Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. 5 5 x dx 5 0.(4 pts) Evaluate the integral by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry x 8 dx.(4 pts) Consider the integral 3x x dx (a) Find the Riemann sum for this integral using right endpoints and n 3. (b) Find the Riemann sum for this same integral, using left endpoints and n 3 3.(4 pts) Evaluate the integral 4 x sin t dt 3.(4 pts) Use part I of the Fundamental Theorem of Calculus to find the derivative of f x 4 t dt f x = [NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary *, (, ), etc. ] 4.(4 pts) Use part I of the Fundamental Theorem of Calculus to find the derivative of h h x x cos t 3 t dt x [NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary *, (, ), etc. ] 5.(4 pts) Use part I of the Fundamental Theorem of Calculus to find the derivative of x u 3 g x u du 4x x 6.(4 pts) Consider the function f x x In this problem you will calculate dx by using the definition b n f x dx lim a n f x i x The summation inside the brackets is R n which is the Riemann sum where the sample points are chosen to be the righthand endpoints of each sub-interval. x Calculate R n for f x 4 8 on the interval 0 4 and write your answer as a function of n without any summation signs. R n i

21 lim n R n 7.(4 pts) The velocity function is v t t 4t 3 for a particle moving along a line. Find the displacement and the distance traveled by the particle during the time interval [-3,6]. displacement = distance traveled = Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

22 Tom Robbins MATH 0- Summer 00 Homework Set 0 due 8//0 at 7:00 AM This is the last WeBWorK based home work set. All questions have equal weight. The maximum point count per problem is 4 points. As usual, please let us know if there are any problems with these questions. Have fun! Peter Alfeld, JWB 7 or 36, (4 pts) Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y 8x and y x 4. Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?.(4 pts) A ball of radius 4 has a round hole of radius 5 drilled through its center. Find the volume of the resulting solid. 3.(4 pts) A ball of radius R has a round hole of radius r drilled through its center. Find the volume of the resulting solid. 4.(4 pts) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y 4x y 3x 5.(4 pts) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y x y x 4 6.(4 pts) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y 6x x y 0 about the x-axis 7.(4 pts) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y ax n x y 0, where n and a 0, about the x-axis 8.(4 pts) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y x y 9 x 0 x 3 about the y-axis 9.(4 pts) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y x 8 y ; about y 4 0.(4 pts) Find the volume of the solid obtained by rotating the region bounded by y x n y ; about the line y c, where n is even, and c. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR