U of U Math Online. Young-Seon Lee. WeBWorK set 1. due 1/21/03 at 11:00 AM. 6 4 and is perpendicular to the line 5x 3y 4 can

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1 U of U Math 0-6 Online WeBWorK set. due //03 at :00 AM. The main purpose of this WeBWorK set is to familiarize yourself with WeBWorK. 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. Print a hard copy of this assignment. Get to work on this 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!.( pt) set/pr.pg The equation of the line that goes through the points and can be written in the form y mx b where m is: and where b is:.( pt) set/pr.pg Find the equation of the line passing through the point (-5,-9) with slope 8. y= 3.( pt) set/pr3.pg Find the equation of the line passing through the point (0,-0) and parallel to the line passing through (-7,4) and (-9,9). y= 4.( pt) set/pr4.pg The equation of the line that goes through the point 6 4 and is perpendicular to the line 5x 3y 4 can be written in the form y and where b is: mx b where m is: 5.( pt) set/pr5.pg If f x 3x 3x 4, find f 4. Use this to find the equation of the tangent line to the parabola y 3x 3x 4 at the point The equation of this tangent line can be written in the form y mx b where m is: and where b is: 6.( pt) set/pr6.pg Find the slope of the curve y 5 m 7x 4 x 3 at the point 7.( pt) set/pr7.pg If an arrow is shot straight upward on the moon with a velocity of 60 m/s, its height (in meters) after t seconds is given by h 60t 0 83t. What is the velocity of the arrow (in m/s) after 7 seconds? After how many seconds will the arrow hit the moon? With what velocity (in m/s) will the arrow hit the moon? 8.( pt) set/pr8.pg If a ball is thrown straight up in such a way that its height t seconds later is s t 6t 3t 6, find the velocity of the ball at t seconds after it is thrown. At what time t does the ball reach its maximum height? (Hint: the velocity will be positive before this time and negative after it). Velocity =. Time at which maximum height is reached =. 9.( pt) set/pr9.pg If a ball is thrown vertically upward from the roof of 48 foot building with a velocity of 64 ft/sec, its height after t seconds is s t 48 64t 6t. What is the maximum height the ball reaches? What is the velocity of the ball when it hits the ground (height 0)? 0.( pt) set/pr0.pg If f x 4x 5 4x, find f x. [NOTE: Your answer should be a function in terms of the variable x and not a number! ]

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

3 U of U Math 0-6 Online WeBWorK set. due /8/03 at :00 AM. The main purpose of this WeBWorK set is to familiarize yourself with WeBWorK. 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. Print a hard copy of this assignment. Get to work on this 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!.( pt) set/pr.pg If find f Find f x. 4. f x x x.( pt) set/pr.pg 6 5 If f x 6 x, find f x. x Find f. 3.( pt) set/pr3.pg Find the x coordinate of the point on the curve y x x x 0 where the tangent line has slope -. x= 4.( pt) set/pr4.pg, find f t 4 If f t 5 t. [NOTE: Your answer should be a function in terms of the variable t and not a number! ] 5.( pt) set/pr5.pg If f x 5x 4 7x 6, find f Find f. 6.( pt) set/pr6.pg Let f x 4x and f is is and f x. 4x. Then f 7.( pt) set/pr8.pg Find 6x x 5 dx. 6x x 5 dx = integration constant. is C, where C is the 8.( pt) set/pr9.pg Find the antiderivative of 6x 3 x that has the value 7 when x. The desired antiderivative is:. 9.( pt) set/pr0.pg Find 0 x 3 dx. 0 x 3 dx =. 0.( pt) set/pr.pg A particle travels along a horizontal line so that its velocity at time t is v t Suppose that at time t t 3t feet per second. the particle is at the origin. What is the location of the particle at time t 3? Particle s location at t 3 is: feet from the origin. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

4 U of U Math 0-6 Online WeBWorK set 3. Due /4/03 at :00 AM..( pt) set3/pr.pg Let f x. f x cos 4x 4 9. Find f.( pt) set3/pr.pg Let y π tangent line to the curve when x 6? 3.( pt) set3/pr3.pg Let f x x 0 f x 3sin x 3 3. What is the slope of the. For what values of x is 0? Write the answers in increasing order.,. 4.( pt) set3/pr4.pg Let f f x 5 x f x 3x 3x 8 5.( pt) set3/pr5.pg If f x sin x 4, find f Find f 4. 6.( pt) set3/pr6.pg If f x sin 5 x, find f Find f. 7.( pt) set3/pr7.pg Let f x 8.( pt) set3/pr8.pg Let f If d dx x 9.( pt) set3/pr9.pg f x 4 6x 3. x Calculate f f x 0.( pt) set3/pr0.pg Let f x f x f 3 x. x. 3sin x f x cos 6x 7 x! 3 x 0! Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

5 U of U Math 0-6 Online WeBWorK set 4. Due /4/03 at :00 AM..( pt) set4/pr.pg Find the slope of the tangent line to the curve x 4xy y 3 at the point ( pt) set4/pr.pg Use implicit differentiation to find the slope of the tangent line to the curve y x 3y x 9 at the point 8 5. m 3.( pt) set4/pr3.pg Find the coordinates of those points on the curve given by the equation x 0 5xy y 6 at which the tangent line has slope. The first point must be the one with the greater x coordinate.,, 4.( pt) set4/pr4.pg Find the slope of the tangent line to the curve given by the equation y xy at the point y 5.( pt) set4/pr5.pg If the variables s and t are related by the equation find ds ds dt dt. st t 3 6.( pt) set4/pr6.pg If f is the focal length of a convex lens and an object is placed at a distance q from the lens, then its image will be at a distance p from the lens, where f, q, and p are related by the lens equation f q p Suppose the focal length of a particular lens is 0 cm. What is the rate of change of q with respect to p when p 6? (Make sure you have the correct sign for the rate.) dq d p 7.( pt) set4/pr7.pg A street light is at the top of a 000 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? 8.( pt) set4/pr8.pg A spherical snowball is melting in such a way that its radius is decreasing at the rate of 0.3 cm/min. At what rate is the volume of the snowball decreasing when the radius is 7 cm? (Note the answer is a positive number). 9.( pt) set4/pr9.pg Sand falls out of the end of a slurry at the rate of 60 cc/sec. The pile forms a circular cone, the ratio of whose base diameter to height is 3. When the pile is of height 80 cm., at what rate is the height of the pile increasing? dh dt 0.( pt) set4/pr0.pg Water is flowing into a balloon at the rate of 0 cc/min. The balloon has a puncture, and the rate at which water flows out of the balloon at this puncture is proportional to the volume of water in the balloon. Let the proportionality constant be 0 6 At what volume do we have equilibrium (that is the volume of water in the balloon remains constant)? Hint: if we let W i represent the amount of water which flows in, and W o the amount flowing out of the puncture, we have dw i dt 0 and dw o dt 0 6V where V is the volume of water in the balloon. V

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

7 U of U Math 0-6 Online 4.( pt) set5/pr4.pg Evaluate the it t 3 t t t" WeBWorK set 5. Due /5/03 at :00 AM..( pt) set5/pr.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 6 6x 4 3x 6. f x x 6 3. f x 5x 6 3x 4 4. f x x 3 x 7 x 9.( pt) set5/pr.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 sum of two even functions is even. The ratio of two odd functions is odd 3. The composition of an odd function and an odd function is even 4. A function cannot be both even and odd. 5. The product of two even function is even 6. The product of two odd function is odd 7. The composition of an even and an odd function is even 8. The sum of an even and an odd function is usually neither even or odd, but it may be even. All of the answers must be correct before you get credit for the problem. 3.( pt) set5/pr3.pg Evaluate the it x" x x 8x 9 5.( pt) set5/pr5.pg Evaluate the it 6.( pt) set5/pr6.pg Evaluate the it 7.( pt) set5/pr7.pg Evaluate the it a" 6 x" 6 a 4 a x 9 x" 9 x 9 8x 3x 6 x 8 8.( pt) set5/pr8.pg Let F be the function below. 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 clearly. Evaluate each of the following expressions. Note: Enter DNE if the it does not exist or is not defined. a) x" # F x = b) x" $ F x =

8 % c) x" F x = d) F = e) x" # F x = f) x" $ F x = g) x" F x = h) x" 3F x = i) F 3 = 9.( pt) set5/pr9.pg Evaluate the it x" 0.( pt) set5/pr0.pg Evaluate the it x x 6x 6 y 3 y" y.( pt) set5/pr.pg Evaluate the it sinx x" 0 6x.( pt) set5/pr.pg Evaluate the it sin9x x" 0 sinx 3.( pt) set5/pr3.pg Evaluate the it tanx x" 0 3x 4.( pt) set5/pr4.pg Evaluate the it 5.( pt) set5/pr5.pg Evaluate the it 3 9x x" 7 7x x" 5 x 9x 6.( pt) set5/pr6.pg Evaluate x" x 7x x 7.( pt) set5/pr7.pg For what value of the constant c is the function f continuous on where f t ct 6 ct 6 if t & if t & 6' 6 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

9 U of U Math 0-6 Online.( pt) set6/pr.pg The function WeBWorK set 6. Due 3/4/03 at :00 AM. f x x 3 6x 44x is decreasing on the interval (, ). It is increasing on the interval (, ) and the interval (, ). The function has a local maximum at..( pt) set6/pr.pg For x &)( 0 0' the function f is defined by f x x x 8 3 On which two intervals is the function increasing? to and to Find the region in which the function is positive: to Where does the function achieve its minimum? 3.( pt) set6/pr3.pg Answer the following questions for the function f x x 4 defined on the interval ( 5 8'. Enter points in ascending order,i.e. smallest x values first. Enter intervals in ascending order also. A. The function f x has vertical asympototes at x 3 and B. f x is concave up on the region to and to 4.( pt) set6/pr4.pg What number exceeds its square by the maximum amount? Begin by convincing yourself that this number is on the interval [0,]. Answer:. 5.( pt) set6/pr5.pg Consider the function f x 5 3x The absolute maximum value is and this occurs at x equals The absolute minimum value is and this occurs at x equals 4 * x *. 6.( pt) set6/pr6.pg Identify the critical points and find the maximum value and minimum value of the following function on the given interval. f x x 3 3x, over ( 3+ 3'. Critical Points:,. Maximum:. Minimum:. Instructions: ) When entering the critical points, please enter them in the order that they appear on the real line. ) If the function has no critical points, enter the string NONE in all answer boxes for critical points. 7.( pt) set6/pr7.pg Consider the function f x, 5 x 4! 3. For this function there are two important intervals: A and A where A is a critical number. Find A For each of the following intervals, tell whether f x is increasing (type in INC) or decreasing (type in DEC). A : A : 8.( pt) set6/pr8.pg Find the length of the shortest line from the origin to the line y 9x d 9.( pt) set6/pr9.pg Answer the following questions for the function defined on the interval ( f x x x '. 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

10 0.( pt) set6/pr0.pg Consider the function f x x 3 4x 4x 3. For this function there are three important intervals: A', ( A B', and ( B where A and B are the critical numbers. 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 f x has an inflection point at x C where C is Finally for each of the following intervals, tell whether f x is concave up (type in CU) or concave down (type in CD). C' : ( C If the answer to a question doesn t exist, then type NO as your answer. If your answer is infinity, type INF..( pt) set6/pr.pg Let y x 7 x 4 a. The graph has a vertical asymptote x=a for a=. b. The horizontal asymptote is y=. c. As x approaches a from the left, y approaches. d. As x approaches a from the right, y approaches. e. The graph has a local maximum at x=. f. The graph has a local minimum at x=. g. The graph is increasing in the intervals (, ) and (, ). h. The graph is concave down in the intervals (, ) and (, ). Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

11 U of U Math 0-6 Online WeBWorK set 7. Due 3/4/03 at :00 AM..( pt) set7/pr.pg A racer can cycle around a circular loop at the rate of 8 revolutions per hour. Another cyclist can cycle the same loop at the rate of 0 revolutions per hour. If they start at the same time (t=0), at what first time are they farthest apart? t hours..( pt) set7/pr.pg Two men are at opposite corners of a square block which is 500 feet on a side. They start to walk at the same time; one man walking east at the rate of 6 feet per second, and the other walks west at the rate of 5 feet per second. At what time are they closest? t seconds. 3.( pt) set7/pr3.pg A rectangle is to be drawn in the first quadrant with one leg on the y-axis, and the other on the x-axis, 0 x Find the coordinates of that vertex which form the rectangle of greatest area. x y and a vertex on the curve y 4.( pt) set7/pr4.pg The illumination at a point is inversely proportional to the square of the distance of the point from the light source and directly proportional to the intensity of the light source. If two light sources are 30 feet apart and their intensities are 00 and 70 respectively, at what point between them will the sum of their illuminations be a minimum? Solution: Let x be the distance at which the sum of the illuminations be minimum. Then x feet. 5.( pt) set7/pr5.pg A rectangle with sides parallel to to the coordinate axes is inscribed in the ellipse 5x y 5 Find the dimensions of the rectangle of greatest area. Answer: x = y = 6.( pt) set7/pr6.pg A drum in the form of a circular cylinder and open at one of the circular ends, is to be made so as to contain one cubic yard. Find the dimensions of the drum (height h and base radius r ) which minimizes the amount of material going into the drum. The surface area of the drum includes the area of the cylinder and the circles at bottom. r = h = 7.( pt) set7/pr7.pg Identify the critical points and find the maximum value and minimum value of the following function on the given interval. f x x 3 3x, over ( 3+ 3'. Critical Points:,. Maximum:. Minimum:. Instructions: ) When entering the critical points, please enter them in the order that they appear on the real line. ) If the function has no critical points, enter the string NONE in all answer boxes for critical points. 8.( pt) set7/pr8.pg Consider the function f x - 8x 7x. For this function there are four important intervals: A', ( A B, B C', and ( C where A, and C are the critical numbers 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' :

12 ( C Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f x is concave up (type in CU) or concave down (type in CD). B : B : 9.( pt) set7/pr9.pg Answer the following questions for the function f x x x 7 5'. 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 defined on the interval ( 0.( pt) set7/pr0.pg Consider the function f x. x 3 33x 68x 6. For this function there are three important intervals: A', ( A B', and ( B where A and B are the critical numbers. 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 f x has an inflection point at x C where C is Finally for each of the following intervals, tell whether f x is concave up (type in CU) or concave down (type in CD). C' : ( C If the answer to a question doesn t exist, then type NO as your answer. If your answer is infinity, type INF..( pt) set7/pr.pg Let y x x 5 a. The graph has vertical asymptotes along the lines x=a for a=,. b. The slant asymptote is y=. c. As x approaches a from the left, y approaches. d. As x approaches a from the right, y approaches. e. The graph has a local maximum at x=. f. The graph has a local minimum at x=. g. The graph is increasing in the intervals (, ) and (, ). h. The graph is concave up in the interval (, ) and (, ). If the answer to a question doesn t exist, then type NO as your answer. If your answer is infinity, type INF..( pt) set7/pr.pg Let y 0x 3 a. The slant asymptote is y=. b. The graph has a local minimum at x=. c. The minimum value of y is y=. d. The graph has a local maximum at x=. e. The graph is concave down in the interval (, ). f. The graph is concave up in the interval (, ). Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

13 U of U Math 0-6 Online WeBWorK set 8. Due 4//03 at :00 AM..( pt) set8/pr.pg Consider the function f x 3x 3 9x 7x 7. 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.( pt) set8/pr.pg Consider the function f x x 0 0x 7 8x 4 0. Enter an antiderivative of f x 3.( pt) set8/pr3.pg Consider 7 the function f x. x x 7 Let F x be the antiderivative of f x with F 0. Then F x 4.( pt) set8/pr4.pg Consider the function f t 0 9sec t 5t. Let F t be the antiderivative of f t with F 0 0. Then F t equals 5.( pt) set8/pr5.pg s s Evaluate the integral: ds. s Answer: + C. 6.( pt) set8/pr6.pg Evaluate the indefinite integral: 3y y 5 dy. Answer: + C. 7.( pt) set8/pr7.pg A car traveling at 4 ft/sec decelerates at a constant 8 ft/sec/sec. How many feet does the car travel before coming to a complete stop? 8.( pt) set8/pr8.pg Consider the differential equation: du dt u t 3 t. a) Find the general solution to the above differential equation. (Instruction: Write the answer in a form such that its numerator is and its integration constant is C rename your constant if necessary.) Answer: u. b) Find the particular solution of the above differential equation that satisfies the condition u 4 at t 0. Answer: u. 9.( pt) set8/pr9.pg Given f x 6. Find f π+ 5 0.( pt) set8/pr0.pg Given and f Find f x and find f 5sin 5x and f f.( pt) set8/pr.pg Find F x 3 x 5x 3 3 and f 0. x x dx Give a specific function for F x. F(x) =.( pt) set8/pr.pg Evaluate the indefinite integral. x x 3 dx 0 4 and f 0 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

14 4 U of U Math 0-6 Online.( pt) set9/pr.pg If f x x 0 then f x.( pt) set9/pr.pg x If f(x) = t5 dt then f x f 0 3.( pt) set9/pr3.pg Given WeBWorK set 9. Due 4//03 at :00 AM. t 3 6t 3 dt x t 6 f x 0 cos dt t At what value of x does the local max of f x occur? x 4.( pt) set9/pr4.pg Let f be an odd function and g be an even function, and suppose that 0 4 f x dx 4 0 g x dx Use geometric reasoning to calculate each of the following: (a) f x dx. (b) g x dx. (c) f x dx. 4 (d) xg x dx. 5.( pt) set9/pr5.pg Suppose that 0 0 g x dx f x dx and 3 f x dx 3 0 g x dx 4 Use properties of definite integrals (linearity, interval additivity, and so on) to calculate the following integral: 0 3 f t 5 g t π dt Answer:. 6.( pt) set9/pr6.pg Let G x 3 Find G x 7.( pt) set9/pr7.pg Find x" x x x xt dt t t dt Answer:. 8.( pt) set9/pr8.pg Find the average value of the following function on the given interval: f x x x 6 on ( 0 3'6 The average value of f on ( 0 3' is. 9.( pt) set9/pr9.pg Use the method of substitution to find the following indefinite integral: zcos 3 3 z 3 z 3 dz Answer: + C. 0.( pt) set9/pr0.pg Use the method of substitution to find the following definite integral: π! π! cosθcos πsinθ dθ Answer:..( pt) set9/pr.pg 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 3x

15 .( pt) set9/pr.pg 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 6x y x 7 3.( pt) set9/pr3.pg 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. x y 30 x y 0 Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

16 U of U Math 0-6 Online WeBWorK set 0. Due 4//03 at :00 AM..( pt) set0/pr.pg Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y x y ; about y 9.( pt) set0/pr.pg You wake up one morning, and find yourself wearing a toga and scarab ring. Always a logical person, you conclude that you must have become an Egyptian pharoah. You decide to honor yourself with a pyramid of your own design. You decide it should have height h 3000 and a square base with side s 090 To impress your Egyptian subjects, find the volume of the pyramid. 3.( pt) set0/pr3.pg A ball of radius 0 has a round hole of radius 4 drilled through its center. Find the volume of the resulting solid. 4.( pt) set0/pr4.pg Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y x 4 y 64x about the x-axis. 5.( pt) set0/pr5.pg Find the volume of the solid generated by revolving about the x-axis the region bounded by the upper half of the ellipse x a y b and the x-axis, and thus find the volume of a prolate spheroid. Here a and b are positive constants, with a b. Volume of the solid of revolution:. 6.( pt) set0/pr6.pg The region bounded by y sinx, y 0, x 0 and π is revolved about the y-axis. Find the volume that results. Hint: xsinx dx sinx xcosx C Volume of the solid of revolution:. 7.( pt) set0/pr7.pg Find the area of the surface generated by revolving the following curve about the axis: x r cost y r sint 0 * t * π Area of the surface:. 8.( pt) set0/pr8.pg The circle x acost y asint 0 * t * π is revolved about the line x b 0 7 a 7 b thus generating a torus (doughnut). Find its surface area. Area of the torus:. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

1.(1 pt) a. Find the slope of the line passing through the points (5,0) and (9,2).

1.(1 pt) a. Find the slope of the line passing through the points (5,0) and (9,2). Peter Alfeld MATH 20-90 Spring 2004 Homework Set due 6//04 at :59 PM.( pt) a. Find the slope of the line passing through the points (5,0) and (9,2). b. Find the slope of the line passing through the points