WeBWorK demonstration assignment

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1 WeBWorK demonstration assignment 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) setdemo/demo pr.pg Evaluate the expression 5 9 =..( pt) setdemo/demo pr.pg Evaluate the expression 4 =. Enter you answer as a decimal number listing at least 4 decimal digits. (WeBWorK will reject your answer if it differs by more than one tenth of percent from what it thinks the answer is.).( pt) setdemo/demo pr.pg Let r Evaluate 4 π r. Next, enter the expression 4 π r and let WeBWorK compute the result. 4.( pt) setdemo/demo pr4.pg Enter here the expression a b. Enter here the expression a b. 5.( pt) setdemo/demo pr5.pg Enter here the expression a b Enter here the expression a b c d If WeBWorK rejects your answer use the preview button to see what it thinks you are trying to tell it. 6.( pt) setdemo/demo pr6.pg Enter here Enter here Enter here a a a b b a b a b 7.( pt) setdemo/demo pr7.pg Enter here Enter here Enter here x y x x y x y x y 8.( pt) setdemo/demo pr8.pg Enter here the expression the expression the expression the expression the expression the expression the expression b b 4ac a Note: this is an expression that gives the solution of a quadratic equation by the quadratic formula.

2 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment due 9/0/0 at :59 PM..( pt) set/c0sp9.pg This problem gives you some practice identifying how more complicated functions can be built from simpler functions. Let f x x and let g x x. Match the functions defined below with the letters labeling their equivalent expressions.. f x. g x. f g x 4. g x f x A. x x x 4 B. x x x C. x x 6 D. x.( pt) 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 5x 8 x. f x x. f x x 8 6x x 4 4. f x x x 7 x.( pt) set/prob.pg Let f x 5x and g x 4x 5x. f g Recall that by convention the absence of an arithmetic operator denotes multiplication. 4.( pt) set/prob4.pg Let f x 4x 5 and g x x x. f g 6 5.( pts) set/composition.pg Let f be the linear function (in blue) and let g be the parabolic function (in red) 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 closely. Note: If the answer does not exist, enter DNE :. (f o g)( ) =. (g o f)( ) =. (f o f)( ) = 4. (g o g)( ) = 5. (f + g)( 4 ) = 6. (f / g)( ) = 6.( pts) set/c0sp.pg Relative to the graph of y x the graphs of the following equations have been changed in what way?. y x 7. y x. y 7x 4. y x A. shifted units down B. shifted units up C. shifted units left D. shifted units right E. stretched vertically by the factor 89 F. compressed vertically by the factor 89 G. compressed vertically by the factor 7 H. stretched vertically by the factor 7

3 7.( pts) set/prob6.pg Relative to the graph of y tan x the graphs of the following equations have been changed in what way?. y tan x 8. y tan x 7. y tan x 7 4. y tan x 7 A. shifted 7 units down B. shifted 7 units up C. shifted 7 units left D. shifted 7 units right E. compressed horizontally by the factor 8 F. stretched horizontally by the factor 8 G. compressed vertically by the factor 8 H. stretched vertically by the factor 8 8.( pt) set/prob.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 to the right and 9 up. 9.( pt) set/srw5 a.pg For each of the following angles, find the degree measure of the angle with the given radian measure: 4π 6 π4 4π 7π 4π 0.( pt) set/srw5 c.pg 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): ( pt) set/srw5 d.pg 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 π π π.( pts) set/srw6 4.pg A plane is flying at an elevation of 4000 feet. It is within sight of the airport and the pilot finds that the angle of depression to the airport is 6. (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: miles Find the distance between a point on the ground directly below the plane and the airport: miles Recall that there are 580 feet in one mile..( pts) set/srw6 54.pg 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 0 and. How high (in feet) is the ballon? 4.( pts) set/s.pg A particle moves along a straight line and its position at time t is given by s t t 4t 4t 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?

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5 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment due 9/8/0 at 5:00 PM..( pt) set/s 8.pg Evaluate the limit x lim x x 4x.( pt) set/s 9.pg Evaluate the limit.( pt) set/s 7.pg Evaluate the limit lim b lim s 4 4.( pt) set/s 6.pg Evaluate the limit 5.( pt) set/s 6.pg Evaluate the limit b b b 4 s s lim t 9 t 9 t 9 4x 8x 5 lim x x 8 6.( pts) set/limits.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 limit does not exist or is not defined. a) lim x F x = b) lim x F x = c) lim x F x = d) F = e) lim x F x = f) lim x F x = g) lim x F x = h) lim x F x = i) F = 7.( pt) set/prob.pg Remember, one of the main applications of limits has to to with derivatives. Can you still do this problem from week?: x h x lim h 0 h is. 8.( pts) set/s.pg The point P 7 lies on the curve y x x 5. If Q is the point x x x 5, find the slope of the secant line PQ for the following values of x. If x, the slope of PQ is: and if x 0, the slope of PQ is: and if x 9, the slope of PQ is: and if x 99, the slope of PQ is:

6 Based on the above results, guess the slope of the tangent line to the curve at P 7. 9.( pts) set/s 5.pg If a ball is thrown straight up into the air with an initial velocity of 65 ft/s, its 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. 0.( pts) set/math0spring00 korevaar p 9.pg The following problem illustrates several of the limit theorems on page 7 of the text, and shows how the limit concept is related to error analysis: Suppose we have the estimates f x g x Using only this information, make the best possible estimates for (a) f x! g x 4 (b) f x g x 48 (c) f x g x 0 75.( pt) set/math0spring00 korevaar p.pg Using your calculator to test values of h approaching zero, you should be able to guess the following limits. (We will see geometric reasons for all of these in class.) Remember, our trig functions eat radians unless otherwise noted. cos h (a) lim h 0 = h sin h (b) lim h 0 = h (c) lim h 0 cos h h =.( pts) set/math0spring00 korevaar p.pg (a) Using the angle addition formulas for sin and cos (which you have memorized!!) Fill in the blanks with appropriate functions of h so that sin x h sin x = sin x + cos x h cos x h cos x = sin x + cos x h Hint: in each blank you will be entering expressions related to the functions of h which you studied in problem. (b) Use your work and part (a), and from problem to find the derivative functions for sin(x) and cos(x): sin x h sin x lim h 0 = h cos x h cos x lim h 0 = h.( pts) set/math0spring00 korevaar p.pg Let f x " x 7 Algebraically simplify the secant line slope % f x h f x, and enter the numerator below: h f x h f x x h 7 x 7 h Let h $# 0 to deduce the derivative, f x The equation of the tangent line passing through the point on the graph of f with x-coordinate 9 can be written in the form y mx b, where m b

7 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment due 9/5/0 at 5:00 PM..( pt) set/psprob4.pg Evaluate the limit sin5x lim x 0 sinx.( pt) set/psprob5.pg Evaluate the limit tanx lim x 0 4x.( pt) set/s 7.pg If f x 7 7x 4x, find f %. 4.( pts) set/korevaar math0spring00 p4 4.pg A city is hit by a flu epidemic. Officials estimate that t days after the beginning of the epidemic the number of persons sick with the flu is given by p t & 90t t when 0 ' t ' 45 At what rate is the flu spreading at time t 5 days? people/day At what rate is the flu spreading at time t 6 days? people/day How many days after the outbreak are the most people sick? t = days. 5.( pts) set/golden math0fall00 p5 7.pg The rate of change of electric charge with respect to time is called current. Suppose that t t coulombs of charge flow through a wire in t seconds. (a) Find the current in amperes (coulombs per second) after seconds. (b) When will a 0-ampere fuse in the line blow? a) Current after seconds: amperes. b) A 0-ampere fuse will blow at: seconds. 6.( pt) set/golden math0fall00 p5 8.pg The radius of a spherical balloon is increasing at the rate of 0.5 inch per second. If the radius is 0 at time t 0, find the rate of change in the volume at time t. Rate of change in volume at t : inch /second. 7.( pt) set/s.pg If f 4 x 7 x, find f % x Find f % 4. 8.( pt) set/s 4.pg x. If f x 7sinx 0cosx, then f % x 9.( pt) set/s 4 4.pg If find f % x. f x sinx cosx 0.( pt) set/s 5.pg x x 4, find f % If f x Find f %..( pts) set/psprob9.pg If f x 6x 4 7x, find f % x. Find f % 5. x. [NOTE: When entering functions, make sure that you put all the necessary *, (, ), etc. in your answer. ].( pts) set/s 4 0a.pg Let f x 4xsinxcosx π f %.( pts) set/ur dr 5 0.pg Let f % x f x 7x 5 5 x ( pts) set/z.pg The purpose of this problem is to show pretty much all of our rules at work at once.

8 If find f % x. f x xsin x x 5.( pts) set/golden math0fall00 p5 0.pg A space traveller is moving from left to right along the curve y x. When she shuts off the engines, she will continue travelling along the tangent line at the point where she is at that time. At what point should she shut off the engines in order to reach the point (4,5)? She should shut off the engine at (, )

9 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment 4 due 0/4/0 at 5:00 PM..( pt) set4/s 5 4.pg If f x sin x 4, find f % x. Find f %..( pt) set4/s 5 5.pg If f x sin 4 x, find f % x. Find f %..( pt) set4/s 5 7.pg If f x tan5x, find f % x. Find f %. 9.( pt) set4/cs8p5.pg 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 45 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing 4 minutes later? 0.( pt) set4/s 8.pg Let xy and let dy dt 5 Find dx dt when x. 4.( pt) set4/cs6p.pg Find the slope of the tangent line to the curve x xy y 690 at the point 7. 5.( pt) set4/s 6 9.pg Use implicit differentiation to find the equation of the tangent line to the curve xy xy 8 at the point 4. The equation of this tangent line can be written in the form y mx b where m is: and where b is: 6.( pt) set4/s 6 4.pg If x y 8 and y 9 " 5, find y% differentiation. 9 by implicit 7.( pt) set4/cs8p.pg A street light is at the top of a ft. tall pole. A man 6 00 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/s 8.pg Let A be the area of a circle with radius r. If dr dt 4, find da dt when r 5..( pt) set4/s 8 5.pg A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.4 cm/min. At what rate is the volume of the snowball decreasing when the diameter is cm. (Note the answer is a positive number)..( pt) set4/s 8.pg Gravel is being dumped from a conveyor belt at a rate of 0 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 7 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V πr h..( pt) set4/cs8p.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 000 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.

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11 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment 5 due 0/9/0 at 5:00 PM..( pts) set5/korevaar-math0spring00-ps6-q.pg A 0 foot ladder is leaning against a building. If the bottom of the ladder is sliding along the level pavement directly away from the building at feet per second, how fast is the top of the ladder moving down when the foot of the ladder is 6 feet from the wall? Answer: ft/sec.( pts) set5/korevaar-math0spring00-ps6-q.pg A large trough 6 feet long has a cross section in the form of an isosceles triangle (with base at the top) 4 feet deep, and 6 feet across. If water is filling the trough at the rate of cubic feet per minute, how fast is the water level rising when the water is feet deep? Answer: ft/min.( pt) set5/s 9.pg Let y 4x. Find the change in y, y when x 4 and x 0 Find the differential dy when x 4 and dx 0 4.( pt) set5/cs9p7.pg Use linear approximation (equivalently, use differentials) to estimate the amount of paint in cubic centimeters needed to apply a coat of paint cm thick to a hemispherical dome with a diameter of meters. 5.( pts) set5/s 9 0.pg Use linear approximation, i.e. the tangent line, to approximate 5 4 as follows: Let f x ( x. The equation of the tangent line to f x at x 5 can be written in the form y mx b where m is: and where b is: Using this, we find our approximation for 5 4 is 6.( pts) set5/nsc4 6.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. 7.( pts) set5/s 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? 8.( pts) set5/cs8p.pg 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 = 9.( pts) set5/nsc4 6 6.pg A fence 5 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.( pts) set5/csp.pg The function f x x x 0x 6 is decreasing on the interval (, ). It is increasing on the interval (, ) and the interval (, ). The function has a local maximum at..( pts) set5/csp4.pg For x )+*, 5- the function f is defined by f x x 4 x 5 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?.( pts) set5/golden-math0fall00-ps6-q.pg Identify the critical points and find the maximum value and minimum value of the following function on the given interval. Recall, critical points are either endpoints, stationary points, or singular points. f x x x, over *. -. Critical Points:,,,. Maximum:. Minimum:.

12 Instructions: When entering the critical points, please enter them in the order that they appear on the real line.

13 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment 6 due 0/6/0 at 0:00 PM..( pt) set6/s 6.pg Consider the function f x / x 45x 00x. 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 :.( pt) set6/s 4.pg The function f x ( x 6x 9x 9 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.( pt) set6/s 6a.pg Consider the function f x 0 x 8x. 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 : 4.( pt) set6/cs4p.pg Answer the following questions for the function f x x x 6 defined on the interval *, 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 has x- coordinate D. The minimum for this function occurs at E. The maximum for this function occurs at 5.( pt) set6/s 4 0.pg Consider the function f x x 5 0x 4 00x 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 : 6.( pt) set6/cs8p.pg Find the point on the line 6x 6y 0 which is closest to the point. (, ) 7.( pt) set6/s 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 4 feet? (The perimeter of this window does not include the top edge of the rectangle.) 8.( pt) set6/p8-.pg Consider the function f x x x The x-intercept for the graph of f is at x=

14 The y-intercept for the graph of f is at y= This function has a vertical asymptote at x a, where a= This function has a horizontal asymptote at y b, where b= For each of the following intervals, tell whether f x is increasing (type INC) or decreasing (type DEC): a a For each of the following intervals, tell whether f x is concave up (CU) or concave down (CD): a a

15 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment 7 due 0/0/0 at :00 PM..( pt) set7/pr.pg Given f %% x 5x and f % and f. Find f % x and find f 4.( pt) set7/pr.pg Evaluate the indefinite integral: y y 5 dy. Answer: + C..( pt) set7/pr.pg Find: sin xdx. Answer: + C. Hint: Consider using a trigonometry identity to express sin x in terms of cos x. 4.( pt) set7/pr4.pg Evaluate the indefinite integral. x 5 x dx 5.( pt) set7/pr5.pg Consider the differential equation: dy dx 54 x y a) Find the general solution to the above differential equation. (Instructions: After you ve separated variables and integrated you will have an integration constant and an implicit equation relating x and y. When you solve this equation explicitly for y your original integration constant may be multiplied or divided by a number, giving you a new constant. Webworks wants you to denote this final constant with the letter C.) Answer: y. b) Find the particular solution of the above differential equation that satisfies the condition y 4 at x. Answer: y. 6.( pt) set7/pr6.pg Consider the differential equation: du dt u t 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 final integration constant is denoted by the letter C, as in problem.) Answer: u. b) Find the particular solution of the above differential equation that satisfies the condition u 4 at t 0. Answer: u. 7.( pt) set7/pr7.pg Consider the integral 8 5 x x dx (a) Find the Riemann sum for this integral using right endpoints and n. (b) Find the Riemann sum for this same integral, using left endpoints and n 8.( pt) set7/pr8.pg 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. 9.( pt) set7/pr9.pg x dx In this problem you will test your understanding of the Riemann sum way of computing area on an example you can check with elementary geometry. Use a geometry formula to find the area of the trapezoid bounded by the y-axis, the line x, the x-axis, and the line y x Area:

16 Now divide the interval * 0 - into n equal-length subintervals. The width of each subinterval is. For ' i ' n, the ith subinterval is [, ]. Use the right-hand endpoint of each subinterval to compute the Riemann sum for the area. Write your answer as a function of n, without any summation signs. You will need the summation formulas in section 5. of the text. R n =. lim n R n =.

17 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment 8 due /6/0 at :00 PM..( pt) set8/c4s4p5.pg x dx is The value of 0.( pt) set8/c4s4p6.pg The value of 6 5 x dx is.( pt) set8/s4 4 7.pg Evaluate the definite integral 7 x 4.( pt) set8/s4 4 0.pg Evaluate the definite integral 8 4 dx x 0x 0 dx 5.( pt) set8/s4 4 7.pg Evaluate the definite integral 6 0x dx x 6.( pt) set8/math0-90 fall00 rossi set0 p-9.pg Evaluate 4 I x x 0dx 0 I= 7.( pt) set8/math0-90 fall00 rossi set0 p-0.pg Evaluate x I dx 7 x I= 8.( pt) set8/math0-90 fall00 rossi set pr.pg Let f 4 x dx f x dx Find f x dx 4 4 and f x dx f x dx 9.( pt) set8/sc5 4.pg Use part I of the Fundamental Theorem of Calculus to find the derivative of f x x t 7 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. ] 0.( pt) set8/sc5 4 8.pg Use part I of the Fundamental Theorem of Calculus to find the derivative of h sin8 x9 x cos t! t dt h% x [NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary *, (, ), etc. ]

18 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment 9 due //0 at 5:00 PM..( pt) set9/pr.pg Use part I of the Fundamental Theorem of Calculus to find the derivative of g 8x u 5 x u du 9x.( pt) set9/set/golden-math0fall00-ps0-q6.pg Let f be an odd function and g be an even function, and suppose that 0 f x dx 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. (d) xg x dx..( pt) set9/set/math0-90 fall00 rossi set0 p7.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 4x 4.( pt) set9/set/math0-90 fall00 rossi set0 p8.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 5x y x 5.( pt) set9/set/math0-90 fall00 rossi set0 p.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 4 x y 0 6.( pt) set9/set/korevaar math0spring00-set- p8.pg Find the volume of the solid obtained by rotating the triangular region bounded by the x-axis, the line y 4x and the line x 0 about the x-axis. Volume = 7.( pt) set9/set/math0-90 fall00 rossi set p5.pg Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y 4x x y 0 about the x-axis 8.( pt) set9/set/math0-90 fall00 rossi set p.pg Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y x 4 y 8x about the x-axis. 9.( pt) set9/set/math0-90 fall00 rossi set p6.pg Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y x x 0 x about the y-axis 0.( pt) set9/set/math0-90 fall00 rossi set p8.pg Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y 0 y x 7 x about the axis x 0

19 Shawn Harrison WW Prob Lib Math 0-, Fall 00 WeBWorK assignment 0 due //0 at 5:00 PM..( pts) set0/benny.pg Find the length of the curve defined by from x 5 to x 7. y 4x :.( pts) set0/ns6 4a.pg Consider the parametric curve given by the equations x t t 0t y t t 0t 8 How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=0, and t=9?.( pts) set0/math0-90 fall00 rossi p-7.pg 0cos θ and g θ is given 0sin θ If f θ is given by: f θ by: g θ Find the total length of the astroid described by f θ and g θ. (The astroid is the curve swept out by f θ ; g θ as θ ranges from 0 to π ) ' ' 4.( pts) set0/korevaar math0spring00-set- p4.pg Find the area of the cone surface obtained by rotating the line segment y x, for 0 x 6 about the x-axis. Surface Area = 5.( pts) set0/golden-math0fall00-ps-q.pg Find the area of the surface generated by revolving the following curve about the x axis: x r cost y r sint 0 ' t ' π Area of the surface:. 6.( pts) set0/golden-math0fall00-ps-q4.pg The circle x acost y asint 0 ' t ' π is revolved about the line x b 0 a b thus generating a torus (doughnut). Find its surface area. Area of the torus:.