Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment due 5/4/06 at 8:00 PM This assignment will cover notes on polynomial calculus and sections and of the book ( pt) The equation of the line with slope 5 that goes through the point (8,) can be written in the form y = mx + b where m is: and where b is: ( pt) The equation of the line that goes through the point (8,7) and is parallel to the x-axis can be written in the form y = mx + b where m is: and where b is: 3( pt) The equation of the line that goes through the point (8, 6) and is parallel to the line 4x + 3y = 3 can be written in the form y = mx + b where m is: and where b is: 4( pt) The equation of the line that goes through the point (7,3) and is perpendicular to the line 3x + y = 4 can be written in the form y = mx + b where m is: and where b is: 5( pt) A line through (-, -6) with a slope of has a y-intercept at 6( pt) An equation of a line through (0, ) which is parallel to the line y = 4x + 4 has slope: and y intercept at: 7( pt) Find the slope of the line through (8, ) and (7, 4) 8( pt) The equation of the line that goes through the points ( 3, 7) and (6,7) can be written in the form y = mx + b where m is: and where b is: 9( pt) The point P(3,0) lies on the curve y = x + x + 8 If Q is the point (x,x + x + 8), find the slope of the secant line PQ for the following values of x If x = 3, the slope of PQ is: and if x = 30, the slope of PQ is: and if x = 9, the slope of PQ is: and if x = 99, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P(3,0) 0( pt) If a ball is thrown straight up into the air with an initial velocity of 55 ft/s, its height in feet after t seconds is given by y = 55t 6t Find the average velocity for the time period begining when t = and lasting (i) 05 seconds (ii) 0 seconds (iii) 00 seconds Finally based on the above results, guess what the instantaneous velocity of the ball is when t = ( pt) Let f (x) = x 3 Find the slope of the curve y = f (x) at the point x = by calculating f (x + h) f (x) and determining what number it approaches as h approaches 0 h f (x + h) f (x) = Slope of f (x) h at x = : ( pt) Let f (x) = x 3 Find the slope of the curve y = f (x) at the point x = by calculating f (x + h) f (x) and determining what number it approaches as h approaches 0 h f (x + h) f (x) = Slope of f (x) h at x = : 3( pt) Let f (x) = x + 5 Find f (x) by calculating and determining what it ap- f (x + h) f (x) h proaches as h approaches 0 f (x + h) f (x) = f (x) = h 4( pt) If f (x) = 3x x 7, find f (x) Find f (4) [NOTE: When entering functions, make sure that you put all the necessary *, (, ), etc in your answer ] 5( pt) Let f (x) = x 3 + x Find f (x) by calculating and determining what it ap- f (x + h) f (x) h proaches as h approaches 0
f (x + h) f (x) = f (x) = h 6( pt) The slope of the tangent line to the parabola y = x + 7x + 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: 7( pt) For what values of x does the curve y = 6x 7x + have positive slope? Negative slope? Zero slope? Positive slope: x Negative slope: x Zero slope: x Instructions: For each line, enter a relational sign (eg =, <, >, etc) in the first answer box and a number in the second 8( pt) Suppose you throw a baseball 5 feet straight up and then catch it at the height you let go What is the net displacement of the baseball? feet What is the total distance traveled? feet 9( pt) If an arrow is shot straight upward on the moon with a velocity of 56 m/s, its height (in meters) after t seconds is given by h = 56t 083t What is the velocity of the arrow (in m/s) after 0 seconds? After how many seconds will the arrow hit the moon? With what velocity (in m/s) will the arrow hit the moon? 0( pt) The domain of the function f (x) = 4x 6 is all real numbers in the interval [A, ) where A equals ( pt) 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 NOTE: You will only have four attempts to get this problem right! f (x) = 5x 4 3x f (x) = x 3 + x 9 + x 7 3 f (x) = x 4 f (x) = x 4 + 3x + x 7 ( pt) Oddly, there is a function that is both even and odd It is f (x) = 3( pt) Let f (x) = x + and g(x) = x+ Then ( f f )(x) =, ( f g)(x) =, (g f )(x) =, (g g)(x) = 4( pt) This problem gives you some practice identifying how more complicated functions can be built from simpler functions Let f (x) = x 3 + and let g(x) = x + Match the functions defined below with the letters labeling their equivalent expressions (g(x)) f (x)/g(x) 3 ( f (x)) 4 g(x ) A + x + x B + x 3 + x 6 C x + x D + x 5( pt) Relative to the graph of y = x 3 the graphs of the following equations have been changed in what way? y = (x 3 )/9 y = (x + 6) 3 3 y = 6859x 3 4 y = x 3 6 A shifted 6 units down B shifted 6 units left C compressed vertically by the factor 9 D compressed horizontally by the factor 9 6( pt) The equation of the line that goes through the point (6, 3) and is parallel to the line 5x + 4y = 3 can be written in the form y = mx + b where m is: and where b is: Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment due 5/3/06 at 8:00 PM This assignment will cover notes on polynomial calculus and sections 3 and 4 of the book ( pt) For each of the following angles, find the degree measure of the angle with the given radian measure: 4π 6 π4 4π 3 3π 4π ( pt) 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): 30 80 50 60 00 3( pt) If θ = 3π 4, then sin(θ) equals cos(θ) equals tan(θ) equals sec(θ) equals 4( pt) If θ = 3π 6, then sin(θ) equals cos(θ) equals tan(θ) equals sec(θ) equals 5( pt) The angle of elevation to the top of a building is found to be from the ground at a distance of 4500 feet from the base of the building Find the height of the building 6( pt) A survey team is trying to estimate the height of a mountain above a level plain From one point on the plain, they observe that the angle of elevation to the top of the mountain is 4 From a point 000 feet closer to the mountain along the plain, they find that the angle of elevation is 9 How high (in feet) is the mountain? 7( pt) For each of the following functions enter E to indicate that the function is even, O to indicate it is odd, and N to indicate that is neither even nor odd f (x) = sinx f (x) = cosx f (x) = tanx 8( pt) For each of the following functions enter E to indicate that the function is even, O to indicate it is odd, and N to indicate that is neither even nor odd f (x) = sin x f (x) = sinx f (x) = sin(cosx) f (x) = sin(sinx) f (x) = sinx + cosx 9( pt) 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) = c) x F(x) = d) F( ) = e) x F(x) =
f) x +F(x) = g) x F(x) = h) x 3 F(x) = i) F(3) = 0( pt) Evaluate the it x x x + 6x 6 ( pt) Evaluate the it t 3 t t y a ( pt) Evaluate the it y 3 y y = 3( pt) Evaluate the it a a = a 4( pt) Evaluate the it 4 a 4 a 4 = 5( pt) Evaluate the it x 8x 7x + 4 x 5 = Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 3 due 6/7/06 at 8:00 PM This assignment will cover notes on polynomial calculus and sections 6-9 of the book ( pt) Evaluate the it sin3x x 0 7x ( pt) Evaluate the it sin7x x 0 sin3x 3( pt) Evaluate the it tanx x 0 4x 4( pt) Evaluate the it 4 + 0x x 5 5x 5( pt) Evaluate the it x + 3x (7 + 4x) 9( pt) For what value of the constant c is the function f continuous on (, ) where { ca + 3 if a (,6] f (a) = ca 3 if a (6, ) 0( pt) For what value of the constant c is the function f continuous on (, ) where { cb + 7 if b (,8] f (b) = cb 7 if b (8, ) ( pt) Calculate the following its If a it does not exist, enter DNE a) x 3x = b) θ π/ tanθ = x c) x+ x sinx d) x + x = e) x f (x) = where x+ = f (x) = 3 8x f) 7 +3x 5 x + = x 7 +6x [ x 3, for x x, for x > ( pt) 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 6( pt) Evaluate x x + 0x + x 7( pt) x 4 x x + 4 = x 4 x x + 4 = 8( pt) For what value of the constant c is the function f continuous on (, ) where { c if x = 0 f (x) = xsin x otherwise At each given value of c, determine which of these is true about F(x):
A - Not continuous because F(c) does not exits (DNE) B - Not continuous because x c F(c) does not exist (DNE) C - Not continuous beacuse x c F(c) F(c) D - It is continuous at c = at c = 3 at c = 4 at c = 3 Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 4 due 6/4/06 at 8:00 PM This assignment will cover sections 3-33 of the book ( pt) If f (x) = 0x + 8, then f ( 3)= ( pt) If f (x) = 6 + 4x x then f ( ) = 3( pt) If f (x) = 3x 8x 8, then f (x) =, and f ()= 4( pt) This problem will help you practice computing tangents in the next problem Let f (x) = x Then f (x) = The tangent to the graph of f through the point (, ) has the slope and the y-intercept It intercepts the x-axis at x = 5( pt) The slope of the tangent line to the parabola y = 3x + x + at the point (3,35) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 6( pt) The slope of the tangent line to the curve y = 3x 3 at the point ( 3, 8) is: The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 7( pt) If f (x) = 5 + 6 x + 5 x, find f (x) Find f () 8( pt) If find f (x) Find f (5) f (x) = x 6 x + 6 9( pt) If f (x) = 6x+8 3x+, find f (x) Find f (4) [NOTE: When entering functions, make sure that you put all the necessary *, (, ), etc in your answer ] 0( pt) Find the equation of the line tangent to the curve y = 5x 7x + 4 at the point (6,4) y = ( pt) If f (t) = 3 t 6, find f (t) [NOTE: Your answer should be a function in terms of the variable t and not a number! ] ( pt) Let f (x) = x 6 x + 8 x x f (x) = [NOTE: Your answer should be a function in terms of the variable x and not a number! When entering functions, make sure that you put all the necessary *, (, ), etc in your answer ] Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 5 due 6//06 at 8:00 PM This assignment will cover sections 34-36 of the book ( pt) Let f (x) = 7cosx + 9tanx Then f (x) = ( pt) If f (x) = 6sinx + 0cosx, then f (x) = 3( pt) If then f (x) = 4( pt) If f (x) = sinx + cosx f (x) = (x + 4x + 3) then f (x) = 5( pt) If f (x) = sin(sin(x)) then f (x) = 6( pt) If f (x) = (x 3 + 4x + 6) 4 then f (x) = 7( pt) If f (x) = (x + 8) then f (x) = and f (5) = 8( pt) If f (x) = sin(x 5 ) then f (x) = 9( pt) If f (x) = sin 3 x then f (x) = and f (4) = 0( pt) If f (x) = 5x + 7 then f (x) = ( pt) If f (x) = tan4x then f (x) = ( pt) If f (x) = x + 5x + 4 then f (x) = and f () = 3( pt) If f (x) = cos(3x + 8) then f (x) = 4( pt) If then f (x) = 5( pt) If f (x) = cos(4x + 4) f (x) = cos(sin(x )), then f (x) = 6( pt) The purpose of this problem is to show pretty much all of our rules at work at once If f (x) = xsin x + x then f (x) = 7( pt) Let f (x) = 3cos(x)+4sin(x) cos(x) Find f (x) f (x) = 8( pt) Find the x coordinate of the point on the curve y = x + 3x,x > 0 where the tangent line has slope -9 x= 9( pt) Let y = x + 7,x 5 x + 5 Find the equation of the line tangent to the curve at the point (9,8/7) y = 0( pt) Let f ( 3π ) = f (x) = 3xsinxcosx
( pt) Let f (x) = (cos(6x) + 5) 6 Find f (x) f (x) = ( pt) 3 Let f (x) = x /3 (x + 5) / f (x) = f (3) = 3( pt) If f (x) = 4tanx x, find f (x) Find f (3) 5( pt) Let f (x) = 8xsinxcosx f ( 3π ) = 6( pt) If f (x) = (4x + 7), find f (x) Find f () 4( pt) If find f (x) f (x) = tanx 5 secx Find f () 7( pt) Let f (x) = (4x 6) 8 (6x + 3) 4 f (x) = Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 6 due 6/8/06 at 8:00 PM This assignment will cover sections 37-30 of the book ( pt) Let f (x) = tanx Then f (x)= f (x)= f (x)= f (x)= ( pt) If f (x) = 5x4 x then f (4) (x) = Note: There is a way of doing this problem without using the quotient rule 4 times 3( pt) This problem is in preparation for the next two problems Suppose you have a cube of length s The volume of that cube is V = s 3 Now let s suppose the dimensions of that cube (and hence its volume) depend on time We are wondering about the relationship between the growth of the length versus the growth of the volume Suppose s(t) = t Then s (t) = and V (t) = Next, suppose V(t) = t Then s (t) = and V (t) = 4( pt) This is problem 6 on page 0 of our textbook The radius of a spherical watermelon is growing at a constant rate of centimeters per week The thickness of the rind is always one tenth of the radius The volume of the rind is growing at the rate cubic centimeters per week at the end of the fifth week Assume that the radius is initially zero 5( pt) Your neighbor is growing a slightly different watermelon It also has a rind whose thickness is one tenth of the radius of that watermelon However, the rind of your neighbor s water melons grows at a constant rate of 0 cubic centimeters a week The radius of your neighbor s watermelon after 5 weeks is and at that time it is growing at centimeters per week 6( pt) Let f (x) = xsinx f (x) = f (x) = 7( pt) Consider these statements written in ordinary language: A The speed of the car is proportional to the distance it has traveled B The car is speeding up C The car is slowing down D The car always travels the same distance in the same time interval E We are driving backwards F Our acceleration is decreasing Denoting by s(t) the distance covered by the car at time t, and letting k denote a constant, match these statements with the following mathematical statements by entering the letters A through E on the appropriate boxes: s < 0 s is constant s < 0 s < 0 s > 0 s = ks 8( pt) Find the slope of the tangent line to the curve 5x xy + 5y 3 = 44 at the point ( 6,) Your answer: 9( pt) suppose 4x +4x+xy = and y() = Find y () by implicit differentiation Your answer: 0( pt) Find y by implicit differentiation Match the expressions defining y implicitly with the letters labeling the expressions for y 4sin(x y) = 3ysinx 4sin(x y) = 3ycosx 3 4cos(x y) = 3ycosx
4 4cos(x y) = 3ysinx A B C D 4cos(x y) 3ycos x 4cos(x y)+3sin x 4cos(x y)+3ysin x 4cos(x y)+3cos x 4sin(x y)+3ysin x 3cos x 4sin(x y) 4sin(x y) 3ycos x 3sin x 4sin(x y) ( pt) Let A be the area of a circle with radius r If dr da dt = 4, find dt when r = Your answer: ( pt) Use implicit differentiation to find the equation of the tangent line to the curve xy 3 +xy = 0 at the point (0, ) The equation of this tangent line can be written in the form y = mx + b where m is: and where b is: 3( pt) Suppose 49 x + y 5 = and y() = 49487 Find y () by implicit differentiation 4( pt) Find the slope of the tangent line to the curve (a lemniscate) at the point ( 3,) m = (x + y ) = 5(x y ) 5( pt) Suppose x + y = 8 and y() = 49 Find y () by implicit differentiation 6( pt) Let and let Find dx dt when x = 3 xy = 5 dy dt = 7( pt) The graph of the equation x + xy + y = 9 is a slanted ellipse illustrated in this figure: Think of y as a function of x Differentiating implicitly and solving for y gives: y = (Your answer will depend on x and y) The ellipse has two horizontal tangents The upper one has the equation y = The right most vertical tangent has the equation x = That tangent touches the ellipse where y = 8( pt) A street light is at the top of a 65 ft tall pole A man 65 ft tall walks away from the pole with a speed of 65 feet/sec along a straight path How fast is the tip of his shadow moving when he is 47 feet from the pole? Your answer: 9( pt) The altitude of a triangle is increasing at a rate of 0 centimeters/minute while the area of the triangle is increasing at a rate of 40 square centimeters/minute At what rate is the base of the triangle changing when the altitude is 5 centimeters and the area is 940 square centimeters? Your answer: 0( pt) 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 6 feet high? Recall that the volume of a right circular
cone with height h and radius of the base r is given by V = π 3 r h Your answer: feet per minute ( pt) Water is leaking out of an inverted conical tank at a rate of 3000 cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate The tank has height 8 meters and the diameter at the top is 35 meters If the water level is rising at a rate of 3 centimeters per minute when the height of the water is 50 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute Your answer: cubic centimeters per minute ( pt) A plane flying with a constant speed of 30 km/min passes over a ground radar station at an altitude of 3 km and cbs at an angle of 40 degrees At what rate, in km/min is the distance from the plane to the radar station increasing 5 minutes later? Your answer: kilometers per minute 3( pt) A spherical snowball is melting in such a way that its diameter is decreasing at rate of 04 cm/min At what rate is the volume of the snowball decreasing when the diameter is 9 cm Your answer (cubic centimeters per minute) should be a positive number 4( pt) You are blowing air into a spherical balloon at a rate of 4π 3 cubic inches per second (The reason for this strange looking rate is that it will simplify your algebra a little) Assume the radius of your balloon is zero at time zero Let r(t), A(t), and V (t) denote the radius, the surface area, and the volume of your balloon at time t, respectively (Assume the thickness of the skin is zero) All of your answers below are expressions in t: r (t) = inches per second, A (t) = square inches per second, and V (t) = cubic inches per second 5( pt) A child is flying a kite If the kite is h feet above the child s hand level and the wind is blowing it on a horizontal course at v feet per second, the child is paying out cord at feet per second when s feet of cord are out Assume that the cord remains straight from hand to kite 6( pt) Use implicit differentiation to find the slope of the tangent line to the curve y x 4y = x9 + 7 at the point (, 33 8 ) m = 7( pt) Use linear approximation, ie the tangent line, to approximate 3 as follows: Let f (x) = 3 x The equation of the tangent line to f (x) at x = can be written in the form y = mx + b where m is: and where b is: Using this, we find our approximation for 3 is 8( pt) Let f (x) = ( x 4 ) For what values of x is f (x) = 0? Write the answers in increasing order, 9( pt) Let f (x) = (x + 0) ( x 9 ) For what value of x is f (x) = 0? 30( pt) Find the slope of the tangent line to the curve given by the equation y + 6xy = 0 at the point ( 83333333333333,) y = 3( pt) If the variables s and t are related by the equation st + 5t 3 = find ds ds dt dt = 3( pt) 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 0 feet from the wall? Answer: ft/sec Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR 3
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 7 due 7/7/06 at 8:00 PM This assignment will cover sections 4-44, 46 and 47 of the book ( pt) The function f (x) = 4x 3 36x + 9x 6 is increasing on the interval (, ) It is decreasing on the interval (, ) and the interval (, ) The function has a local maximum at ( pt) Find the point on the line x + 8y = 0 which is closest to the point (, 5) (, ) 3( pt) A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 4 x What are the dimensions of such a rectangle with the greatest possible area? Width = Height = 4( pt) A cylinder is inscribed in a right circular cone of height 5 and radius (at the base) equal to 3 What are the dimensions of such a cylinder which has maximum volume? Radius = Height = 5( pt) A fence 4 feet tall runs parallel to a tall building at a distance of 7 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? 6( pt) If 900 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( pt) The function f (x) = 4x + 3x 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 8( pt) Consider the function f (x) = 8(x 3) /3 For this function there are two important intervals: (,A) and (A, ) where A is a critical point Find A For each of the following intervals, tell whether f (x) is increasing (type in INC) or decreasing (type in DEC) (,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, ): 9( pt) A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle What is the area of the largest possible Norman window with a perimeter of 3 feet? 0( pt) A rancher wants to fence in an area of 500000 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? ( pt) 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 = 68 s, is given by v(t) = 000638000t 3 008038t + 49t + 46 (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 ( pt) Consider the function f (x) = x 3 + 3x + 3x 4 Find the average slope of this function on the interval (0,3) By the Mean Value Theorem, we know there exists a c in the open interval (0,3) such that f (c) is equal to this mean slope Find the value of c in the interval which works:
3( pt) Answer the following questions for the function f (x) = x x + 9 defined on the interval [ 4,5] A f (x) is concave down on the interval to B f (x) is concave up on the interval to C The inflection point for this function is at x = D The minimum for this function occurs at x = E The maximum for this function occurs at x = 4( pt) The function f (x) = x 3 33x + 80x + 0 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 = with value f (x) =, and a local maximum at x = with value f (x) = 5( pt) The function f (x) = x 3 + 39x 40x + 8 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 = with value f (x) =, and a local maximum at x = with value f (x) = 6( pt) Consider the function f (x) = x 3 5x on the interval [ 3,3] 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 ( 3,3) 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 7( pt) Consider the function f (x) = x 3 x 30x + 5 on the interval [ 5,9] 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 ( 5,9) 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( pt) Consider the function f (x) = x 3 + 36x 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, ): 9( pt) Consider the function f (x) = 8x + 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, ): 0( pt) Consider the function f (x) = x 5 + 45x 4 00x 3 + 3 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, ):
( pt) 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? ( pt) 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 3( pt) This is problem 30 on page 67 of the textbook, see the Figure there A huge conical tank is to be made from a circular piece of sheet metal of radius 0 meters by cutting out a sector with vertex angle θ and then welding together the straight edges of the remaining piece The angle θ = radians will result in the cone with the largest possible volume There is a very detailed solution for this problem Make sure you check it out after the set closes 4( pt) Here s an interesting example in which the graph crosses its horizontal asymptote If you think an answer below is, enter INF If you think an answer is, enter -INF If the function does not have the requested attribute, answer no If more intervals are given than are needed, enter no in all extra blanks Let y = x + 6 (x 0) 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 (, ) 5( pt) Answer the following questions for the function f (x) = x3 x 5 defined on the interval [ 8,8] Enter points, such as inflection points in ascending order, ie smallest x values first Enter intervals in ascending order also A The function f (x) has vertical asympototes at and B f (x) is concave up on the region to and to C The inflection points for this function are, and Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR 3
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 8 due 7/5/06 at 8:00 PM This assignment will cover sections 5 of the book ( pt) Given f (x) = x + 5 and f (0) = and f (0) = 6 Find f (x) = and find f (4) = ( pt) A car traveling at 50 ft/sec decelerates at a constant 3 feet per second squared How many feet does the car travel before coming to a complete stop? 3( pt) A ball is shot straight up into the air with initial velocity of 44 ft/sec Assuming that the air resistance can be ignored, how high does it go? (Assume that the acceleration due to gravity is 3 ft per second squared) 4( pt) A ball is shot at an angle of 45 degrees into the air with initial velocity of 49 ft/sec Assuming no air resistance, how high does it go? How far away does it land? (Assume that the acceleration due to gravity is 3 ft per second squared) 5( pt) Consider the function f (x) = 8x 3 8x + 9x 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( pt) Consider the function f (x) = 36x 3 x + 0x Enter an antiderivative of f (x) 7( pt) Consider the function f (x) whose second derivative is f (x) = 8x + 3sin(x) If f (0) = 4 and f (0) = 3, what is f (5)? 8( pt) Consider the function f (x) whose second derivative is f (x) = 0x + 9sin(x) If f (0) = 3 and f (0) =, what is f (x)? 9( pt) Given that the graph of f (x) passes through the point (4,9) and that the slope of its tangent line at (x, f (x)) is 4x + 5, what is f (4)? 0( pt) Consider the function f (x) = 8x 9 +6x 7 0x 4 Enter an antiderivative of f (x) ( pt) Consider the function f (x) = 9 x 0 x 6 Let F(x) be the antiderivative of f (x) with F() = 0 Then F(3) equals ( pt) Consider the function f (x) = 3 x 3 4 x 5 Let F(x) be the antiderivative of f (x) with F() = 0 Then F(x) = 3( pt) Suppose f (x) = x + sinx and F is an antiderivaive of f that satisfies Then F(x) = 4( pt) Suppose F(π) = 0 f (x) = (x ) and F is an antiderivaive of f that satisfies Then F(x) = 5( pt) Suppose F(0) = f (x) = x + and F is an antiderivaive of f that satisfies F(0) = Then F(x) = 6( pt) This problem revisits our familar formulas for vertically moving objects Suppose we know that h (t) = g, h (0) = V, and h(0) = H Then h(t) =
R R s(s + ) 7( pt) Evaluate the integral: ds s Answer: + C 8( pt) Evaluate the indefinite integral: 3y y + 5 dy Answer: + C 9( pt) Find: R sin xdx Answer: + C 0( pt) Consider the function f (t) = 9sec (t) 9t Let F(t) be the antiderivative of f (t) with F(0) = 0 Then F() = ( pt) Consider the function f (x) whose second derivative is f (x) = 4x + 9sin(x) If f (0) = 3 and f (0) = 3, what is f ()? Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 9 due 7/9/06 at 8:00 PM This assignment will cover sections 5 and 53 of the book ( pt) Compute the sum 6 i= i = ( pt) Compute the sum 7 (i ) = i= 3( pt) Compute the sum n (i ) = i= 4( pt) Consider the differential equation: dy dx = x y a) Find the general solution to the above differential equation (Instruction: Call your integration constant C) Answer: y = b) Find the particular solution of the above differential equation that satisfies the condition y = 4 at x = Answer: y = 5( pt) Consider the differential equation: du dt = u3 (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 = 6( pt) An object is moving along a coordinate line subject to acceleration a (in centimeters per second per second) as follows a = ( +t) 4 with initial velocity v 0 = 0 (in centimeters per second) and directed distance s 0 = 0 (in centimeters) Find both the velocity v and the directed distance s after seconds Velocity after seconds: centimeter(s) per second Directed distance after seconds: centimeter(s) 7( pt) The wolf population P in a certain state has been growing at a rate proportional to the cube root of the population size The population was estimated at 000 in 980 and at 700 in 990 a) Find the differential equation for P(t) and the corresponding conditions (Instruction: Use C for the constant of proportionality) dp dt = P( ) = and P( ) = b) Solve your differential equation P = c) When will the wolf population reach 4000? The population will reach 4000 by the year 8( pt) Find 7 k=3 ( ) k k k + = 9( pt) Find 6 k= k sin(kπ/) = 0( pt) Find the value of the following collapsing sum: 0 k= (k k ) = ( pt) Use the Special Sum Formulas (see Section 53 of Varberg, Purcell and Rigdon) to find: ((i )(4i + 3)) = 0 i= ( pt) In statistics, we define the mean x and the variance s of a sequence of numbers x,,x n by x = n n i= x i s = n n i= (x i x) Find x and s for the sequence of numbers, 5, 7, 8, 9, 0, 4 x = s =
Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment 0 due 7/6/06 at 8:00 PM This assignment will cover sections 55-6 of the book ( pt) If f (x) = R x t5 dt then f (x) = f ( 5) = ( pt) The value of R 6 0 (x + 6) dx is 3( pt) The value of R 7 5 x dx is is 4( pt) The value of (8x + 3)dx 5( pt) The value of R 6 (3x 6x + 6)dx is 6( pt) The value of R 4 4 (6 x )dx is R 7( pt) The value of 63 0x +9 x dx is R 8( pt) The value of π0 8sin(x)dx is 9( pt) 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 R 4 4 6 x dx = 0( pt) 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 R 80 9x dx = R 9 4 ( pt) Consider the integral Z 8 (3x + x + 3)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 ( pt) Evaluate the integral R 45 sin(t)dt = 3( pt) Find the derivative f (x) = of f (x) = Z x 4( pt) Find the derivative h (x) = of h(x) = Z sin(x) 5 ( ) 0 t dt (cos(t 3 ) +t) dt 5( pt) Find the derivative g (x) = of Z x u + g(x) = 4x u du 6( pt) If R R 0 f (x)dx = 4, 0 f (x)dx =, and R 0 g(x)dx = 3, evaluate each integral (a) R f (x)dx = (b) R 0 f (x)dx = (c) R 0 3 f (x)dx = (d) R 0 [g(x) 3 f (x)]dx = (e) R 0 f ( x)dx = 7( pt) Evaluate: n ( + i ) n n n 8( pt) Evaluate I = I= 9( pt) Evaluate I = I= 0( pt) Given f (x) = i= Z 8 0 Z 9 Z x 0 x x 3 + dx x (x + ) dx t 6 + cos (t) dt At what value of x does the local max of f (x) occur?
x = ( pt) 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 = 7x ( pt) 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 + 7 3( pt) 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 =,x + y = 0 Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR
Peter Alfeld Math 0-, Summer 006 WeBWorK Assignment due 8//06 at 8:00 PM This assignment will cover sections 6-66 of the book ( pt) Find the volume of the solid obtained by rotating the triangular region bounded by the x-axis, the line y = x and the line x = 6 about the x-axis Volume = ( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y = 3x,x =,y = 0, about the x-axis 3( pt) Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = x 4 y = 5x about the x-axis 4( pt) 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 = 3, about the y-axis 5( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y = 0,y = x(3 x) about the axis x = 0 6( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y = x 6,y = ; about y = 4 7( pt) Find the length of the curve defined by from x = 3 to x = 7 y = x 3/ 3 8( pt) Find the length of the following curve: y = Z x π/6 64sin ucos 4 u du, π 6 x π 3 Length of the curve: 9( pt) 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 = 40 and a square base with side s = 70 To impress your Egyptian subjects, find the volume of the pyramid 0( pt) A ball of radius 7 has a round hole of radius 4 drilled through its center Find the volume of the resulting solid ( pt) 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: ( pt) The masses and coordinates of a system of particles are given by the following: 5,( 3,); 6,(, );,(3,5); 7,(4,3);,(7, ) Find the moments of this system with respect to the coordinate axes, and find the coordinates of the center of mass Moment with respect to the x-axis: Moment with respect to the y-axis: Center of mass: (, ) 3( pt) Find the centroid of the region bounded by the following curves: y = x,y = x + 3 Hint: Make a sketch and use symmetry where possible Centroid: (, ) Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR