Homework Assignments Math /02 Spring 2015

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1 Homework Assignments Math 1-01/0 Spring 015 Assignment 1 Due date : Frida, Januar Section 5.1, Page 159: #1-, 10, 11, 1; Section 5., Page 16: Find the slope and -intercept, and then plot the line in problems: #, ; Find verte, ais of smmetr, focus, focal width, and then plot the parabola in: #6, 8; Section.1, Page 1: #,, 7, 8; Section., Page 151: #,, 6, 1, 1, 15, 9, 0,, ; Required Additional Eercises #1 5 (below). Instructions In problems 1-, plot the function and indicate its domain and range. 1. f ( ) = 1 0 < 5. f ( ) = 1-0 < < 5. f ( ) = for all 0.. Determine the value of f [ f ( ) ] for the function f in problem above. 5. Find the eact coordinates of the square DEFG inscribed in ABC given that A = ( 0, 0 ), B = (, ) and C = ( 5, 0 ). Assignment Due date : Monda, Februar Section.1, Page 75: #,, 7, 8, 9,1, 15, 17, 18,, 5, 6, 8. Required Additional Eercises #1 (below) (a) Sketch a graph of the function: f ( ) = 1 0. > 0 (b) Use the formula for f from part (a) to determine the value of each of the following limits, the eists. (i) lim f ( ) 0 (ii) lim f ( ) 0 (iii) lim f ( ) < 1. (a) Sketch a graph of the function: f ( ) = 1 < 1. = 1 > 1 (b) Use the formula for f from part (a) to determine the value of each of the following limits, the eists. (i) lim f ( ) 1 (ii) lim f ( ) 1 (iii) lim f ( ) 1 (iv) lim f ( ) 1 0 (v) lim f ( ) 1 (vi) lim f ( ) 1. A graph of the function = g ( ) is shown in the figure. (a) State the domain and range of g. (b) Use the graph to determine each of the limits: ( i ) lim g( ) 1 (ii) lim g( ) 1 (iii) lim g( ) 1 (iv) lim g( ) 0 (v) lim g( ) 0 (vi) lim g( ) 0.

2 . (a) Write a piecewise definition for the function = g ( ) shown in the figure. (b) State the domain and range of g. (c) Determine the limits: lim g( ), lim g( ). 1 1 Assignment Due date : Monda, Februar 9 Section., Page 758: Use the method of Eample 1, Page 751, to find the derivative of = f ( ) in each of the following problems: #1, 6, 8, 9, 11, 1. To find the slope of the graph of = f ( ) at the given value, = a, evaluate the limit: d f ( a Δ ) f ( a ) = f ( a ) = lim d Δ 0 = a Δ in each of the following problems: #1, 15, 17, 19. Required Additional Eercises #1 7 (below). Instructions In problems 1-, estimate the slope of the curve at the point P Assign one of the following descriptors to each point A, B, C, D, E, F in the figure: large positive slope small positive slope zero slope small negative slope large negative slope. 6. Estimate the value of f (1) in the figure below. 7. Find the equation of the tangent line to f() at the point P.

3 Assignment Due date : Wednesda, Februar 18 Eercises # A stone is thrown verticall upward into the air with an initial velocit of 96 ft / s. On Mars, the height of the stone above the ground after t seconds is h = 96t 6t, and on Earth, h = 96t 16t. How much higher will the stone travel on Mars than on Earth?. The graph displas the position s = f ( t ) of a freight train locomotive t hours after 5:00pm relative to its starting point s = 0, where s is measured in miles. (a) Describe the speed of the train. Specicall, when is it speeding up and when is it slowing down? s = f ( t ) (b) At approimatel what time of da is it traveling the fastest? The slowest? Wh? In problems - 8, determine whether the given functions is: (a) continuous at = 1 lim f ( ) = f (1). (b) dferentiable at = 1 1 f is continuous at =1 and (c) Then sketch a graph of = f ( ). lim Δ 0 f (1 Δ ) f ( 1 ) Δ f (1) = f (1) f (1 Δ ) f ( lim Δ Δ 0 1 ).. f ( ) = 1. f ( ) = f ( ) = = 1 1 > < 6. f ( ) = 1-1 < 1 7. f ( ) = 1 1 < 1 8. f ( ) = = 1 In problems 9 and 10, f is defined for all ecept for one value. If possible, define f () at that eceptional point in order to make it continuous there. 9. f ( ) = 1, 10. f ( ) = (6 ) 6, For what number a will the function f ( ) = ( a) a 1 be both continuous and a > 1 dferentiable at = 1? Sketch the graph of = f ( ) using this particular value of a. 1. Eplain in words wh the function described in each case below is either continuous everwhere on its domain, or instead, it possesses discontinuities. Then sketch a rough graph of the function. (a) The temperature, T, (as a function of time) of a pitcher of ice water left to stand on a table for several hours in a 70 o F room. (b) The cost, C, (as a function of time) for parking at the Millennium Park Garage, 5 S. Columbus Drive, in downtown Chicago for anwhere from 0 to hours. The parking rates are: $ for 0 to hours $6 for to 8 hours $8 for 8 to 1 hours $0 for 1 to hours.

4 Assignment 5 Due date : Frida, Februar 7 Section., Page 76: #1,, 8, 1, 1, 18, 0,, 8, 5, 6, 9, 61. Section.5, Page 769: #, 9, 1, 17, 0, 1. Section.6, Page 775: #, 1, 1, 17, 1, 5, 0, 1,, 7, 8,. Required Additional Eercises #1 (below). 1. The acceleration due to gravit, g, varies with height above the surface of the earth in a certain wa. If ou go down below the surface of the earth, then g varies in a dferent wa. It can be shown that g is given b: g ( r ) = GMr R GM r for for 0 r < R r R where R is the radius of the earth, M is the mass of the earth, G is the gravitational constant, and r is the distance to the center of the earth. (a) Sketch a graph of g against r for r > 0. (Note: Since r is the independent variable, ou should measure r along the horizontal ais.) (b) Is g continuous at r = R? Eplain our answer. (c) Compute the piecewise formula for g ( r ) and investigate whether or not g is dferentiable at r = R. 0 = f H L. Look at the graph of f ( ) = ( ) in the figure. This graph appears to have a sharp corner at = 0. Does it? Is f dferentiable at = 0? Just our answers Eam 1 Review Problem Set : ( ) Eam 1 Review Topics : ( ) Date for Midterm Eam Monda, March nd Assignment 6 Due date : Wednesda, March 11 Section.7, Page 78: #15, 16, 0, 1,,, 5, 6, 7, 8, 9. Required Additional Eercises # 1 16 (below). Instructions: In problems 1 15, find the derivative of in each case. 1. = ( 7). = ( 1). = ( 9). = ( ) (1 ) 5. = ( 9 6. = 7. = = 1 9. = 10. = 1. = = ) = 1. = ( 1) 15. = 5 1 ( 1) Use calculus to find the coordinates of the two points along the graph of = whose tangent lines pass through point P = ( 1, - ). Then sketch a graph which displas both the parabola and its tangent lines which intersect at P.

5 Assignment 7 Due date : Wednesda, March 5 Eercises # 1 (below). Refer to the lecture notes: and for details on implicit dferentiation, the conversion of equations from rectangular to polar coordinates, and plotting in polar coordinates. 1. (a) Use implicit dferentiation to find the slope and inclination of the tangent line to the graph of the cardioid: ( ) = at the points P = ( 1, 0 ) and Q = ( - 1, 0 ). (b) Set = r cos θ and = r sin θ in the equation from (a) and simpl it, to obtain the polar equation of the cardioid: r = 1 - sin θ. (c) Sketch a careful graph which displas the cardioid and its tangent lines at P and Q.. (a) Use implicit dferentiation to find the slope and inclination of the tangent line to the graph of the circle: ( - 1 ) = 1 at the point P = ( 1, ). (b) Set = r cos θ and = r sin θ in the equation from (a) and simpl it, to obtain the polar equation of the circle: r = cos θ. (c) Sketch a careful graph which displas the circle and its tangent line at P.. (a) Use implicit dferentiation to find the slope and inclination of the tangent line to the graph of the parabola: = at the points P = ( 1, ) and Q = (, - ). (b) Set = r cos θ and = r sin θ in the equation from (a) and simpl it, to obtain the polar equation of the cardioid: r = cot θ csc θ. (c) Sketch a careful graph which displas the parabola and its tangent lines at P and Q. d. (a) Use implicit dferentiation to find d for the lemniscate: ( ) = -. (b) Set = r cos θ and = r sin θ in the equation from (a) and simpl it, to obtain the polar equation of the lemniscate. H L = - Etra Credit Eercise In each of the following problems, f is dferentiable on an interval containing the arbitrar points a and b, P = ( a, f ( a ) ) and Q = ( b, f ( b ) ) are distinct points on the graph of f, and c is the coordinate of the point where the lines tangent to the curve at P and Q intersect, assuming that the tangent lines are not parallel (as in the figure shown to the right). R = f ( ) Q P a c b (a) If f ( ) =, draw a figure which displas the graph of f, two points P = ( a, f ( a ) ) and Q = ( b, f ( b ) ) on its graph, and the point R = ( c, f ( c) ) where the tangents to the graph of f at P and Q intersect. Then prove that c = a b, this is the arithmetic mean of a and b. (b) If f ( ) =, draw a figure which displas the graph of f, two points P = ( a, f ( a ) ) and Q = ( b, f ( b ) ) on its graph, (where a, b > 0 ), and the point R = ( c, f ( c) ) where the tangents to the graph of f at P and Q intersect. Then prove that c = ab, this is the geometric mean of a and b. (c) If f ( ) = 1, draw a figure which displas the graph of f, two points P = ( a, f ( a ) ) and Q = ( b, f ( b ) ) on its graph, (where a, b > 0 ), and the point R = ( c, f ( c) ) where the tangents to the graph of f at P and Q intersect. Then prove that c = ab, this is the harmonic mean of a and b. a b

6 Assignment 8 Due date : Wednesda, April 1 Section 5. Page 85: #6, 8, 10, 1. Required Additional Eercises # 1 16 (below). Refer to the lecture notes ( for details on relative and absolute etreme values and associated theorems and results. On Relative and Absolute Etreme Values and Critical Numbers 1. Eplain the dference between an absolute minimum and a relative minimum value.. Suppose that f is a continuous function defined on the closed interval [a, b]. (a) What theorem guarantees the eistence of an absolute maimum value and an absolute minimum value for f? (b) What steps would ou take to find those absolute etreme values? In problem 5, sketch the graph of a function f that is continuous on [1, 5] and has the given properties.. Absolute minimum at, absolute maimum at, relative minimum at.. Absolute maimum at 5, absolute minimum at, relative maimum at, relative minima at and. 5. f has no relative maimum or minimum but and are critical numbers. Recall that c is a critical number for f c is in the interior of dom( f ) and either f (c)= 0 or f (c) does not eist. 6. (a) Sketch the graph of a function that has a relative maimum at and is dferentiable at. (b) Sketch the graph of a function that has a relative maimum at and is continuous but not dferentiable at. (c) Sketch the graph of a function that has a relative maimum at and is discontinuous at. In problems 7 10 sketch the graph of f over the indicated interval. Then locate all absolute and relative maimum and minimum values of f that eist. 7. f () = 8, > 1 8. f () =, 1 < < 9. f () =, 1 < < 10. f () = 1/, 0 < < In problems find all critical numbers for the function, all the intervals on which f is increasing or decreasing, and all relative and absolute etreme values. Then use this information to help ou sketch a graph of = f (). 11. f ( ) = 1 1. f () = 1. f ( ) = 1. f () = 15. f () = f () = (8 ) Assignment 9 Due date : Wednesda, April 8 Section 5. Page 86: #17, 1,, 9. Section 9.1 Page 97: #,,, 7, 11, 1, 19, 6, 9, 6. Required Additional Eercises # 1 7 below. In problems 1 6 find all etreme points and all inflection points for the graph of the function. Then find all the intervals on which f is increasing, decreasing, concave up, and concave down and sketch a graph of = f (). (Note that problems 1 5 were initiall eamined in the previous assignment.) 1. f ( ) = 1. f () =. f ( ) =. f () = 1 5. f () = 1 / ( 8 ) 6. f () = / 7. A marble of radius r, 0 < r <, is dropped into a container having the shape of a circular clinder. The base of the container has radius. What is the radius of the marble that requires the most water to cover it completel? Recall that V Sphere = π r.

7 Eam Review Problem Set : ( ) Eam Review Topics : ( ) Date for Midterm Eam Monda, April 1 th Assignment 10 Due date : Monda, April 0 Section 9. Page 951: #1,, 9, 1, 1, 1, 6, 7. Required Additional Eercises # 1 below. In problems 1 do the following: (a) Sketch a rectangular plot of the given equation in the θ r plane over the indicated θ interval. (b) Sketch a polar graph of the equation in the plane, where = r cos θ, = r sin θ. (c) Find the slope and inclination of the tangent line to the polar curve at the point P = (, ) = ( r cos( π ), r sin( π ) ) b using the formula: d d dθ =. d d dθ (Refer to the notes at for reference on this topic.) 1. r = π π ; < θ < π. r = 1 sin θ ; 0 < θ < π. r = sin ( θ ) ; 0 < θ < π θ 6 Assignment 11 Due date : Monda, April 7 Section 6.1 Page 850: #, 7, 1, 19,, 7, 1, 5, 6, 8, 9. Section 6. Page 857: #1,, 6, 7, 8, 10, 17, 0, 1,. Section 6. Page 86: #,, 8, 1, 1,17. Section 9.7 Page 97: #1,, 9, 10, 19. Required Additional Eercises # 1 below. In problems 1 the graph of a function f is displaed. Sketch the graph of an antiderivative function, F, in each case. Include an analsis of the intervals on which F is increasing/decreasing and concave up/down based on our knowledge of f = F Assignment 1 Due date : Wednesda, Ma 6 Section 6. Page 866: #, 5, 6, 9, 11, 1, 1. Section 6.6 Page 87: # 1, 6, 1. Section 9.7 Page 97: # 7, 9. Required Additional Eercises # 1 7 below.

8 1. A unorm horizontal beam of length L having a simple point support at its left end and a fied support at its right end will be distorted, due to its own weight, into a curve: = f () as shown in the figure below. O L This curve is called the deflection curve of the beam and it satisfies the dferential equation: E I d = w along with the four boundar conditions: d ( 0 ) = 0, ( 0 ) = 0 ( L ) = 0, ( L ) = 0 B substituting each boundar condition into either equation (), () or () from page of the article: and then solving for the coefficients, C i, i = 1,,,, ver that the beam s equation is: = w 8EI [ L L ] Mean Value Theorem: If F is continuous on the closed interval [a, b] and dferentiable on the open interval (a, b) then there must eist at least one number c in (a, b) for which F (c) = F(b) F(a) b a In problems and do the following: (a) Appl the Mean Value Theorem to the function: F () = on the indicated interval I to find all numbers, c, in the interior of I, which satisf the conclusion of this theorem. I.e., find all numbers c interior to I for which the average change in F over I is equal to the instantaneous rate of change in F at c. (b) Sketch a graph of the function = F () on the interval I and interpret our results from part (a) geometricall in terms of the appropriate tangent and secant lines to our graphs.. I = [ 0, ]. I = [ 1, 1 ] In problems 7 use the definite integral to find the area of the shaded region in each case

9 Eam Review Topics : ( ) Eam Review Problem Set : ( ) (Final) Eam dates and times: Math 1-01, 8:00am 10:00 am, Wednesda, Ma 6, Room 106 E1 Math 1-0, 10:0am 1:0 pm, Wednesda, Ma 6, Room 106 E1.