E3 Fall 2011 MAT 111 /44pw Grade NAME Banner IDN00

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1 i E3 Fall 2011 MAT 111 /44pw Grade NAME Banner IDN00 [2p] 1) Complete the table for the following transformations of f(x). Place an X in the cells that do not have solutions on the given domain. x f (x) f ( x) 7 3 ( x 2) f ) Referring to the Ferris wheel examples from class, use the sketch to answer the questions. Let H(t) be the height above the ground as a function of time. Your initial boarding occurs at the lowest possible height. H(t) 18 meters 6 meters 30 t (minutes) Assuming counterclockwise rotation of the Ferris wheel determine the following: [5p] a) Period a) b) diameter of the Ferris wheel b) c) maximum height above the ground c) d) height above the ground when you board d) e) generate a formula for the sketch [2p] e) i

2 ii 4 [6p ] 3) Given the trigonometric function f ( ) 2sin 3 2, algebraically solve completely f 1 (3). Answer [6p] 4) Use a trigonometric identity to show algebraically when the two functions are equal 1 on the interval [0,2) 2 1 sin 2 sec ii

3 iii (3p) 5) Using the sketch, generate a possible minimum degree equation P (x). Include the k value. Show work. ANSWER: P (x) (2p) 6) Find a possible power function formula for g (x), if g (1) = 6 and g (3) = 54. Show work. ANSWER: g (x) = iii

4 iv [7p] 7) The temperature, measured in degrees, of a room temperature Christmas turkey, t 375 minutes after being placed in a hot oven is given by the equation H( t) t 0 1 4(0.7) t a) What is the original temperature of the room? b) What is (approximately) the final temperature of the turkey? c) Find a formula for: H ( t 15) d) Find a formula for: H ( t) 15 e) Approximate a sketch (label intercepts, asymptotes, etc.) for H (t), H( t 15) and H ( t) 15 t t t H (t) H ( t 15) H ( t) 15 f) Describe in terms of the problem a situation that might be modeled by the function H ( t 15) g) Describe in terms of the problem a situation that might be modeled by the function H ( t) Ht ( 15) 1 4(0.7) t Ht ( ) (0.7) t iv

5 v 8) In a camera factory the daily cost C, in dollars of producing n camera lens covers is given by 2 C( n) 2.89n 0.095n for 0 n 30 and the number of covers produced in t hours is given by n( t) 4. 5t for t 10 (1p) a) Generate ( C n)( t) and simplify Answer (1p) b) Interpret ( C n)( t) (1p) c) If the factory operates for 6.2 hours, calculate the daily production cost to the nearest penny. Answer (2p) d) If the daily production cost is $18, how many hours did the factory operate? Answer v

6 vi (7p) 9) Suppose it costs a company $30,000 in start up costs to begin production of pints of good ice cream plus $3 for each pint produced. Let p be the number of pints produced by the company. a) Find a formula for T ( p), the total cost incurred by the company for the production of p pints. b) If the companies average cost per pint, C ( p), is defined to be the total cost T ( p), divided by the number of pints p produced, then find a formula for C( p) in terms of T ( p). c) Graph C( p) for x: [ 0, 50000] and y: [ 0, 10 ] and label the horizontal asymptote. lim C( p d) Explain in some economical terms the ). p e) Explain in some economical terms what is happening near the vertical asymptote. f) Find a formula for 1 C. g) What is the economic interpretation of 1 C? ANSWERS: a) b) c) sketch here d) e ) f) g) vi

7 vii E3 Fall 2011 MAT 111 HOME PORTION Due Friday before 2pm Grade NAME Banner IDN00 (2p) 1) a) In the power function y Kx P if p > 1 the sketch is ( circle the correct answers ) increasing / decreasing and concave up / down. b) In the power function y Kx P if 1 > p > 0 the sketch is ( circle the correct answers ) increasing / decreasing and concave up / down. 2) A boat is anchored in calm waters 150 feet above the sea floor. Rough waters arrive and continually lower the boat down to 141 feet. The boat is raised past the original calm water level by an equal amount to its maximum height above the sea floor, every 6 seconds. Generate one cycle of a sketch for the periodic function D(t), the distance the boat is above the ocean floor in rough waters. Label the maximum and minimum heights, midline and period on the sketch. [4p] a) [4p] b) Using nonmathematical terminology completely describe (interpret) the new motion of the small boat in the water according to the transformation 3D( 2t ). vii

8 viii (3p) 3) Decompose order of the composition. y 1 0.8e 0.216x0.5, into at least two functions and then state the Many answers as: f ( x) 1 x and g( x) 0.8e 0.216x0.5 as y f ( g( x)) (3p) 4) Show work to determine the horizontal and vertical asymptotes if they exist. R x) (5x 4 3x 7 2)( x { 2 2 7) Answer: same exponent in highest degree 3x 5x 4 4 so, HA 3 5 viii

9 ix [5p] 5) Three adjacent rectangular pigpens are to be built by using 500 feet of fence. Calculate the maximum area possible to be enclosed [3p] 6) Generate a quadratic equation for the sketch. (-6,9) (-15,0) ix

10 x lim f ( x (5p) 7) Consider the ) a) x 1/ 3 5/ 4 f ( x) 4x b) f ( x) 27.5x c) for each of the following: f ( x) 13x d) f ( x) 34x e) Using the letters ( a, b, c, d, e ) place each function in order such that: f X ) f ( x) f ( x) f ( X ) f ( x) f ( ) 1( x f ( x) 5e i) ii) iii) iv) v) X x