. Show the work that leads to your answer. (c) State the equation(s) for the vertical asymptote(s) for the graph of y f( x).

Transcription

1 Chapter 1 1. (AB/BC, non-calculator) The function f is defined as follows: f( ) (a) State the value(s) of for which f is not continuous. (b) Evaluate f ( ). Show the work that leads to your answer. 3 (c) State the equation(s) for the vertical asymptote(s) for the graph of y f( ). (d) State the equation(s) for the horizontal asymptote(s) for the graph of y f( ). Show the work that leads to your answer.

2 . (AB/BC, calculator neutral) y y Graph of f Graph of g The graphs of functions f and g are shown above. Evaluate each it using the graphs provided. Show the computations that lead to your answer. (a) f( ) 4 (c) f ( ) g( ) 1 (b) 5 g 3 ( ) (d) f ( ) 3 g ( ) 1 (Assume that f and g are linear on the interval [,3]).

3 3. (AB/BC, calculator neutral) A hot cup of tea is placed on a counter and left to cool. The temperature of the tea, in degrees Fahrenheit (correct to the nearest degree), minutes after the cup is placed on the counter is modeled by a continuous function T( ) for Values of T( ) at various times are shown in the table below T( ) (a) Evaluate: T( ). Justify your answer. 4 (b) Using the data in the table, find the average rate of change in the temperature of the tea for 3 8. Include units on your final answer. (c) Identify, using the times listed in the table, the shortest interval during which there must eist a time for which the temperature of the tea is166.5? Justify your answer. (d) Use the data in the table to find the best estimate of the slope of the line tangent to the graph of T at 8.

4 4. (AB/BC, non-calculator) Find the value of each of these its, or else eplain why the it does not eist. Show the computations which lead to your answers. (a) (b) (c) cos 0 sin (d) 5 5

5 5. (AB/BC, non-calculator) The position function st ( ) 4.9t fallen from a height of meters after t seconds., gives the height (in meters) of an object that has (a) Eplain why there must eist a time t, 1 t, at which the height of the object must be 38 meters above the ground. (b) Find the time at which the object hits the ground. (c) Find the average rate of change in s over the intervalt 8,9. Include units of measure. Eplain why this is a good estimate of the velocity at which the object hits the ground. How can this estimate be improved? (d) Evaluate: st () s(3). Show the work that leads to your answer. Include units. t3 t 3

6 6. (AB/BC, calculator neutral) Let a and b represent real numbers. Define a b if f( ) ab if 5. a 7 if 5 (a) Find the values of a and b such that f is continuous everywhere. (b) Evaluate: f ( ). 3 (c) Let f ( ) g ( ). 1 Evaluate: g ( ). 1

7 7. (AB/BC, calculator neutral) y The graph of function g is shown above. Which of the following is true? I. g ( ) 1 II. g ( ) g() III. g is continuous at 3. (a) I only (b) I and II only (c) I and III only (d) III only (e) I, II and III

8 8. (AB/BC, calculator neutral) y The graph of the function f is shown above. The line 1is a vertical asymptote. Which of the following statements about f is true? (a) 1 (b) 1 3 (c) f 3 3 ( ) f( ) (d) f ( ) does not eist 4 (e) f ( ) f( ) 0 3

9 9. (AB/BC, non-calculator) Define 4 3 if, 4 8 f( ). 8 if 4 Which of the following statements about f are true? I. f is not continuous at 4. II. f( ) 4 III. 4 is a vertical asymptote of the graph of y f( ). (a) None (b) I only (c) I and II only (d) I and III only (e) I, II and III

10 10. (AB/BC, calculator neutral) y The figure above shows three rectangles each with a verte on the graph of of the areas of these rectangles is y 16. The sum (a) 4 sq. units (b) 40 sq. units (c) 34 sq. units (d) 33 sq. units (e) 9 sq. units

11 Chapter 1 (Solutions) Question 1 The function f is defined as follows: f( ) (a) State the value(s) of at which f is not continuous. (b) Evaluate f ( ). Show the work that leads to your answer. 3 (c) State the equation(s) for the vertical asymptote(s) for the graph of y f( ). (d) State the equation(s) for the horizontal asymptote(s) for the graph of y f( ). Show the work that leads to your answer. (a) f( ) (b) f is discontinuous at 1 3 ( 1) 15 1 ; 3 and 0 3: 1 per answer : 1: reduced fraction 1: answer (c) 1 ; 0 : 1 per answer

12 Question 1 (cont.) (d) 56 1 : 1: it epression 1: answer 7 3

13 Question y y Graph of f Graph of g The graphs of functions f and g are shown above. Evaluate each it using the graphs provided. Show the computations that lead to your answer. (a) f( ) 4 1 (b) 5 3 g ( ) (c) f ( ) g( ) (d) f ( ) 3 g ( ) 1 (Assume that f and g are linear on the interval [,3]). (a) f( ) 4 1 = 34 7 : 1: breaking up it 1: answer (b) 5 3 g ( ) = : 1: breaking up it 1: answer

14 Question (cont.) (c) f ( ) g( ) = ()(0) 0 : 1: breaking up it 1: answer (d) f( ) 6 g ( ) 1 ( ) : : algebraic representations 1: factoring and answer

15 Question 3 A hot cup of tea is placed on a counter and left to cool. The temperature of the tea, in degrees Fahrenheit (correct to the nearest degree), minutes after the cup is placed on the counter is modeled by a continuous function T( ) for Values of T( ) at various times are shown in the table below T( ) (a) Evaluate: T( ). Justify your answer. 4 (b) Using the data in the table, find the average rate of change in the temperature of the tea for 3 8. Include units on your final answer. (c) Identify, using the times listed in the table, the shortest interval during which there must eist a time for which the temperature of the tea is 166.5? Justify your answer. (d) Use the data in the table to find the best estimate of the slope of the line tangent to the graph of T at 8. (a) Since T is continuous for 0 10, T( ) T(4). 4 T( ) 17 : 1: justification 4 1: answer (b) T(8) T(3) F 8 3 min 1: setup 3: 1: answer 1: units

16 Question 3 (cont.) (c) By the Intermediate Value Theorem, the temperature must be for some time 6 8 : 1: answer 1: justification since T(6) T(8). (d) T(9) T(8) 98 : 1: setup 1: answer

17 Question 4 Find the value of each of these its, or else eplain why the it does not eist. Show the computations which lead to your answers. (a) (b) (c) cos 0 sin (d) 5 5 (a) 1 1: answer 3 (b) (c) (d) 1 1 3: 0 0 :simplify : answer sin : 1: simplify 0 1: answer ( ) 4 3: : simplify 1: answer

18 The position function st ( ) 4.9t Question 5 fallen from a height of meters after t seconds., gives the height (in meters) of an object that has (a) Eplain why there must eist a time t, 1 t, at which the height of the object must be 38 meters above the ground. (b) Find the time at which the object hits the ground. (c) Find the average rate of change in s over the intervalt 8,9. Include units of measure. Eplain why this is a good estimate of the velocity at which the object hits the ground. How can this estimate be improved? (d) Evaluate: st () s(3). Show the work that leads to your answer. Include units. t3 t 3 (a) s(1) 38 s(). The function st ( ) is a polynomial, and is therefore continuous. Therefore by the Intermediate Value Theorem, 1: eplanation there eists a value t,1t, for which st ( ) 38. (b) (c) 0 4.9t t 9 s 1: answer s(9) s(8) m This is the average velocity on 98 sec the interval 8t 9. Since the object hits the ground at t 9, 1: answer 4 : 1: units : 1 per eplanation this average velocity is close to the instantaneous velocity s at t = 9. The estimate (9) s ( t ) can be improved by allowing the value of t to approach 9. 9 t

19 Question 5 (cont.) (d) st () s(3) 4.9( t3)( t3) m 9.4 t3 t3 t3 sec 1: work 3: 1: answer 1: units

20 Question 6 Let a and b represent real numbers. Define a b if f( ) ab if 5. a 7 if 5 (a) Find the values of a and b such that f is continuous everywhere. (b) Evaluate: f ( ). 3 (c) Let f ( ) g ( ). 1 Evaluate: g ( ). 1 (a) 4 a b a b a ; b 3 5ab10a7 1: its 4 : 1: equations : answers (b) 9 : 1: correct interval 1: answer (c) : 1: correct interval 1: answer

21 Questions c g() 3 g( ) 1 8. e both of these its equal 9. b f( ) 4 f ( ) 1 so the horizontal asymptote is y c = 34