**UFRGS**

# Exercícios Resolvidos - Cálculo - James Stewart

(Parte **1** de 7)

(c) is equivalent to | When , we have and . |

(e) The domain of consists of all values on the graph of | For this function, the domain is |

, or | The range of consists of all values on the graph of . For this function, |

(f) As increases from to , increases from to | Thus, is increasing on the interval |

1. (a) The point is on the graph of , so . (b) When , is about , so . (d) Reasonable estimates for when are and . the range is , or . .

2. (a) The point is on the graph of , so | The point is on the graph of , so |

(c) is equivalent to | When , we have and . |

(d) As increases from to , decreases from to | Thus, is decreasing on the interval |

. (b) We are looking for the values of for which the values are equal. The values for and are equal at the points and , so the desired values of are and . .

(e) The domain of consists of all values on the graph of | For this function, the domain is |

, or | The range of consists of all values on the graph of . For this function, |

the range is , or . (f) The domain of is and the range is .

3. From Figure 1 in the text, the lowest point occurs at about | The highest point |

occurs at about | Thus, the range of the vertical ground acceleration is . In |

Figure 1, the range of the northsouth acceleration is approximately | In Figure 12, |

the range of the eastwest acceleration is approximately .

of the time | The domain of the function is , where is measured in hours. The range of |

4. Example 1: A car is driven at mi h for hours. The distance traveled by the car is a function the function is , where is measured in miles.

Example 2: At a certain university, the number of students on campus at any time on a particular day is a function of the time after midnight. The domain of the function is , where is

measured in hours. The range of the function is , where is an integer and is the largest number of students on campus at once.

Stewart Calculus ET 5e 0534393217;1. Functions and Models; 1.1 Four Ways to Represent a Function

0,30 0,2.0,4.0, | ,238.0,240.0{ } |

P h

Example 3: A certain employee is paid per hour and works a maximum of hours per week. The number of hours worked is rounded down to the nearest quarter of an hour. This employee’s

gross weekly pay is a function of the number of hours worked | The domain of the function is |

and the range of the function is .

5. No, the curve is not the graph of a function because a vertical line intersects the curve more than once. Hence, the curve fails the Vertical Line Test.

6. Yes, the curve is the graph of a function because it passes the Vertical Line Test. The domain is and the range is .

7. Yes, the curve is the graph of a function because it passes the Vertical Line Test. The domain is and the range is .

8. No, the curve is not the graph of a function since for , , and , there are infinitely many points on the curve.

9. The person’s weight increased to about pounds at age and stayed fairly steady for years. The person’s weight dropped to about pounds for the next years, then increased rapidly to about pounds. The next years saw a gradual increase to pounds. Possible reasons for the drop in weight at years of age: diet, exercise, health problems.

10. The salesman travels away from home from to A.M. and is then stationary until : | The |

: , at which time the distance from home decreases until : | Then the distance starts |

increasing again, reaching the maximum distance away from home at : | There is no change from |

salesman travels farther away from until noon. There is no change in his distance from home until until , and then the distance decreases rapidly until : P.M., at which time the salesman reaches home.

1. The water will cool down almost to freezing as the ice melts. Then, when the ice has melted, the water will slowly warm up to room temperature.

Stewart Calculus ET 5e 0534393217;1. Functions and Models; 1.1 Four Ways to Represent a Function

12. The summer solstice (the longest day of the year) is around June 21, and the winter solstice (the shortest day) is around December 2.

13. Of course, this graph depends strongly on the geographical location!

14. The temperature of the pie would increase rapidly, level off to oven temperature, decrease rapidly, and then level off to room temperature.

16. (a)

(b) (c)

Stewart Calculus ET 5e 0534393217;1. Functions and Models; 1.1 Four Ways to Represent a Function

(d)

17. (a) (b) From the graph, we estimate the number of cellphone subscribers in Malaysia to be about 540 in

1994 and 1450 in 1996.

18. (a) (b) From the graph in part (a), we estimate the temperature at 1:0 A.M. to be about 84.5 C.

Stewart Calculus ET 5e 0534393217;1. Functions and Models; 1.1 Four Ways to Represent a Function

20. A spherical balloon with radius has volume | We |

wish to find the amount of air needed to inflate the balloon from a radius of to | Hence, we need |

to find the difference .

21. , so , , and

2. , so , , and

23. is defined for all except when , so the domain is .

24. is defined for all except when or

, so the domain is .

Stewart Calculus ET 5e 0534393217;1. Functions and Models; 1.1 Four Ways to Represent a Function

is defined when | These values of give real number results for , whereas any |

value of gives a real number result for | The domain is . |

26. is defined when and | Thus, the domain is . |

27. is defined when | Note that since that would |

result in division by zero. The expression is positive if or | (See Appendix A for |

methods for solving inequalities.) Thus, the domain is .

28 | Now |

, so the graph is the top half of a circle of

radius with center at the origin. The domain is . From the graph, the range is , or .

29. is defined for all real numbers, so the domain is , or | The graph of is a |

horizontal line with intercept .

30. is defined for all real numbers, so the domain is , or | The graph of is |

a line with intercept and intercept .

31. is defined for all real numbers, so the domain is , or | The graph of is a |

solve for | and . The coordinate of the vertex is halfway between the |

parabola opening upward since the coefficient of is positive. To find the intercepts, let and 6

Stewart Calculus ET 5e 0534393217;1. Functions and Models; 1.1 Four Ways to Represent a Function

intercepts, that is, at | Since , the vertex is . |

32. , so for , | The domain is . So the graph of is |

the same as the graph of the function (a line) except for the hole at .

3. is defined when or , so the domain is | Since |

, we see that is the top half of a parabola.

The domain is , or .

Stewart Calculus ET 5e 0534393217;1. Functions and Models; 1.1 Four Ways to Represent a Function

Note that is not defined for | The domain is . |

36 | Since , we have |

Note that is not defined for | The domain is . |

. Since , we have

37. Domain is , or .

(Parte **1** de 7)