11. Maximum Height A toy rocket is fired straight up into the air. Let s(t)=-612 + 72r denote its position in feet after seconds. (a) Find the velocity after 1 seconds. (b) Find the acceleration after seconds. (c) When does the rocket reach its maximum height? [Hint: What happens to the velocity when the rocket reaches its maximum height?] (d) What is the maximum height reached by the rocket?
11. Maximum Height A toy rocket is fired straight up into the air. Let s(t)=-612 + 72r denote its position in feet after seconds. (a) Find the velocity after 1 seconds. (b) Find the acceleration after seconds. (c) When does the rocket reach its maximum height? [Hint: What happens to the velocity when the rocket reaches its maximum height?] (d) What is the maximum height reached by the rocket?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![120 CHAPTER 1
The Derivative 170.1
3. Average and Instantaneous Rates of Change Suppose that
f(x) = 4x².
(a) What is the average rate of change of f(x) over each of
the intervals 1 to 2, 1 to 1.5, and 1 to 1.1?
(b) What is the (instantaneous) rate of change of f(x) when
x=1?
4. Average and Instantaneous Rates of Change Suppose that
f(x)=-6/x.
(a) What is the average rate of change of f(x) over each of
the intervals 1 to 2, 1 to 1.5, and I to 1.2?
(b) What is the (instantaneous) rate of change of f(x) when
x=1?
5. Average and Instantaneous Rates of Change Suppose that
f(t)=1²+31-7.
(a) What is the average rate of change of f(t) over the inter-
val 5 to 6?
(b) What is the (instantaneous) rate of change of f(t) when
1 = 5?
6. Average and Instantaneous Rates of Change Suppose that
f(t)-31+2- 12.0) (2
urin ja
(a) What is the average rate of change of f(t) over the inter-
val 2 to 3?
(b) What is the (instantaneous) rate of change of f(t) when
t=2?
7. Motion of an Object An object moving in a straight line travels
s(t) kilometers in t hours, where s(t) = 21² +41.
(a) What is the object's velocity when = 6?
(b) How far has the object traveled in 6 hours?
(c) When is the object traveling at the rate of 6 kilometers
per hour?
8. Effect of Advertising on Sales After an advertising campaign,
the sales of a product often increase and then decrease. Sup-
pose that days after the end of the advertising, the daily sales
are f(t)=-31²+ 32r + 100 units. What is the average rate of
growth in sales during the fourth day, that is, from time r = 3
to t= 4? At what (instantaneous) rate are the sales changing
when = 2?
9. Average Daily Output An analysis of the daily output of a
factory assembly line shows that about 601 + ²-³ units
are produced after r hours of work, 0 st≤ 8. What is the
rate of production (in units per hour) when = 2?
10. Rate of Change of Volume after Liquid Liquid is pouring into
a large vat. After r hours, there are 5t + Vi gallons in the vat.
At what rate is the liquid flowing into the vat (in gallons per
hour) when t= 4?
11. Maximum Height A toy rocket is fired straight up into the
air. Let s(t)=-61² + 721 denote its position in feet after
seconds.
(a) Find the velocity after 1 seconds.
(b) Find the acceleration after t seconds.
(c) When does the rocket reach its maximum height? [Hint:
What happens to the velocity when the rocket reaches its
maximum height?]
(d) What is the maximum height reached by the rocket?
12. Analysis of a Moving Particle Refer to Fig. 6, where s(t) repre-
sents the position of a car moving in a straight line.
(a) Was the car going faster at A or at B
(b) Is the velocity increasing or decreasing at B? What does
this say about the acceleration at B?
(e) What happened to the car's velocity at C
(d) In which direction was the car moving at D?
(e) What happened at ET
(f) What happened after F?
0
LEB
C
V=8(1)
D
E
Figure 6
13. Position of a Toy Rocket A toy rocket fired straight up into the
air has height s(t) = 160r16r2 feet after t seconds.
(a) What is the rocket's initial velocity (when r = 0)?
(b) What is the velocity after 2 seconds?
(c) What is the acceleration when = 3?
(d) At what time will the rocket hit the ground?
(e) At what velocity will the rocket be traveling just as it
smashes into the ground?
14. Height of a Helicopter A helicopter is rising straight up in the
air. Its distance from the ground r seconds after takeoff is s(t)
feet, where s(t)= 1² +1.
(a) How long will it take for the helicopter to rise 20 feet?
(b) Find the velocity and the acceleration of the helicopter
when it is 20 feet above the ground.
15. Height of a Ball Let s(1) be the height (in feet) after t seconds
of a ball thrown straight up into the air. Match each question
with the proper solution.
Questions
A. What is the velocity of the ball after 3 seconds?
B. When is the velocity 3 feet per second?
C. What is the average velocity during the first 3 seconds?
D. When is the ball 3 feet above the ground?
E. When does the ball hit the ground?
F. How high is the ball after 3 seconds?
G. How far did the ball travel during the first 3 seconds?
Solutions
a. Set s(t)= 0 and solve for t.
b. Compute s'(3).
c. Compute s(3).
d.
Set s'(t)= 3 and solve for t.
Find a value of a for which s(a) = 3.
e.
f. Compute [s(3)-s(0)]/3.
g. Compute s(3) - s(0).
16. Average Speed Table 2 gives a car's trip odometer reading
(in miles) at 1 hour into a trip and at several nearby times.
What is the average speed during the time interval from 1 to
1.05 hours? Estimate the speed at time 1 hour into the trip.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe83b972a-c189-49af-9357-b08dbf716629%2F1bd4e725-a3d3-483b-b3c5-6f28b2338658%2Fitvo3z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:120 CHAPTER 1
The Derivative 170.1
3. Average and Instantaneous Rates of Change Suppose that
f(x) = 4x².
(a) What is the average rate of change of f(x) over each of
the intervals 1 to 2, 1 to 1.5, and 1 to 1.1?
(b) What is the (instantaneous) rate of change of f(x) when
x=1?
4. Average and Instantaneous Rates of Change Suppose that
f(x)=-6/x.
(a) What is the average rate of change of f(x) over each of
the intervals 1 to 2, 1 to 1.5, and I to 1.2?
(b) What is the (instantaneous) rate of change of f(x) when
x=1?
5. Average and Instantaneous Rates of Change Suppose that
f(t)=1²+31-7.
(a) What is the average rate of change of f(t) over the inter-
val 5 to 6?
(b) What is the (instantaneous) rate of change of f(t) when
1 = 5?
6. Average and Instantaneous Rates of Change Suppose that
f(t)-31+2- 12.0) (2
urin ja
(a) What is the average rate of change of f(t) over the inter-
val 2 to 3?
(b) What is the (instantaneous) rate of change of f(t) when
t=2?
7. Motion of an Object An object moving in a straight line travels
s(t) kilometers in t hours, where s(t) = 21² +41.
(a) What is the object's velocity when = 6?
(b) How far has the object traveled in 6 hours?
(c) When is the object traveling at the rate of 6 kilometers
per hour?
8. Effect of Advertising on Sales After an advertising campaign,
the sales of a product often increase and then decrease. Sup-
pose that days after the end of the advertising, the daily sales
are f(t)=-31²+ 32r + 100 units. What is the average rate of
growth in sales during the fourth day, that is, from time r = 3
to t= 4? At what (instantaneous) rate are the sales changing
when = 2?
9. Average Daily Output An analysis of the daily output of a
factory assembly line shows that about 601 + ²-³ units
are produced after r hours of work, 0 st≤ 8. What is the
rate of production (in units per hour) when = 2?
10. Rate of Change of Volume after Liquid Liquid is pouring into
a large vat. After r hours, there are 5t + Vi gallons in the vat.
At what rate is the liquid flowing into the vat (in gallons per
hour) when t= 4?
11. Maximum Height A toy rocket is fired straight up into the
air. Let s(t)=-61² + 721 denote its position in feet after
seconds.
(a) Find the velocity after 1 seconds.
(b) Find the acceleration after t seconds.
(c) When does the rocket reach its maximum height? [Hint:
What happens to the velocity when the rocket reaches its
maximum height?]
(d) What is the maximum height reached by the rocket?
12. Analysis of a Moving Particle Refer to Fig. 6, where s(t) repre-
sents the position of a car moving in a straight line.
(a) Was the car going faster at A or at B
(b) Is the velocity increasing or decreasing at B? What does
this say about the acceleration at B?
(e) What happened to the car's velocity at C
(d) In which direction was the car moving at D?
(e) What happened at ET
(f) What happened after F?
0
LEB
C
V=8(1)
D
E
Figure 6
13. Position of a Toy Rocket A toy rocket fired straight up into the
air has height s(t) = 160r16r2 feet after t seconds.
(a) What is the rocket's initial velocity (when r = 0)?
(b) What is the velocity after 2 seconds?
(c) What is the acceleration when = 3?
(d) At what time will the rocket hit the ground?
(e) At what velocity will the rocket be traveling just as it
smashes into the ground?
14. Height of a Helicopter A helicopter is rising straight up in the
air. Its distance from the ground r seconds after takeoff is s(t)
feet, where s(t)= 1² +1.
(a) How long will it take for the helicopter to rise 20 feet?
(b) Find the velocity and the acceleration of the helicopter
when it is 20 feet above the ground.
15. Height of a Ball Let s(1) be the height (in feet) after t seconds
of a ball thrown straight up into the air. Match each question
with the proper solution.
Questions
A. What is the velocity of the ball after 3 seconds?
B. When is the velocity 3 feet per second?
C. What is the average velocity during the first 3 seconds?
D. When is the ball 3 feet above the ground?
E. When does the ball hit the ground?
F. How high is the ball after 3 seconds?
G. How far did the ball travel during the first 3 seconds?
Solutions
a. Set s(t)= 0 and solve for t.
b. Compute s'(3).
c. Compute s(3).
d.
Set s'(t)= 3 and solve for t.
Find a value of a for which s(a) = 3.
e.
f. Compute [s(3)-s(0)]/3.
g. Compute s(3) - s(0).
16. Average Speed Table 2 gives a car's trip odometer reading
(in miles) at 1 hour into a trip and at several nearby times.
What is the average speed during the time interval from 1 to
1.05 hours? Estimate the speed at time 1 hour into the trip.
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