Let f be defined as follows.
(a) Find the average rate of change of y with respect to x in the following intervals.
from x = 6 to x = 7
from x = 6 to x = 6.5
from x = 6 to x = 6.1
(b) Find the (instantaneous) rate of change of y at x = 6.
The demand for Sportsman 5 X 7 tents is given by the following function where p is measured in dollars and x is measured in units of a thousand. (Round your answers to three decimal places.)
p = f(x) = ?0.1x^2 ? x + 40
(a) Find the average rate of change in the unit price of a tent if the quantity demanded is between the following intervals.
between 4400 and 4450 tents $ per 1000 tents
between 4400 and 4410 tents $ per 1000 tents
(b) What is the rate of change of the unit price if the quantity demanded is 4400?
$ per 1000 tents
Under a set of controlled laboratory conditions, the size of the population of a certain bacteria culture at time t (in minutes) is described by the following function.
Find the rate of population growth at t = 11 min.
bacteria per minute
The position function of an object moving along a straight line is given by
s = f(t).
The average velocity of the object over the time interval [a, b] is the average rate of change of f over [a, b]; its (instantaneous) velocity at t = a is the rate of change of f at a.
A ball is thrown straight up with an initial velocity of 144 ft/sec, so that its height (in feet) after t sec is given by s = f(t) = 144t ? 16t^2.
(a) What is the average velocity of the ball over the following time intervals?
(b) What is the instantaneous velocity at time t = 4?
(c) What is the instantaneous velocity at time t = 8?
Is the ball rising or falling at this time?
(d) When will the ball hit the ground?
t = sec
Here we present multiple examples as to how to use derivative of f to solve the problems related to velocity and rate of change.
Physics: velocity, speed, force, temperature, energy, magnitude, acceleration, wave
1) A particle moves according to the equation x=10t2, where x is in meters and t is in seconds. Find the average velocity for the time interval from 2.0 second to 2.1 second.
A) 41 m/s
B) 100 m/s
2) If A = (12i -16j) and B = (-24i + 10j), what is the magnitude of the vector C = (2A - B)?
3) A room measures 3.0 m by 4.5 m by 6.0 m. The heating and air conditioning ducts to and from the room are circular with diameter 0.30 m, and the air in the room is to be exchanged every 12 minutes. What is the necessary flow speed in the duct? (Assume that the density of the air is constant)
A) 0.6 m/s
B) 1.6 m/s
C) 3.2 m/s
D) 4.1 m/s
5) A 50-kg ice skater is moving at 5 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius 1 m around the pole. Determine the force exerted by the rope on her arms.
A) 1250 N
B) 250 N
C) 50 N
D) 1100 N
6) Steam at 1000C is added to ice at O0C. The mass of the steam is 10.0 g and the mass of the ice is 50.0 g. What is the final temperature of the mixture?
7) The speed of a 4.0-kg object is given by v = (2t) m/s, where t is in seconds. At what rate is the resultant force on this object doing work at t = 1 second?
A) 48 W
B) 40 W
C) 56 W
D) 16 W
8) A 10-kg object is initially at the top of a rise, at point A. It then moves down to point B. Choose point B to be the zero level for gravitational potential energy. Find the change in potential energy of the object if point B is 10 m lower than point A.
A) - 980
B) - 10
C) - 100
D) - 98
9) A 3-kg particle has a velocity of (3i - 4j) m/s. Find the magnitude of its momentum.
A) 9 kg.m/s
B) -12 kg.m/s
C) 15 kg.m/s
D) 3 kg.m/s
10) At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration has an angular velocity of 2 rad/s. Two seconds later it has turned 5 complete revolutions. What is the angular acceleration of this wheel?
A) 17 rad/s^2
B) 14 rad/s^2
C) 20 rad/s^2
D) 23 rad/s^2
11) A car of mass 1000 kg moves with a speed of 50 m/s on a circular track of radius 100 m. What is the magnitude of its angular momentum (in kg.m^2/s) relative to the center of the racetrack?
A) 5 x 10^2
B) 5 x 10^6
C) 2.5 x 10^6
D) 2.5 x 10^4
14) How many significant figures are in the following number? 0.0064
15) In order to understand the concept of temperature it is useful to understand
A) the zeroth law of thermodynamics
B) the first law of thermodynamics
C) the second law of thermodynamics
D) none of the above
16) An archer shoots an arrow with a velocity of 45 m/s at an angle of 50 degrees with the horizontal. What is the height of the arrow at a point 150 meters downrange?
A) 4.7 m
B) 47.0 m
C) 5.6 m
D) 56.0 m
17) A sinusoidal wave is described by y=(0.30 m)sin(0.20x-40t ). Determine the wave speed.
A) 100 m/s
B) 133 m/s
C) 150 m/s
D) 200 m/s
18) The exhaust temperature of a Carnot heat engine is 350oC. What is the intake temperature if the efficiency is 25.0%?
20) A house has well-insulated walls. It contains a volume of 100 m3 of air at 300 K. Calculate the energy required to increase the temperature of this diatomic gas by 2oC. Assume it is heating at constant pressure and use Cp=7R/2.
A) 118 kJ
B) 236 kJ
C) 354 kJ
D) 472 K
21) If the speed at some point in a fluid changes with time, the fluid flow is not __________.
22) A bowling ball rolls without slipping on a flat surface. The ball has ___________.
A) rotational kinetic energy
B) translational kinetic energy
C) both a and b
D) neither a or b
23) In uniform circular motion, there is a ____________.
A) constant velocity
B) constant angular velocity
C) zero acceleration
D) net tangential acceleration
24) Linear momentum is ____________.
A) always conserved
B) a scalar quantity
C) a vector quantity
D) unrelated to force
25) If the net force on an object is zero, the object must _______________.
A) be at rest
B) be in motion with constant velocity
C) have zero acceleration
D) none of these