Many applications of the derivative involve determining the rate of change at a given instant of a function with the independent variable time—which is why the term
instantaneous is used. Consider the height of a ball tossed upward with an initial velocity of 64 feet per second, given by
where
is measured in seconds and
is measured in feet. We know the path is that of a parabola. The derivative will tell us how the height is changing at any given point in time. The height of the ball is shown in
[link] as a function of time. In physics, we call this the “
s -
t graph.”
Finding the instantaneous rate of change
Using the function above,
what is the instantaneous velocity of the ball at 1 second and 3 seconds into its flight?
The velocity at
and
is the instantaneous rate of change of distance per time, or velocity. Notice that the initial height is 6 feet. To find the instantaneous velocity, we find the
derivative and evaluate it at
and
For any value of
,
tells us the velocity at that value of
Evaluate
and
The velocity of the ball after 1 second is 32 feet per second, as it is on the way up.
The velocity of the ball after 3 seconds is
feet per second, as it is on the way down.
Using graphs to find instantaneous rates of change
We can estimate an instantaneous rate of change at
by observing the slope of the curve of the function
at
We do this by drawing a line tangent to the function at
and finding its slope.
Given a graph of a function
find the instantaneous rate of change of the function at
Locate
on the graph of the function
Draw a tangent line, a line that goes through
at
and at no other point in that section of the curve. Extend the line far enough to calculate its slope as
Estimating the derivative at a point on the graph of a function
From the graph of the function
presented in
[link] , estimate each of the following:
To find the functional value,
find the
y -coordinate at
To find the
derivative at
draw a tangent line at
and estimate the slope of that tangent line. See
[link] .
is the
y -coordinate at
The point has coordinates
thus
is the
y -coordinate at
The point has coordinates
thus
is found by estimating the slope of the tangent line to the curve at
The tangent line to the curve at
appears horizontal. Horizontal lines have a slope of 0, thus
is found by estimating the slope of the tangent line to the curve at
Observe the path of the tangent line to the curve at
As the
value moves one unit to the right, the
value moves up four units to another point on the line. Thus, the slope is 4, so
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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