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
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.