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
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
