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Calculate velocity and rate of change with derivative of f

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Let f be defined as follows.
f(x)=x^2-3x
(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.
P=f(t)=3t^2+2t+1
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?
[4,5] ft/sec

[4,4.5] ft/sec

[4,4.1] ft/sec

(b) What is the instantaneous velocity at time t = 4?
ft/sec

(c) What is the instantaneous velocity at time t = 8?
ft/sec
Is the ball rising or falling at this time?
rising falling

(d) When will the ball hit the ground?
t = sec

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Here we present multiple examples as to how to use derivative of f to solve the problems related to velocity and rate of change.

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