Average Rate of Change
For f(x)=x2−4x+7, determine the average rate of change from x=1 to x=5.
[f(5)−f(1)]/(5-1)
=(12−4)/4
=2
What does the slope of a secant line represent on the graph of a function?
Answer: The average rate of change of the function over an interval.
Using the definition of the derivative, find f′(x) for
f(x)=6x2+5/x
12x-5/x2
The position of a particle is s(t)=t2+4t+1.
Find its instantaneous velocity at t=3.
10
If the average rate of change of a function on an interval is zero, does this necessarily mean that the function is constant on that interval?
Answer: No. The function can increase and decrease while having the same values at the endpoints.
Find the average rate of change of
f(x)=2x3−3x2+4
on the interval [1,3].
f(1)=3,f(3)=31
f(1)=3,f(3)=31
(31−3)/(3-1)=14
Find the equation of the secant line joining the points on
f(x)=x2+1
at x=1 and x=
y=5x-3
Using first principles, determine the derivative of
f(x)=3x4−2x2+5.
12x3-4x
A particle's position is
s(t)=2t3−9t2+12t.
At what times is the particle momentarily at rest?
v=0
t=1,2
A function has a positive average rate of change on an interval. Does this guarantee that the function is increasing throughout the entire interval?
Answer: No. The function could decrease over part of the interval and still have an overall positive average rate of change.
The position of an object is modeled by
s(t)=t3−6t2+9t+2.
Find the average velocity from t=1 to t=4.
s(1)=6,s(4)=6
(6−6)/(4-1)=0
For
f(x)=x2−3x+2,
find the slope of the secant line between x=2 and x=2+h.
1+h
Use first principles to find the derivative of
f(x)=x1/2/2x.
The position of an object is
s(t)=t3−6t2+9t.
Determine the intervals on which the object is moving forward and backward.
v(t)=3(t−1)(t−3)
Moving forward:
t<1 and t>3
Moving backward:
1<t<3
What is the geometric meaning of the instantaneous rate of change of f(x) at x=a?
Answer: It is the slope of the tangent line to the graph at x=a.
For
f(x)=x1,
find the average rate of change from x=2 to x=2+h.
−1/(2(2+h))
For f(x)=x3, write the difference quotient used to determine the instantaneous rate of change at x=2.
lim[(2+h)3-8]/h,
h to 0
Using first principles, determine the derivative of
f(x)=x1/2.
1/2x-1/2
A particle has position
s(t)=t3−6t2+9t+4.
Find the instantaneous velocity at t=2 and interpret your answer.
-3
The particle is moving in the negative direction at 3 units per second.
A function has a horizontal tangent at x=3. What can you conclude about its instantaneous rate of change at x=3?
Answer:
f′(3)=0
Find the average rate of change of
f(x)=(x+4)1/2
from x=5 to x=9.
(131/2-3)/4
Use the limit definition of the derivative to determine the instantaneous rate of change of
f(x)=x2+3x
at x=2.
7
Use the limit definition of the derivative to find the instantaneous rate of change of
f(x)=x3−4x
at x=−2.
8
The position of a particle is
s(t)=t3−9t2+24t.
At what time does the particle change direction?
v(t)=3t2−18t+24
v(t)=3(t−2)(t−4)=0
Critical times:
t=2,t=4
The velocity changes sign at both values, so the particle changes direction at
t=2 and t=4
The function
f(x)=x3−3x2−9x+5
has two points where its tangent is horizontal. Find the x-coordinates of these points and determine the corresponding y-coordinates.
(−1,10) and (3,−22)