Download 2.2 Basic Differentiation Rules and Rates of Change Objective: Find

Ms. Battaglia
AB/BC Calculus
Thm 2.2 The Constant Rule
The derivative of a constant function is
0. That is, if c is a real number, then
d
[c] = 0
dx
Examples:
Function
a.y = 7
b.f(x) = 0
c. s(t) = -3
d.y = kπ2, k is constant
Derivative
dy/dx =
f’(x) =
s’(t) =
y’ =
Thm 2.3 The Power Rule
If n is a rational number, then the function
f(x) = xn is differentiable and
d/dx[xn]=nxn-1
For f to be differentiable at x=0, n must be
a number such that xn-1 is defined on an
interval containing 0.
a.
f (x) = x
3
b. g(x) = 3x
1
c. y = 2
x
Find the slope of the graph of f(x) = x4 when
a. x = -1
b. x = 0
c. x = 1
Find an equation of the tangent line to the
graph of f(x) = x2 when x = -2
Thm 2.4 The Constant Multiple Rule
If f is a differentiable function and c is a real
number, then cf is also differentiable and
d/dx [cf(x)] = cf’(x)
Thm 2.5 The Sum and Difference Rules
The sum (or difference) of two differentiable
functions f and g is itself differentiable. Moreover,
the derivative of f+g (or f-g) is the sum (or
difference) of the derivatives of f and g.
d
[ f (x) + g(x)] = f '(x) + g'(x)
dx
d
[ f (x) - g(x)] = f '(x) - g'(x)
dx
a. y = 2
x
2
4t
b. y =
5
c. y = 2 x
d.
y=
1
2 3x 2
e.
-3x
y=
2
Original
Function
Rewrite
5
y= 3
2x
5 -3
y= x
2
y=
5
( 2x )
3
7
y = -2
3x
y=
7
(3x )
-2
Differentiate
y=
5
-4
-3x
(
)
2
Simplify
-15
y= 4
2x
a. f(x) = x3 – 4x + 5
b. g(x) = -x + 3x 3 -2x
4
2
Theorem 2.6
d
[sin x] = cos x
dx
d
[cos x] = -sin x
dx
a.
y = 2sin x
b. y = sin x
2
c. y = x + cos x
distance
Rate =
time
the average velocity is
Change in distance ∆s
=
Change in time
∆t
The position function for a
projectile is
s(t) = –16t2 + v0t + h0, where
v0 represents the initial velocity
of the object and h0 represents
the initial height of the object.
An object is dropped from the second-highest floor
of the Sears Tower, 1542 feet off of the ground. (The top
floor was unavailable, occupied by crews taping for the new ABC special "Behind the Final Behind the Rose Final Special, the
Most Dramatic Behind the Special Behind the Rose Ever.")





(a) Construct the position and velocity equations for the
object in terms of t, where t represents the number of
seconds that have elapsed since the object was released.
(b) Calculate the average velocity of the object over the
interval t = 2 and t = 3 seconds.
(c) Compute the velocity of the object 1, 2, and 3 seconds
after it is released.
(d) How many seconds does it take the object to hit the
ground? Report your answer accurate to the thousandths
place.
(e) If the object were to hit a six-foot-tall man squarely on
the top of the head as he (unluckily) passed beneath, how
fast would the object be moving at the moment of impact?
Report your answer accurate to the thousandths place.
 AB:
Read 2.2, Page 115 #3-30,
31-57 odd
 BC:
Read 2.2, Page 115 #3-30,
31-57 odd, 97-100