Section 2.2 - Basic Differentiation Rules and Rates of Change

1
Section 2.2 - Basic Differentiation Rules and
Rates of Change
2
The Constant Rule

• A constant function has derivative
  , or
• Note The constant function is a horizontal line
  with a constant slope of 0.

3
Examples

• Differentiate both of the following functions.

The function is a horizontal line at y 13.
Thus the slope is always 0.
The function is a horizontal line because e, Pi,
and i all represent numeric values. Thus the
slope is always 0.
4
The Power Rule

• For any real number n, the power function
  has the derivative , or
  
• Ex

5
Examples

• Differentiate all of the following functions.

1. Bring down the exponent
2. Leave the base alone
3. Subtract one from the original exponent.
The procedure does not change with variables.
Make sure the function is written as a power
function to use the rule.
6
The Constant Multiple Rule

• The derivative of a constant times a function, is
  the constant times the derivative of the
  function. In other words, if c is a constant and
  f is a differentiable function, then
• Objective Isolate a power function in order to
  take the derivative. For now, the cf(x) will
  look like .

7
Examples

• Differentiate all of the following functions.

Pull out the coefficient
Take the derivative
Make sure the function is written as a power
function to use the rule.
Make sure the function is written as a power
function to use the rule.
8
The Sum/Difference Rule

• The derivative of a sum or a difference of
  functions is the sum or difference of the
  derivatives. In other words, if f and g are
  both differentiable, then
• OR
• Objective Isolate an expressions in order to
  take the derivative with the Power and Constant
  Multiple Rules.

9
Example 1

• If find if
  and .

Evaluate the derivative of k at x 5
Find the derivative of k
10
Example 2

• Evaluate

Sum and Difference Rules
Constant Multiple Rule
Power Rule
Simplify
11
Example 3
First rewrite the absolute value function as a
piecewise function

• Find if .

Find the Left Hand Derivative
Find the Right Hand Derivative
Since the one-sided limits are not equal, the
derivative does not exist at the vertex
12
Example 4

• Find the constants a, b, c, and d such that the
  graph of contains
  the point (3,10) and has a horizontal tangent
  line at (0,1).

1. f(x) contains the points (3,10) and (0,1)
What do we know
2. The derivative of f(x) at x0 is 0
Use the points to help find a,b,c
Find the Derivative
We need another equation to find a and b
We know the derivative of f(x) at x0 and x1 is 0
13
Example 4 (Continued)

• Find the constants a, b, c, and d such that the
  graph of contains
  the point (3,10) and has a horizontal tangent
  line at (0,1).

1. f(x) contains the points (3,10) and (0,1)
AND
What do we know
2. The derivative of f(x) at x0 is 0
Find a
Use the Derivative
a 1, b 0, and c 1
14
Derivatives of Sine and Cosine

• We will assume the following to be true
• AND

15
Example 1

• Differentiate the function

Rewrite the ½ to pull it out easier
Rewrite the radical to use the power rule
Sum and Difference Rules
Constant Multiple Rule
Power Rule AND Derivative of Cosine/Sine
Simplify
16
Example 2

• Find the point(s) on the curve
  where the tangent line is horizontal.

Horizontal Lines have a slope of Zero.
First find the derivative.
Find the x values where the derivative (slope) is
zero
Find the corresponding y values
17
Calculus Synonyms

• The following expressions are all the same
• Instantaneous Rate of Change
• Slope of a Tangent Line
• Derivative
• DO NOT CONFUSE AVERAGE RATE OF CHANGE WITH
  INSTANTANEOUS RATE OF CHANGE.

18
Position Function

• The function s that gives the position (relative
  to the origin) of an object as a function of time
  t. Our functions will describe the motion of an
  object moving in a horizontal or vertical line.

TIME
2
0
1
3
2
4
5
6
7
Origin
Description of Movement
-2
Upward
Downward
No Movement
-4
19
Displacement

• Displacement is how far and in what direction
  something is from where it started after it has
  traveled. To calculate it in one dimension,
  simply subtract the final position from the
  initial position. In symbols, if s is a position
  function with respect to time t, the displacement
  on the time interval a,b is

EX Find the displacement between time 1 and 6.
20
Average Velocity

• The position function s can be used to find
  average velocity (speed) between two positions.
  Average velocity is the displacement divided by
  the total time. To calculate it between time a
  and time b

Two Points Needed
It is the average rate of change or slope.
EX Find the average velocity between time 2 and
5.
21
Instantaneous Velocity

• The position function s can be used to find
  instantaneous velocity (often just referred to as
  velocity) at a position if it exists. Velocity is
  the instantaneous rate of change or the
  derivative of s at time t

One Point Needed
EX Graph the objects velocity where it exists.
v(t)
Corner
Slope 0
2
Slope -4
t
1
2
3
4
5
6
Slope 2
-2
Slope 4
-4
Slope -2
Slope 0
22
Position, Velocity,

• Position, Velocity, and Acceleration are related
  in the following manner
• Position
• Velocity

Units Measure of length (ft, m, km, etc)
The object is Moving right/up when v(t) gt
0 Moving left/down when v(t) lt 0 Still or
changing directions when v(t) 0
Units Distance/Time (mph, m/s, ft/hr, etc)
Speed absolute value of v(t)
23
Example 1

• The position of a particle moving left and right
  with respect to an origin is graphed below.
  Complete the following
• Find the average velocity between time 1 and time
  4.
• Graph the particles velocity where it exists.
• Describe the particles motion.

24
Example 2

• Sketch a graph of the function that describes the
  motion of a particle moving up and down with the
  following characteristics
• The particles position is defined on 0,10
• The particles velocity is only positive on (4,7)
• The average velocity between 0 and 10 is 0.

25
Example 3

• The position of a particle is given by the
  equation
• where t is measured in seconds and s in meters.
• (a) Find the velocity at time t.

The derivative of the position function is the
velocity function.
26
Example 3 (continued)

• The position of a particle is given by the
  equation
• where t is measured in seconds and s in meters.
• (b) What is the velocity after 2 seconds?
• (c) What is the speed after 2 seconds?

m/s
m/s
27
Example 3 (continued)

• The position of a particle is given by the
  equation
• where t is measured in seconds and s in meters.
• (d) When is the particle at rest?

The particle is at rest when the velocity is 0.
After 1 second and 3 seconds