Basic Differentiation Rules and Rates of Change

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Basic Differentiation Rules and Rates of Change

• Section 2.2

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After this lesson, you should be able to

• Find the derivative using the Constant Rule.
• Find the derivative using the Power Rule.
• Find the derivative using the Constant Multiple
  Rule and the Sum and Difference Rules.
• Find the derivative of sine and cosine.
• Use derivatives to find rates of change.

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Basic Rules For Computing Derivatives
The Constant Rule The derivative of a constant
function is zero.
The slope of a horizontal line is zero.
Examples
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Basic Rules For Computing Derivatives
The Power Rule If n is a rational number, then
the function f(x) xn is
differentiable and
Examples
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Basic Rules For Computing Derivatives
The Constant Multiple Rule If f is a
differentiable function and ,
then cf is also differentiable.
Examples. Find the derivative of the function.
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Basic Rules For Computing Derivatives
The Sum and Difference Rules For f and g
differentiable functions,
Examples

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Basic Rules For Computing Derivatives
Derivatives of sine and cosine
Examples Function Derivative
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Finding the Equation of a Tangent Line
Example Find the equation of the tangent line
to the graph of f at x 1. Verify using
calculator.
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Finding an Equation of a Horizontal Tangent Line
Example Find an equation for the horizontal
tangent line to the graph of
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Rates of Change

• Rates of Change Used to determine the rate at
  which one variable changes with respect to
  another.
• For example,
• velocity is the change in position w/ respect to
  time
• acceleration is the change in velocity w/
  respect to time
• Water flow involves the change in height of the
  water w/ respect to time

Average Velocity
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Motion Along a Straight Line
Position function, s(t) gives the position of
the object at time t relative to the origin.

• If s is positive, the object moved to the right
  (or upward)
• If s is negative, the object moved to the left
  (or downward)
• s 0 is the starting position (origin)

Velocity function, v(t) gives the instantaneous
velocity of the object at time t.

• If v is positive, the object is moving forward or
  upward.
• If v is negative, the object is moving backward
  or downward.
• v 0 means the object is stopped (at that very
  instant)

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Velocity, Speed, and Acceleration
Velocity is the rate of change of position with
respect to time, thus we have v(t) s(t)
Measured in units of position over units of
time. e.g. ft/s, m/s
The speed of an object is the absolute value of
the velocity.
speed v(t)
Acceleration is the rate of change of velocity
with respect to time, thus we have a(t) v(t)
s(t) Measured in units of position
over units of time squared. e.g. ft/s2,
m/s2
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Free Falling Object
The position of a free falling object t seconds
after its release can be represented by the
equation
For the position, measured in feet, of a free
falling object, we have
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Example of Free Falling Object
94 A ball is thrown straight down from the top
of a 220-foot building with an initial velocity
of 22 feet/second. a) What is the average
velocity of the ball on the interval 1, 2? b)
What is its velocity after 3 seconds? c) What is
its velocity after falling 108 feet? d) Find the
velocity of the ball at impact.
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Example of Free Falling Object
a) What is the average velocity of the ball on
the interval 1, 2?
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Example of Free Falling Object
b) What is its velocity after 3 seconds?
Ahanow we need calculus!
The velocity of the ball after 3 seconds can be
expressed as v( ). Thus, we need an
expression for the velocity function.
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Example of Free Falling Object (continued)
c) What is its velocity after falling 108 feet?
Note If the ball falls 108 ft, its position,
s(t) 220 108 _______
i) We need to find the time at which the object
is ______ feet above the ground. That is, we need
to find the value of t for which s(t)
_____.Solve
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Example of Free Falling Object (continued)
c) What is its velocity after falling 108 feet?
(continued)
ii) Now we need to find the velocity at time t
_______.
Verify your solution using the graphing
calculator.
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Example of Free Falling Object (continued)
d) What is its velocity at impact?

1. We need to find the time at which the object is
   ______ feet above the ground. That is, we need to
   find the value of t for which s(t)
   _____.Solve
   Round to nearest tenth
   of second.

ii) Now we need to find the velocity at time t
_______.
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Homework
Section 2.2 page 115 1-59 odd, 83 - 88 all, 93,
95, 103