DIFFERENTIATION

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AP Calculus BC Class Notes Ms. Nickles
DIFFERENTIATION
Fall 2009
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Learning Objectives

• Understand the formulation and definition of a
  derivative
• Learn about the difference quotients
• 2. Know how to find derivatives using the
  definition
• 3. Applying the derivative (Applications)

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Definition
The derivative of a function f is the function f
whose value at x is
Memorize
provided the limit exists.
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Example
Find the slope of the tangent line to
at x2. Stated differently, find f (2). Write
the equation of the tangent line.
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Definition
Memorize
An alternate definition of a derivative at a
point xa is
provided the limit exists.
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Formal Definition of Derivative
Note that the derivative of f(x) is actually the
slope of the tangent line.
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Derivative Notation
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Example
Find the slope at .
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If we find derivatives with the difference
quotient
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Important Idea
if the limit exists.
If the limit does not exist, if there is a
discontinuity at x or if the left derivative is
not equal to the right, there is no derivative at
x.
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Example of functions that are not differentiable
because theyre not continuous
1.
2.
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3.
4.
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Important Idea
Theorem If f has a derivative at xa, then f is
continuous at xa.
The converse if f is continuous at a, then f has
a derivative at a, is not true.
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Calculating Derivatives

• Develop techniques for computing derivatives
• Addition rule
• Product rule
• Quotient rule...

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Important Idea
There are differentiation rules that allow a
shorter, easier way to find derivatives. You
will need to memorize these rules.
Were now ready to use the Differentiation Rules
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Derivative of constant function

• f(x)c has tangent line with slope 0 at every
  point

Note the derivative of a constant is always 0.
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Power Rule

• If n is a positive integer then

Example Find dy/dx of
First write as then use the power rule

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Example
Find the derivative
Now find the slope of the graph of f(x) x3
when a. x -1 b. x 0 c. x 2
f(-1) 3(-1)2 3 f(0) 3(0)2 0 f(2)
3(2)2 12
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Try This
Find the derivative
First write as_____, then
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Constant Multiple Rule
Examples

Where c is a constant.
Note alternate notation
is the same as
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Sums and Differences
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Example
Find the horizontal tangents of
Plugging the x values into the original equation,
we get
(The function is even, so we only get two
horizontal tangents.)
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Product Rule
Notice that this is not just the product of two
derivatives.
This is sometimes memorized as


Note this can also be written as d f(x)g(x)
f(x)g(x) g(x)f(x) dx
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Example
Find the derivative, if it exists
f(x) (x2 2x 1)(3x2) (2x 2)(x3
1) f(x) 3x4 6x3 3x2 2x4 -2x 2x3
2 f(x) 5x4 8x3 3x2 - 2x 2
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Example

• Compute the derivative of the following function
  by (1) Using product rule, (2) multiplying first
  and then doing term-by-term differentiation

y 84x6 20x4 12x
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Quotient Rule
or
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Important Idea
Sometimes it is easier to re-write the function
and find the derivative using rules other than
the quotient rule. However, for some functions,
the quotient rule must be used.
Example
Must use the quotient rule on this one
f (x2 1)(5) (5x 2)(2x) 5x2 5
10x2 4x -5x2 4x 5 (x2 1)2 (x2
1)2 (x2 1)2
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Try This
Find the derivative using the quotient rule and
simplify your answer
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Try This
Find an equation of the line tangent to s(t) at
t2
s _2_ (t 1)2
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Try This
Find the derivative (hint re-write and use the
quotient rule)
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Try This
Find the derivative, if it exists
Solution
h -2x3 9x2 -2
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Example

• Compute the derivative of the following function
  and determine where it has a horizontal tangent
  line

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Higher Order Derivatives
(y double prime)
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Examples
1. Find the second derivative
2. Find the second derivative
3. Find the second derivative
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Average Velocity
Average velocity change in distance change
in time To find the average velocity over an
interval plug the interval values into the
position function. Then find s(t2 ) s(t1
) t2 t1
Position Function s(t) gt2 v0 t s0
2
Where, s0 is the initial height V0 initial
velocity g is the acceleration due to gravity
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Instantaneous Velocity
To find the instantaneous velocity of an object
when t c. Find the derivative of the position
function v(t) s(t)
Acceleration
To find acceleration take the derivative of the
velocity function. a(t) v(t) s(t)
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Example
If an object is dropped, its height above the
ground is given by
1. Find the average velocity between 1 and 3
seconds.
2. Find the instantaneous velocity at 3 seconds.
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Example
The height above the ground of an object dropped
from altitude is
Find the velocity and acceleration of the object
after 10 seconds.
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Chain Rule
If f and g are differentiable, then so is their
composition f(g(x))
Y (x2 1)3 Y 3 (x2 1)2 (2x) 6x
(x2 1)2
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Example
Find using the chain rule
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Example
Differentiate
Let
and
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Try This
Find a function for the slope of the tangent line
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Example
Find
Let and
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Try This
1)
Find
2)
Find the derivative
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(No Transcript)
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Definition
A function is in explicit form if the independent
variable is stated in terms of the dependent
variable. Example
A function is in implicit form if the independent
variable is not stated in terms of the dependent
variable. Example
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Important Idea
When a function is in implicit form, it may be
difficult to solve for a dependent variable and
differentiate explicitly. In this case, implicit
differentiation is used.

• Implicit differentiation is a method that is used
  to find the derivative of y when given an
  implicit function.
• Steps
• Take the derivative of both sides of the equation
  with respect to x.
• Solve the resulting equation for y.

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Example
Differentiate xy1 implicitly.
1. Use product rule on left.
2. Differentiate x normally differentiate y
using chain rule.
3. Solve for and simplify.
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Example
Differentiate
Variables disagree-use power rule differentiate
implicitly
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Example
Find given that
Steps
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Try This
Find the slope, if it exists
1.
at the points (0,1) and (1,0)
at (0,1), slope0 and at (1,0), slope is undefined
2.
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Practice
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Derivatives of Trig functions
Derivative of tangent
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Derivatives of all trig functions
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Example
Find the derivatives of the following
1.
2.
3.
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Home work
Use the identity
and the quotient rule to show that the derivative
of
is
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Examples
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Derivative of exponential functions
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The exponential function, exp (x)
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Derivative of log functions
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Differentiating Inverse Functions
Find the derivative of y f -1(x) Step 1 Apply
f to both sides to get x f(y) Step 2
Differentiate Step 3 Conclude that
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Example
Consider y f(x) x13 2x 5. Compute the
derivative of its inverse x f -1(y) using
above formula. Compute the derivative of its
inverse using implicit differentiation.
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Examples
Compute the derivatives of the following
functions
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Logarithmic Differentiation
General Strategy Differentiate complicated
function y f(x) by simplifying and (implicitly)
differentiating both sides of ln y ln
f(x)
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Derivatives of irrational powers of x
Let y xr , xgt0, where r is a real
number Differentiate ln y ln xr Conclude that
for all real numbers have power law
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More facts
If f is differentiable, then it is
continuous Converse not true There are
continuous functions which are not
differentiable Example x is not differentiable
at x0
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Local Extrema
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Example

• Let f(x) (x-1)2(x2), -2 ? x ? 3
• Use the graph of f(x) to find all local extrema
• Find the global extrema

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Example

• Consider f(x) x2-4 for 2.5 ? x lt 3
• Find all local and global extrema

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Fermats Theorem

• Theorem If f has a local extremum at an
  interior point c and f? (c) exists, then f? (c)
  0.
• Proof Case 1 Local maximum at interior point c
• Then derivative must go from ?0 to ?0 around c
• Proof Case 2 Local minimum at interior point c
• Then derivative must go from ?0 to ?0 around c

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Cautionary notes

• f? (c) 0 need not imply local extrema
• Function need not be differentiable at a local
  extremum (e.g., earlier example x2-4)
• Local extrema may occur at endpoints

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Summary Guidelines for finding local extrema

• Dont assume f? (c) 0 gives you a local extrema
  (such points are just candidates)
• Check points where derivative not defined
• Check endpoints of the domain
• These are the three candidates for local extrema
• Critical points points where f? (c)0 or where
  derivative not defined

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What goes up must come down

• Rolles Theorem Suppose that f is
  differentiable on (a,b) and continuous on a,b.
  If f(a) f(b) 0 then there must be a point c
  in (a,b) where f ?(c) 0.

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Proof of Rolles theorem

• If f 0 everywhere its easy
• Assume that f gt 0 somewhere (case flt 0 somewhere
  similar)
• Know that f must attain a maximum value at some
  point which must be a critical point as it cant
  be an endpoint (because of assumption that f gt 0
  somewhere).
• The derivative vanishes at this critical point
  where maximum is attained.

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Average Velocity Must be Attained

• Mean Value Theorem Let f be differentiable on
  (a,b) and continuous on a,b. Then there must
  be a point c in (a,b) where

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Consequences

• f is increasing on a,b if f ?(x) gt 0 for all x
  in (a,b)
• f is decreasing if f ?(x) lt 0 for all x in (a,b)
• f is constant if f ?(x)0 for all x in (a,b)
• If f ?(x) g ?(x) in (a,b) then f and g differ
  by a constant k in a,b

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Example
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Increasing/Decreasing functions

• If f is defined on an interval then
• f is increasing if f(x1) lt f(x2) whenever x1ltx2
• f is decreasing if f(x1) gt f(x2) whenever x1ltx2
• f is constant if f(x1) f(x2) for all points x1
  and x2

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Sign of first derivatives

• Fact If f is continuous on a,b and
  differentiable in (a,b), then
• f is increasing on a,b if f ?(x)gt0 for all x in
  (a,b)
• f is decreasing on a,b if f ?(x)lt0 for all x in
  (a,b)
• f is constant on a,b if f ?(x)0 for all x in
  (a,b)

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Examples

• Find the intervals on which the following
  functions are increasing or decreasing

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Concavity

• Concavity measures the curvature of a function.
  Concave upholds water. Concave downspills
  water.
• If f is differentiable on an open interval I then
  f is concave up if f ? is increasing there, and f
  is concave down if f ? is decreasing there
• Conclusion f concave up on I if f ?? gt0 on I and
  f concave down on I if f ?? lt0 on I

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Examples

• Find the intervals on which the following
  functions are concave up and concave down

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Inflection Points

• If f changes concavity at x0 then f has an
  inflection point at x0 (i.e. f ?? changes sign
  at x0)
• Examples Find the inflection points of the
  following functions

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More examples

• Find the inflection points of the following
  functions and sketch their graphs

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Definition of relative extrema

• Say f has a relative maximum at x0 if f(x0) ?
  f(x) for x near x0
• Say f has a relative minimum at x0 if f(x) ?
  f(x0) for x near x0
• Say f has a relative extremum at x0 if one of the
  above occurs there
• Relative max/min may not be a global max/min

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Critical points

• If f ?(x)0 or if f is not differentiable at x,
  say that x is a critical point for f
• Importance Relative extrema can only occur at
  critical points
• Remark If f ?(x0)0, call x0 a stationery point

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1st Derivative Test

• First Derivative Test Suppose that f is
  continuous at a critical point x0
• If f ? changes signs from ? to ? at x0 then f has
  a relative maximum at x0
• If f ? changes signs from ? to ? at x0 then f
  has a relative minimum at x0
• If f ? does not change signs at x0 then f cannot
  have a relative extrema at x0

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Examples

• Find the local extrema (if any) of the following

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Second Derivative Test

• Suppose that f is twice differentiable and that
  f ? (x0) 0
• If f ?? (x0) ? 0 then f has a relative minimum at
  x0
• If f ?? (x0) ? 0 then f has a relative maximum at
  x0
• If f ?? (x0) 0 the test is inconclusive

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Examples

• Use the 2nd derivative test to locate the extrema
  of the following functions

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Key Steps in curve sketching

• Find critical points and intervals on which f is
  increasing/decreasing
• Determine nature of critical points relative
  extrema?
• Find inflection points and intervals on which f
  is concave up/down
• Locate x y intercepts if possible

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Examples

• Sketch the graph of the following functions and
  label the coordinates of the stationery points
  and inflection points

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Optimization Applied max/min problems
Analyze using earlier techniques Two main cases
Problems reducing to max/min of functions on
closed bounded intervals Problems reducing to
max/min of functions on other types of intervals
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Boxers

• A box which is open on top is made from a 16 inch
  by 30 inch cardboard by cutting out
• squares of equal size from the four corners and
  bending up the sides. What size should
• the squares be to obtain a box with largest
  possible volume?

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Campbells Soup

• Find the dimensions of a can with the smallest
  possible surface area that encloses a volume of
  16? cm3 of chicken noodle soup.

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Profit Revenue - Cost

• Microsoft sells PowerPoint at a price of 100 per
  program. If the daily production cost in dollars
  for x copies of the program is C(x)
  100,000 50x 0.0025x2and if the daily
  production capacity is at most 7000 copies, how
  many copies must be manufactured and sold to
  maximize the profit?
• Should Microsoft expand the daily production
  capacity?

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Bay Watch

• A pretty girl is drowning. She is 100 m
  down-shore from a lifeguard and she is 9 m
  offshore. The lifeguard can swim 1 m/sec and he
  can run 2 m/sec. What path should he take to get
  to the girl as quickly as possible?

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Different choices for optimization function

• Find a point on the curve y x2 that is closest
  to the point (18,0).

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Definition
Acceleration is the derivative of the velocity
function
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Definition
The position function of a free falling body is
This discovery is attributed to Galileo Galilei
(1564-1642)
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Example
Given that the position function for a free
falling body is
where g is a constant, find the velocity
function, v(t), and the acceleration function
a(t).
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Solution
v(t) is the first derivative of s(t). a(t) is the
second derivative of s(t)
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Definition
Free-fall constant, g, is the acceleration of an
object due to earths gravity. Its value is
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Anti-differentiation By Parts
Learning Goals
Use integration by parts to evaluate definite and
indefinite integrals.
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Important Idea
Integration by parts works with integrals such as
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Review
The product rule for differentiation
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Try This
Differentiate
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Important Idea
leads to...
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Analysis
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...therefore
or...
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Example
Evaluate
???
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Example
Second Try
Evaluate
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Example (cont.)
c
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Example
Evaluate
Let uln x dvdx, then integrate by parts...
Check your answer.
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Example
Evaluate
Let u x dvcos x dx, then integrate by parts...
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Try This
Evaluate
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Important Idea
Guidelines for Integration by Parts
1. Try letting dv be the most complicated part of
the integrand that fits an integration rule.
Integrate to obtain v and let u be the remaining
part of the integrand. Find du.
2. Try letting u be the part of the integrand
whose derivative is a simpler function than u. dv
is then the remaining part of the integrand.
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3. Arbitrarily let dv be one factor of the
integrand and u be the other and see if it works.
Sometimes either choice will work.
Example Evaluate
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Example
Sometimes we must use integration by parts more
than once in the same problem.
Evaluate
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Example
Evaluate
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Try This
Evaluate
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Logistic Growth
Learning Goals
Anti-differentiate partial fraction
decompositions
Solve problems involving logistic population
growth.
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Example The population in Arlington grew from
7692 in 1950 to 332,969 in 2000. If population
continues to grow at this rate, what will be the
population in 2050?
Important Idea Exponential growth models are
valid only for a limited time. There are
constraints such as available food, habitat, and
living space that impose limits on exponential
growth. Logistic growth is a model that
recognizes these constraints.
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Example
From Algebra
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Important Idea
The reverse process
is called decomposition to partial fractions. It
is a useful anti-differentiation technique and
needed for solving logistic grow models.
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Important Idea
Instead of integrating
The degree of the numerator must be less than the
degree of the denominator.
we integrate the equivalent partial fraction
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Practice
Integrate
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Example
Integrate using partial fractions
1. Change the integrand to partial fractions
verify the result.
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Step 1
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Step 2
Since A2 B1
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3. Integrate the partial fractions
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Practice
Integrate using partial fractions
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Practice
Integrate using partial fractions
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Example
Integrate using partial fractions
Hint Divide then use partial fractions.
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Important Idea
When the degree of the numerator is greater than
or equal the degree of the denominator, divide
then using partial fraction decomposition.
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Practice
Integrate using partial fractions
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Example
Integrate using partial fractions
How does this problem differ from previous
problems?
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Example
1. Factor completely the denominator
2. Write the partial fractions using one fraction
for each factor a constant for each fraction.
3. Integrate the fractions.
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Important Idea
Partial fractions work only when you can factor
the denominator of the rational function.
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Definition
The Logistic Differential Equation Rate of
change is proportional to the existing quantity p
and the difference between the existing quantity
and a limit M called the carrying capacity.
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Definition
The Logistic Differential Equation (Logistic
D.E.)
initial quantity
carrying capacity
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Example
The change in population of bears in a national
park can be modeled by the logistic differential
equation
What is the present number of bears?
What is the carrying capacity?
What is k?
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Example
Solve the Logistic D.E using separation of
variables and decomposition of partial fractions.
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Definition
If we solve the logistic D.E., we obtain the
Logistic Growth Model
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Practice
For the logistic grow model
What is
M
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Definition
The Logistic Growth Model
,
M is the max population or carrying capacity. It
is the upper limit past which growth cannot
occur. Some texts use L instead of M.
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Definition
The Logistic Growth Model
where
p0 is the present population (usually at time
t0)
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Definition
The Logistic Growth Model
where
P is the population after time t has passed
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Definition
The Logistic Growth Model
where
k is the constant of proportionality
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Important Idea
Logistic Grow
Exponential Grow
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Example
A population of 40 bacteria grows according to
the Logistic Growth Model. The carrying capacity
is 1000 and k0.00007. Find the population after
25 hours, after 100 hours and after 1000 hours.
Graph the model on your calculator.
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Example
A park can support no more than 100 bears. Ten
bears are in the park today. Write the D.E. for
logistic growth with k0.001. Use the model for
logistic growth to find when the bear population
will reach 50.
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Example
P
100
Slope Field with solution curve-note the S shape
t
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Try This
P
What is the limit as
100
?
t
100
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P
Try This
100
How many bears when they are growing the fastest?
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Try This
The number of infected persons y in a population
after t days follows the logistic differential
equation
a) What is the maximum number that can be
infected?
b) If 10 people are infected at t0, find a
solution for the D.E.
c) How many days until half the carrying capacity
is infected? Hint half is 150.
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Solution
a) 300
b)
c) 56 days
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Slope Fields Eulers Method
Learning Goals

• Solve initial value problems
• Construct slope fields
• Use Eulers method

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Definition
Equations involving derivatives are called
differential equations. The order of a
differential equation is the order of the highest
derivative involved in the equation.
Example
Find all functions that satisfy the
differential equation
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Definition
In the last example, we found a general solution.
You cannot find a particular (unique) solution to
a differential equation unless we are given the
value of the function at a single point, called
an initial condition.
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Example
Find the particular solution to the differential
equation
whose graph passes through the point (1,0)
initial condition
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Practice
Find the particular solution to the differential
equation
whose graph passes through the point (1,0)
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Example
Solution
Find the particular solution to the equation
(0,3)
whose graph passes through (0,3).
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Important Idea
A discontinuity in the solution to a differential
equation requires the domain of the solution be
specified. The domain must be the continuous part
of the solution that contains the initial
condition.
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Practice
Find the particular solution to the differential
equation
whose graph passes
through (1,1). Specify any restrictions on the
domain of the solution.
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Example
Suppose we have the initial value problem
Find a formula for y.
Problem I cant find an anti-derivative
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Analysis
The slope
(rate of change) at any point (x,y) on the
solution curve is the x coordinate of the point
minus the y coordinate.
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Solution Curve
(0,1)
Rate of change at (0,1)x-y-1
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Solution Curve
(2,1)
Rate of change at (2,1)x-y1
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can be represented by tangent line segments
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Definition
All such segments represent the slope field or
direction field for
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Example
Using the slope field, sketch the solution curve
through (0,1)
Hint start at (0,1). Sketch right then left,
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Important Idea
Since a differential equation gives the slope
(rate of change) at any point (x,y), we can use
that information to approximate the solution
curve when we are not able to solve the
differential equation.
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Practice
Using the slope field, sketch the solution curve
through (1,0)
(1,0) is the initial condition. Estimate the
solution to the initial value problem at x3.
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Example
For
Sketch the tangent line segments (slope field) at
each integer coordinate
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Important Idea
If we are given a differential equation
and an initial condition ,
we can also approximate a solution by a process
called Eulers Method.
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Example
Use Eulers Method to approximate the solution of
y y with initial conditions y(0)1 and a
step size of
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Important Idea
the slope of the tangent line at
y is y
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Example
Move to right 1 now yy2
Use Eulers Method to approximate the solution of
y y with initial condition y(0)1 and a step
size of
(1,2)
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Example
Move (again) to right, now yy4
Use Eulers Method to approximate the solution of
y y with initial condition y(0)1 and a step
size of
(2,4)
(1,2)
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Example
Move (again) to right, now yy8
Use Eulers Method to approximate the solution of
y y with initial condition y(0)1 and a step
size of
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Important Idea
m
therefore
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Practice
(3,8)
What is the Euler approximation to
(2,4)
(1,2)
with initial condition
initial condition
at x3.
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Try This
How could the Euler approximation be improved?
Let be smaller
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Example
5
Use Eulers Method to approximate the solution of
y y at x2 with initial condition y(0)1 and a
step size of
4
(2,5.1)
3
2
1
2
1
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Example
yy
y(0)1
Can you see a pattern?
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Example
Use Eulers Method to approximate the solution of
y xy at x2 with initial condition y(1)1 and
a step size of
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Example
y xy at x2,
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Practice
For the differential equation
Starting at (0,2), use Eulers method with 2
steps of size 1 to estimate