2.2a Derivatives The Shortcut Objective: To learn the Power Rule for derivatives

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2.2a Derivatives The Shortcut!Objective To
learn the Power Rule for derivatives
Basic Rules for Differentiation ? Constant y
c (horizontal)
? y 0 Why?
Ex1) y 7 ? y
? Power Rule f(x) xn
? f(x) n xn-1
?Constant Multiple f(x) cxn
? f(x) cn xn-1 ( what does this mean?)
DO NOT LEAVE NEGATIVE EXPONENTS!!
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See Ex 5 6 from the book for more common
re-writes!
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Why do we need this ?
Ex11) y 3x3 5x2 2x 9 ?
? Simplify if possible We do not have rules
for products or quotients (YET!)
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Basic Trig Derivatives
y sin x ? y cos x ? y tan x ? y cot x
? y sec x ? y csc x ?
Help! How can you remember which is negative?
Ex15) y 3x2 cos x Ex16) y 5tan x 4x3
Assign 3-30 all Label answers and try all of
this set!
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2.2b More Basic DerivativesObjective To use
the shortcut to write equations for tangent lines
? Questions? ? The quiz is after section 2-4.
Tangent lines 1)Find f(x) ? slope equation (use
the shortcut) 2) f(c) ? slope of the
tangent line at x c 3) f(c) ? the
y-value at x c 4) y y1 f(c) (x
x1)
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Determine whether the function is differentiable
at x c.
At x 2?Is the function continuous?How can you
tell?Is the function differentiable? How can
you tell?
At x 1?Is the function continuous?How can you
tell?Is the function differentiable? How can
you tell?
For Piecewise The function must be continuous
(equal at x c) and differentiable (the
derivatives of each branch must be equal at x
c.)
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Find k such that the the line is tangent to the
graph of the function. What does this mean??????
? They share a common point and slope!
Function Line F(x) k x2 y
-4x 7
Determine where the function has horizontal
tangents.What does this mean????
? m 0 ? f0
Ex5) y 3x3 2x2 4 Ex 6)y x3 x
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Vertical Motion x(t) -½gt2 vot xo v(t)
a(t)
? Constant acceleration for vertical motion
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Note Do not use physics on the AP Calculus
Exam!! ? You must use Calculus!

• Ex 10 from page 112
• The position of the diver is given by s(t)
  -16t2 16t 32,
• When does the diver hit the water?
• b. What is the divers velocity at impact?

• s(t) 0 when the diver hits the water.
• Need v(t) to get velocity at time of impact.

What is the difference between speed and velocity?
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Linear Approximation This uses the equation for
the tangent line to approximate the value of the
function close to the point of tangency.
The closer the point chosen is to x c, the
better the approximation of f(x).
Ex Given f(x) x3 at (1, 1)

• Use the tangent line to approximate the value of
  the function for f(.5),f(.9), f(1.1), f(1.5).
• Compare these values to the values of
  f(.5),f(.9), f(1.1), f(1.5).

( )
These could also be called local linear approx
or tangent line approx.Assign 2-2 31-51
odd, 57-62 63, 65, 67-72, 74, 81-88 all, 107, 111
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2.3a Product and Quotient RulesObjective
To apply the product and quotient rules for
derivatives.
? Questions from 2.2 ?????????

• Where does f(x) have a positive slope?
• 2) Where does f(x) have a negative slope?
• 3) Where is the slope 0?
• 4) Give the extrema for f(x).

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Caution It may be simpler to multiply before
you take a derivative.
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If the quotient can be simplified ?YOU DO NOT
NEED QR!
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Extended PR If y f . g . h, find y
Use your results to find y for y 3x2 . sin
x . cos x .
Memorize the PR and the QR ? ASAP Assign 1-37
odd, 64-66, 69, 70, 75, 79, 103
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Please let your students know that MEGSL is
hosting a Math competition with St. Louis
Community College at Forest Park. Two 1000
scholarships will be awarded to the highest
scoring seniors in each division. They are also
giving away a TI calculator (I assume it is
graphing, but it doesn't say) through a random
drawing, and a memento for each
participant... The Scholar Bowl team is thinking
about going, and we are wondering if others might
be interested. Please send any kids you think
might want to go down to my room in the next
couple of school days... if we are going I have
to turn the form in by Tuesday. The date is
Saturday November 12th - 9am - 2pm
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2.3b MORE PR, QR AND HIGHER ORDER
DERIVATIVESObjective To continue using PR and
QR and to learn the notation and procedures for
higher order derivatives
? Questions from 2.3a?
Note the change that occurs at the 4th derivative
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Ex1) Given y 4x5 x4 3x2
How many derivatives will it take for this to
constant?
How many derivatives will it take for this to 0?
Given f(x) x sin x, find f (x)
Recall Trig derivativesy sin x ? y cot x
? y cos x ? y sec x ? y tan x ? y
csc x ?
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Page 115 True or False 81) If f(x) g(x) ,
then f(x) g(x). 82) If f(x) g(x) c, then
f(x) g(x). 83) If y p2 , then y 2p
. 84) If y x/p, then dy/dx 1/p . 85) If
g(x) 3f(x) , then g(x) 3 f(x) . 86) If
f(x) 1/xn , then f(x) 1/(nxn-1) .
Make sure you have your PR and QR formulas
memorized! Assign 2-3 39-53 odd, 59-62, 67, 68,
83-92, 93-100, 109-114
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2.4a Chain Rule
Objective To take derivatives on composition of
functions by using the chain rule.
Questions from 2.3?
? The chain rule (CR) is used to take the
derivative of composite functions. You need to
identify the outermost to innermost function.
General h(x) f(g(x)) ? h(x) f(g(x)) g(x)
OR ? h(x) f(g(k(x))) ? h f(g(k(x)) g(k(x))
k(x)
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What is the chain rule?
Assign 2-4 1-26 all Try all of these!
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2.4b More Chain Rule
Objective To use CR with PR and QR and to write
tangent lines for composite functions.
Questions 2.4a?
How do you know if PR or CR? y tan (sin x)
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• f(-1)
• f(-1)
• 3)

Can you use PR and QR with CR?
Assign2-4 27-30, 31-39 odd, 47-59 odd, 67-71
odd, 79, 81
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2.4c Last Day of CR
Objective To do higher order derivatives with CR
Questions 2.4b??
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Absolute Value If you can change the function
to piecewise, that is generally best. If you
cannot ? memorize the following.
4. y cos x ? y
3. y 3x 1 ? y

• y x2 6x 5 ? y

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• A 15 cm pendulum moves according to the
  equationq 0.2cos 8t where q is the angular
  displacement from the vertical in radians and t
  is the time in seconds. Determine the maximum
  angular displacement and the rate of change when
  t 3 sec.

• Ask for a copy of the review so you can start
  studying over the weekend.
• Make sure you review page 133.
• Assign 2-4 74-82, 92, 93, 102, 105-108, 111, 112

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Review for Quiz

• Quiz over 2.2-2.4
• Power Rule
• Product Rule y f(x) g(x) ? y fg g f
• Quotient Rule y f(x) / g(x) ? (fg
  gf)/g2
• Chain Rule y f(g(x)) ? y f(g(x)) g(x)
• Can you find horizontal tangents, write
  equations for tangent lines, write equations for
  normal lines, find where 2 curves have parallel
  slopes, find slopes at any given point, find
  slopes of piecewise functions, make a piecewise
  function differentiable???
• Do you know your trig derivatives? When to
  simplify? How to find common denominators? Do
  you remember the notation for higher order
  derivatives?

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2.5a Implicit DifferentiationObjective To find
the derivative for implicitly defined functions.
Questions?
? If you cannot solve for y ? you must use
implicit differentiation.
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Recall 6x. ? Power Rule
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?Find the derivative implicitly. ?Solve for dy/dx
(y)
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Implicit allows us to find slope for circles. ?
x2 y2 25 ? Circle centered at the origin
with radius 5.Find the tangent line to the
circle at the following points. DERIVATIVE 1.
(3, 4) 2. (-5, 0) 3. (-3,
4) 4. (0, 5) 5. (-3, -4)
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Just a few more to try!!! Ex) 4xy y2
5x Ex) 2x2 3xy y2 7 Ex sin(xy) y2
Do you understand when and how to use implicit
differentiation?Assign2-51-29 odd
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2.5b Implicit Day 2Objective To use implicit
differentiation on higher order derivatives and
to write tangent lines
Questions??
ReviewIf the variable is x ? use power rule. If
you cannot solve for y ? you must use implicit.
Second Derivatives ?Take the 1st derivative
implicitly ?Solve for dy/dx ?Take the 2nd
derivative implicitly ?Substitute dy/dx into
the original equation and simplify.
Find y for x2 y2 25 ?
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Find y 1 xy x - y
Find y y2 4x
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Write the equation for the tangent and normal
line to the curve at the point (1, 2). x2
y3 -7
45) Show that any normal line to x2 y2 r2
passes through the origin.
Must learn 2.5 for section 2.6!! Can you
take second derivatives on implicitly defined
functions?Assign 2-5 6-16 even, 35-45 odd,
59, 60
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Chapter 2 Review
2-1 ?Definition of derivatives (both
versions) ?Compare msec to mtan ?Sketch
derivatives ?Discuss where derivatives do not
exist and why they do not exist
2-3 ? Product Rule ? Quotient Rule ? Higher
Order Derivatives (notation Procedures)
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2-5 ? Implicit Differentiation When to use
and why this method is necessary Second
derivatives notation and procedures
All Sections ? Equations for tangent lines to
curves ? s(t) position ? Equations for normal
lines to curves ? s'(t) v(t) velocity (m/s
has direction) ? f'(c) ? s''(t) v'(t)
a(t) acceleration (m/s2) ? Horizontal tangent
lines ?v(t) speed (m/s) ? Exact answers or
nearest thousandth ? LABEL everything ? Limits