Cluster(B,1)The First derivative
Worksheet(1,1)
Objective 12A(Revision for grade 11)
Find the first derivative (the slope of the tangent) by usind First principles.
Vocabulary.
= ……………………
……………………
………………………………………………………………………………… ……………………
….
12A(8.1) Find from first principles …………………….
.
Examples:
1. . Find y' from first principles if y = If (c is constant)
2. a) Find y' from first principles if y =
b) Find the slope of the tangent where x = 1 and also where x = -6.
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Cluster(B,1)The First derivative
Worksheet(1,2)
1. . Find y' from first principles if y =
1. . Find y' from first principles if y=
……………………………………………………………………………………………………
Practice:(Pairs Activity)
………………………………………………………………………………………………………
Practice(individual Activity) Lesson Ending
……………………………………………………………………………………………………
Home work : Find y' from first principles if y =
http://www.youtube.com/watch?v=2wH-g60EJ18
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Cluster(B,1)The First derivative
Worksheet(1,3)
Rules of Derivatives
12A(8.1) Use the rules to find the derivatives
Examples: Find the derivatives
1)
2)
3)
4)
…………………………………………………………………………………………
Practice (Pairs Activity)
1)
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Cluster(B,1)The First derivative
2) Worksheet(1,4)
………………………………………………………………………
Practice(individual Activity) Lesson Ending
a) y = -5x3
b) y = 8x4 - 1
c)
……………………………………………………………………………
Homework : Find the derivatives
1)
2)
3)
4) +
5)
Home work page ( 603) 1- 2-3
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Cluster(B,2) The product rule
Objective :12A(8.11)
Understand that given a function f(x) = u(x) v(x) then the derivative of this product of two
functions is given by
use this result in calculating the derivative of the product of two functions; know the special case
of this result that if y = a f(x), where a is constant, then
Example(1) : Find If
Practice(1) Find If
……………………………………………………………………………………….
Example(2)
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Practice(2)
…………………………………………………………………………………………….
Practice(3)
If show that
………………………………………………………………………………………
Homework in textbook page(609) (1 -2)
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Cluster(B,3) The Quotient Rule
Objective:12A(8.12)
Understand that given a function f(x) = u(x) / v(x) then the derivative of this
quotient of two – vu′) / v2; use this result in
calculating the derivative of the quotient of two functions
The Quotient Rule
Example(with Teacher)
………………………………………………………………………………
Practice1(Pairs Activity)
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Example 2 (with Teacher)
Find the slope of the tangent to: at
………………………………………………………………………………………………………………………………………………..
Practice 2 (Pairs Activity)
Find the slope of the tangent to:
………………………………………………………………………
Practice 3 (Individual Activity):
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Reference:
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/quotientruledirectory/QuotientRule.ht
ml
P: (Home work(The product and the Quotient Rules )
………………………………………………………………………………….
………………………………………………………………………………….
……………………………………………………………………………………
………………………………………………………………
…………………………………………………………………………….
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Derivative of composite functions
Worksheet(B,1)
Objective: - 12A 8.13
1- Understand that given composite function then the
derivative of this composite functions is given by
2- Use this result in calculating the derivative of the composite of two
functions
dy dy du
Chain Rule:
dx du dx
The derivative of the composite function ( f g )( x ) f ( g ( x )) is
( f g )' ( x ) f ' ( g ( x )) g ' ( x ).
This is most easily remembered as “take the derivative of
the outside function and multiply by the derivative of the
inside function”
………………………………………………………………
Example(1)
If and find
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Practice1(Pairs Activity)
find for each
……………………………………………………
Example(2) Find
For any functions if
Practice2(Pairs Activity)
Find
Practice3(Pairs Activity)
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Lesson Ending
1) Find
.
2)Find
1)
………………………………………………………………………..
2)
……………………………………………………………………..
Homework in textbook page 607 (1-2-3)
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Differentiation of Implicit Functions
Objective: - 12-A (8.15)
to find the derivative of an implicitly defined function Vocabulary
using implicit differentiation; …………………
to find the equation of a tangent line to a curve which …………………
implicitly defines a function …………………
..
Remember That
If
replace g(x) of y we get
Lesson starter (Watch power point)
Example:- solve with teacher
Find if
1)
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2)Find the slope for at the point (2,1)
Practice (Pairs Activity):
a)
b)
c)
d)
e)
f)
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g)
Lesson Ending(Individual Activity)
1) find then the turning point of this curve
**************************************************************
2)Find the slope
at the point ( 1 , 2 )
****************************************************************
3)Find the slope
at the point (-2 , 5 )
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Reference:http://www.intmath.com/Differentiation/8_Derivative-implicit-
function.php
Cluster (B,2)The First derivative Test
Worksheet(1,7)
Objective(12A 8.3)
Understand that stationary points of any function may correspond to a local maximum or
minimum of the function, or may be a point of inflexion; understand how the derivative
changes as the point at which the derivative is calculated moves through the local
maximum or minimum, or through a Inflexion; understand that not all points of
inflexion are stationary points.
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Cluster (B,2)The First derivative Test
Worksheet(1,3)
Objective(12A 8.2)
Interpret the numerical value of the derivative at a point on the curve of the
function; know that:
• when the derivative is positive the function is increasing at the point;
• when the derivative is negative the function is decreasing at the point;
• when the derivative is zero the function is stationary at the point.
Vocabulary
…………………
…………………
…………………
…………………
…….
Example(1)
Find The interval of the increasing or decreasing function:
1)
2)
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Example(2)
Find The interval of the increasing or dedreasing function
1)
2)
Practice(Pairs Activity)
Find The interval of the increasing or decreasing function
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8.1A Find the second derivatives
Example(1) with Teacher
Find given that:
………………………………………………………………
…………………………………………………………
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Practice(Pairs Activity)
given that
……………………………………………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………………………………………………..
…………………………………………………………………
Lesson Ending
given that
1)
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2)Find x If =54 given that
Cluster(B,4)The Second derivatives test
Worksheet(1,2)
Objective(12A8.4)
Understand and use the second derivative to test whether a stationary point
is a local maximum, or a local minimum, or a point of inflexion.
……………………………
……………………………
……………………………
……………………………
…………………………..
centered at x = a. Then
x = a is a locally a minimum if f"(a) > 0,
x = a is a locally a maximum if f"(a) < 0; and
Here if f"(a) = 0, the sign of f"(x) gives no result.
Example(1)
Use the second derivative test to find the local maximum or minimum point.
1)
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Example(2)
Use the second derivative test to find the local maximum or minimum point.
1) 2)
Practice(1)Pairs Activity
Use the second derivative test to find the local maximum or minimum point
1)
..........................................................................................................
2)
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...........................................................................................................
3)
Lesson Ending(Individual Activity)
Find all Critical Numbers and then find the local maximum or minimum point
1)
2)
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Derivative of exponential functions
Objective 8.6 (12-A)
Vocabulary
Find the derivative of exponential functions from first
principles
………………
Lesson starter :- ………………
………………
1) Find for a) b) y= ………….
**************************************
2) Show the student power point to find the derivative of
exponential functions
The first principles method can be used to prove the
derivative of y= ex.
However, the limit is hard to evaluate. Given the limit, the proof is
easy.
note
Proof
Let
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= = =
In general:
for
Example ( solve with teacher )
Differentiate with respect to x:
(i) e3x
(ii) e2x + e-x
Practice (pairs activity)
Differentiate with respect to x:
1) e2x (x - 1)
2) Find the derivative of y =
3) Find the derivative of
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………………………………………………………………………………
lesson ending
text book page 673 ( 1-a ,b ) (2-e , f)
Home work
Solve exercise in text book page 674 ( 2 , 3 )
Derivative of the Logarithm Function y = ln x
Objective: - 8.8 (12-A)
Know that the derivative of the natural logarithm function
Lesson starter
Find if
If y = lnx then
If y = ln f(x) then the derivative of y is given by:
Example 1 (solve with teacher )
a) F(x) = 2 ln (3x2 − 1).
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B)
Practice: - pairs activity
Find the derivative of
a) y = ln x2
b)
c) y = x lnx
……………………………………………………………………………………………………………………………………………………
lesson ending( individual activity )
1)Find the derivative of
a)
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b) y = ln(1 − 2x)4
home work
1. Find the derivative of
a) y = ln(2x3 −x)2. B) ) y = ln(
Reference :-
http://www2.warwick.ac.uk/services/elearning/mathsfit/differentiation/dif
ferentiatingexponentialsandlogarithms/
http://www.teachnet.ie/apatton/2007/lessons.html
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Derivative of Trigonometric Functions
Objective:8.9
to derive the formulas for the derivatives of the trigonometric functions; and
to differentiate functions which involve trigonometric functions.
Original Rule Generalized Rule (Chain Rule)
d d du
sin x = cos x sin u = cos u
dx dx dx
d d du
cos x = sin x cos u = sin u
dx dx dx
d d du
2 2
tan x = sec x tan u = sec u
dx dx dx
Example 1: Find the first derivative of
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Practice : 1) Find the first derivative a)
2)Find the slope to at the point where =
2 (b) y = sin(x2)
(a) y = sin x
c) y = tan(x2 1) d) y = cos (e3x)
Differentiate.
(a) Example 2: Find the first derivative of
1) =
2)
3)
4)
5)
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4) y=
Find the first derivative of =
Home work
Find the derivatives of the following functions.
2)
3)
4)
5) 6)
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http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/trigderivdirectory/TrigDerivatives.ht
ml#PROBLEM%2013
http://people.hofstra.edu/Stefan_Waner/trig/trig3.html
http://archives.math.utk.edu/visual.calculus/2/trig.1/index.html
http://www.analyzemath.com/calculus/Differentiation/trigonometric.html
http://www.intmath.com/Differentiation/7_Derivative-powers-of-function.php (text book bage 691 – 692
Applications of derivatives
Objective :12A( 8.20)
1-Use polynomial and other functions to model arrange of phenomena , including
some relating to mechanics and motion
2- Knowing that the derivative of distance with respect to time is acceleration
If a particle p moves in straight line and its position is given by the
displacement function S(t) , t≥ 0 then
The velocity of p at time t is given by the derivative of the displacement
function
The acceleration of p at time is given by the derivative of the velocity
S(0) , v(0) and a(0) give us the position , velocity and acceleration of the particle at
time t=0 , and these are called the initial conditions
Example : 1) Let s(t) = 16t2 - 128t + 8. Answer each question.
a) Find v(t)
b) Find the velocity at t = 3
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c) Find a(t)
d) Find the acceleration at t = 1
2) A particle move sin straight line with position relative to some
origin O given by S(t) = where t is the time in
seconds (t≥ 0 )
1) Find expressions for the particle's velocity and acceleration
2) Find the initial conditions and hence describe the motion at
this instant
………………………………………………………………………
…
3-a) Find the velocity function and the acceleration function for the
function s(t) = 2t3 + 5t – 7
b) Find the velocity and acceleration at t = 2 for the above function
………………………………………………………………………
…
4) If a ball is thrown vertically upward with an initial velocity of 128
ft/sec, the ball's height after t seconds is s(t) = 128t - 16t2
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a) What is the velocity function?
b) What is the velocity when t = 4
c) At what time is the velocity 48 ft/sec?
d) When is the velocity zero?
.
.
e) What is the acceleration function?
f) What is the acceleration at t = 3
………………………………………………………………………
…
Lesson Ending (Individual Activity)
5) A ball is hit straight upward with an initial velocity of 256 feet per
second. the ball's height after time t seconds is h(t) = 256t - 16t2
a) What is the velocity function?
b) What is the velocity at t = 6, t = 8, t = 10?
c) What is the acceleration function?
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http://home.windstream.net/okrebs/page205.html
Optimization problems
Objective: 18.7
Vocabulary Vocabulary
Use the derivative to explore arrange of optimization …………………
…………………
problems in which function is maximized or minimized
…………………
…………………
…………………
Problems 1-
……
Find two nonnegative numbers whose sum is 9 and so that the product of
one number and the square of the other number is a maximum
…………………………………………………………………………………
Problem2
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Problem3
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**http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindire
ctory/MaxMin.html#PROBLEM%201
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