Slide 2 / 165 Pre-Calculus -...
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Slide 1 / 165 Pre-Calculus Intro to Derivatives www.njctl.org 2015-03-24 Slide 2 / 165 Table of Contents Derivatives Using a Calculator to find a Derivative Derivative Rules Velocity, Speed, and Other Rates of Change Limits Continuity Difference Quotient click on the topic to go to that section Slide 3 / 165
Transcript of Slide 2 / 165 Pre-Calculus -...
Slide 1 / 165
Pre-Calculus
Intro to Derivatives
www.njctl.org
2015-03-24
Slide 2 / 165
Table of Contents
DerivativesUsing a Calculator to find a DerivativeDerivative RulesVelocity, Speed, and Other Rates of Change
LimitsContinuityDifference Quotient
click on the topic to go to that section
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Limits
Return to Table of Contents
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A rubber ball is dropped and bounces back up to half the height it was on the previous bounce. Given this scenario, does the ball ever come to rest? What is the approximate height as time goes to ∞?
Limits
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A limit allows a function, f(x), to have a value as the function approaches x.
In the previous example, the balls height was zero as time went to ∞, even if the object never stopped moving.
Limits
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Consider the functions as x approaches 3.
Limits
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As the function f(x) approaches x=3 what is the value f(x)?
As the function g(x) approaches x=3 what is the value g(x)?
A limit describes what happens to the function as it gets closer and closer to a certain value of x. The function doesn't have to have a value at that x for the limit to exist.
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Limits
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We can find a limit by substituting into the function the value of x.
So in the previous example f(x)= x-3, f(3)=0, which the graph shows.
What about ?
Substitution doesn't work. But since a limit is defined by getting close to that value we can look at the graph and see that the function has a value approaching 3 as x approaches 3 from both the left and right. Algebraically, we can factor and reduce and we also will get a limit of 3.
Limits
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LimitsLimits describe what happens to a function as x approaches a value.
is read "The limit of f of x, as x approaches c, is L.
For the previous example of f(x)= x-3:
Limits
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Limits of a Composite Function can be found by finding the limits of the individual terms.
Limits
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Ex: a) b)
c) d)
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Ex: a)
b)
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1 Find the indicated limit, if it exists. If it doesn't exist enter 0.0 Te
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2 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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Limits
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3 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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Use graphing calculator to determine:
Using the second function, find the value of x=2.What does this demonstrate?
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Limits
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Using a graphing calculator with
Go to 2nd -> TBLSET TblSTART= -2 Auto #Tbl=.5 Auto
Now use 2nd -> TABLE What do you observe? What happens at x = 2?
Limits
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8 Find the indicated limit, if it exists. If it doesn't exist enter 0.0 Te
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9 Find the indicated limit, if it exists. If it doesn't exist enter 0.0 Te
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10 Find the indicated limit, if it exists. If it doesn't exist enter 0.0 Te
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Continuity
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a b c d e f g
At what points do you think the graph is continuous?At what points do you think the graph is discontinuous?
What should the definition of continuous be?
Continuity
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AP Calculus Definition of Continuous
1) f(a) exis ts
2) exis ts
3)
This definition shows continuity at a point on the interior of a function.
For a function to be continuous, every point in its domain must be continuous.
Continuity
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Continuity at an EndpointReplace step 3 in the previous definition with:
Left Endpoint:
Right Endpoint
Continuity
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Infinite
Jump
Removable
Essential
Types of DiscontinuityContinuity
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Removable discontinuities come from rational functions.
A piecewise function can be used to fill the "hole".
What should k be so that g(x) is continuous?
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Continuity
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11 What value(s) would remove the discontinuity(s) of the given function?
A -3B -2C -1D -1/2E 0
F 1/2
G 1
H 2
I 3
J DNE
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Continuity
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12 What value(s) would remove the discontinuity(s) of the given function?
A -3B -2C -1D -1/2E 0
F 1/2
G 1
H 2
I 3
J DNE
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Continuity
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13 What value(s) would remove the discontinuity(s) of the given function?
A -3B -2C -1D -1/2E 0
F 1/2
G 1
H 2
I 3
J DNE
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Continuity
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14 What value(s) would remove the discontinuity(s) of the given function?
A -3B -2C -1D -1/2E 0
F 1/2
G 1
H 2
I 3
J DNE
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Continuity
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Making a Function Continuous
, find a so that f(x) is continuous .
Both 'halves' of the function are continuous. The concern is making
Solution:
Continuity
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15 What value of k will make the function continuous? Te
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16 What value of k will make the function continuous?
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Continuity
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Characteristics of a function on closed continuous interval.1) Somewhere on the interval there will be a maximum and a minimum.i ii
iii iv
Continuity
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When discussing Maximum and Minimum, there is a difference between Absolute Maximum and Relative Maximum.
Think of dropping a ball.The starting point is absolute max.The starting point and each bounce are relative max.
Continuity
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Characteristics of a function on closed continuous interval.*2) Intermediate Value Theorem- if f(x) is a closed continuous interval on [a,b], then f(x) takes on every value between f(a) and f(b).
* IVT is a theorem that can be used as a justification on the AP exam.
a
f(a)
bf(b) This comes in handy when looking for zeros.
Continuity
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Can you use the Intermediate Value Theorem to find the zeros of function?
(Assuming there is a smooth curve running through the points)
X Y
-2 -6
-1 -2
0 3
1 5
2 -1
3 -4
4 -2
5 2
Continuity
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Can you use the Intermediate Value Theorem to find the zeros of function?
X Y
-3 4
-2 1
-1 -1
0 1
1 2
2 3
3 -2
4 2
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Continuity
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17 Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y -2 -1 0.5 2 3
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Continuity
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18 Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y 4 3 2 1 -1
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Continuity
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19 Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y -2 -1 -0.5 2 3
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Continuity
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Difference Quotient
Return to Table of Contents
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Draw a possible graph of traveling 100 miles in 2 hours.
Distance
Time
100
t
d
2
Difference Quotient
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Using the graph on the previous slide: What is the average rate of change for the trip?
Is this constant for the entire trip?
What formula could be used to find the average rate of change between 45 minutes and 1 hour?
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Difference Quotient
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The slope formula of represents the Velocity or
Average Rate of Change. This is the slope of the secant line from (t1,d1) to (t2,d2).
Suppose we were looking for Instantaneous Velocity at 45 minutes, what values of (t1,d1) and (t2,d2) should be used?
Is there a better approximation?
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Difference Quotient
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The Difference Quotient gives the instantaneous velocity, which is the slope of tangent line at a point.
A Derivative is used to find the slope of a tangent line. So Difference Quotient can be used find a derivative algebraically.
Difference Quotient
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Example of the Difference Quotient
Find the slope of the tangent line to the functionat x=3.
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Difference Quotient
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Example of the Difference QuotientFind an equation that can be used to find the slope of the tangent line at any point on the function
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23 What expression can used to find the slope of any tangent line to
A
B
C
D
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Now try finding the slope of a tangent of a graph.
y1=sin(x)Find the slope at x=# / 4
Try ZOOM IN at the point till it looks like a line in the viewing window. (To get calculator to zoom at x=# /4, use 2nd CALC 1:Value, then zoom once the cursor is on π/4.)
Difference Quotient
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The cursor is on the function at x=# /4 . We don't want to round on the AP exam until the end but who wants to copy down all those numbers?
2nd QUIT will take you to the computation screen with x and y still equal to the point on the graph. STORE x->A and y->B using the STORE key and ALPHA.
Go back to graph and "bump" the cursor over to a point close by.2nd QUIT again and this time STORE x->C and y->D.
Calculate the slope (D-B)/(C-A)= .707 (now you can round)
Difference Quotient
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The calculator does have a built in derivative key for graphs.
It is dy/dx. When the cursor is where you want it, use 2nd CALC 6: dy/dx (You may need to zoom in a few times to take the derivative where you want it).
These both stand for the derivative of a function.
Difference Quotient
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Derivatives
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The derivative gives the instantaneous rate of change.In terms of a graph, the derivative gives the slope of the tangent line.
The Difference Quotient is used to take the derivative of a function.
f '(x) is the notation for derivative. So is dy/dx.
Derivatives
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Find f '(x) given f(x)= x3 + x2 - 2.
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27 Which of the following is the derivative of
A
B
C
D
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28 Which of the following is the derivative of
A
B
C
D
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29 Which of the following is the derivative of
A
B
C
D
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30 Find f '(3) given
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31 Which of the following equations is the tangent line to at x=3?
A
B
C
D
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32 Which of the following equations is the normal line to at x=3?
A
B
C
D
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Is differentiable at x=0?
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Is differentiable at x=0?
What is slope of the tangents when x<0?What is slope of the tangents when x>0?Are they the same?
Derivatives
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33 Is the following function differentiable? If not, why not?A Differentiable
B No, Discontinuous
C No, Vertical Tangent
D No, Corner
E No, Cusp
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34 Is the following function differentiable? If not, why not?
A Differentiable
B No, Discontinuous
C No, Vertical Tangent
D No, Corner
E No, Cusp
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35 Is the following function differentiable? If not, why not?
A Differentiable
B No, Discontinuous
C No, Vertical Tangent
D No, Corner
E No, Cusp
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36 Is the following function differentiable? If not, why not?
A Differentiable
B No, Discontinuous
C No, Vertical Tangent
D No, Corner
E No, Cusp
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Derivatives
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37 Is the following function differentiable? If not, why not?
A Differentiable
B No, Discontinuous
C No, Vertical Tangent
D No, Corner
E No, Cusp
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Derivatives
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Using a Calculator to find a Derivative
Return to Table of Contents
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38 Given the table below, find the approximate value of f '(2). (Enter 0.0 if derivative does not exist.)
x 1 1.9 1.97 2 2.02 ... 3.99 4 4.01
f(x) 2.5 6.6 6.905 7 7.059 ... 8.98 9 9.2
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39 Given the table below, find the approximate value of f '(4). (Enter 0.0 if derivative does not exist.)
x 1 1.9 1.97 2 2.02 ... 3.99 4 4.01
f(x) 2.5 6.6 6.905 7 7.059 ... 8.98 9 9.2
Using a Calculator
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Find the derivative ofusing the Difference Quotient Te
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Using a Calculator
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y1
y2
y3
The blip of the third graph is tracing the second. So our derivative was correct.
You take other derivatives and just change y1 and y3.
Your calculator also has a built in function that will let you take a derivative at a point.
Using a Calculator
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To find a derivative at a point: MATH-> 8: nDeriv(
EX: y=3x2+2x-1 find f '(3).
Once you hit the nDeriv key you will need to enter the equation, the variable you're taking the derivative of (x), and the value of x that want the derivative at.Depending on the version of your operating system it will look like one of the following:
nDeriv(3x2+2x-1,x,3) 20
or
Can check this by finding the value at x=3 on graph y3 from the previous example.
Using a Calculator
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Try it.y= x2 - 1, write the equation of tangent line and the normal line at x=4.
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40 What is f '(2) givenTe
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42 What is f '(0) given
*Make sure calculator is in radians
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Using a Calculator
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The Intermediate Value Theorem applies to differentiable closed intervals. For example if f(x) is differentiable [-2,5] and f '(-2)= 3 and f '(5)= -2 then there is some point between (-2, 5) that will give us a value between (3,-2).
Using a Calculator
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43
Yes
No
f(x) is differentiable on [2,5] and f '(2)= -3 and f '(5)= 4 then there is some point between (2, 5) that has a rate of change of 0.
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Using a Calculator
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No
f(x) is differentiable on [-6,2] and f '(-6)= 8 and f '(2)= 1 then there is some point between (-6,2) that has a rate of change of 0.
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Using a Calculator
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Derivative Rules
Return to Table of Contents
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Examples: Take the derivatives of the following.Te
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Rules for Derivatives
5. Product Rule: y = f(x)g(x) then dy/dx=f(x)g '(x)+f '(x)g(x)
Using Product Rule: Using distribution and the Power Rule:
As function becomes more complex, distribution will no longer be an option.
Derivative Rules
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Example: Use the product rule to find the derivative of g(x). Check your answer by using distribution.
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Rules for Derivatives
6. Quotient Rule: then
Mnemonic for quotient rule: lo d hi minus hi d lo over the denominator square it goes! (clap-clap-clap)
Example:
Derivative Rules
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A Short Cut for Some Quotient Rule Derivatives
rewrite as
This short cut works because the rational has a denominator that is a monomial.
Derivative Rules
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Ex: Let h(3)= 1, h '(3)= 3, f(3)=5, and f '(3)=4. Find g'(3) if:
g(x)=f(x)h(x) g(x)=f(x)/h(x)
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53 Given g(4)=3, g'(4)=4, f(4)=6,and f '(4)= -2. Find h'(4) if h(x)=f(x)+g(x).
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Derivative Rules
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54 Given g(4)=3, g'(4)=4, f(4)=6,and f '(4)= -2. Find h'(4) if h(x)=3f(x).
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55 Given g(4)=3, g'(4)=4, f(4)=6,and f '(4)= -2. Find h'(4) if h(x)=f(x)g(x).
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Derivative Rules
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56 Given g(4)=3, g'(4)=4, f(4)=6,and f '(4)= -2. Find h'(4) if h(x)=f(x)/g(x).
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Derivative Rules
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57 Given g(4)=3, g'(4)=4, f(4)=6,and f '(4)= -2. Find h'(4) if h(x)=1/f(x).
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Derivative Rules
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Rules for Derivatives
7. Chain Rule: then
Example:
Derivative Rules
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A Tip to Remember
rewrite as
Derivative Rules
Ex:
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D
find
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Try it:
Ex: Find Ex: Find
Derivative Rules
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A Tip to Remember
Ex: Find
Derivative Rules
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Now that we've seen all of the Derivative Rules, we should be able to combine them all together into larger problems.
Derivative Rules
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Ex:
Derivative Rules
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Velocity, Speed, and Other Rates of Change
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Free FallObjects fall toward the Earth at the same rate. Okay, we don't live in a vacuum so not exactly the same rate, but we'll think of it that way. The force that causes the acceleration is called gravity. When gravity is the only force on a falling object, we call that free fall.
Gravity (g) = -32 ft/sec 2 = -9.8 m/sec2 = -980 cm/sec2
(This is Earth's gravity, g will change for other planets.)
Rates of Change
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Position of a Falling Object
s is distance traveledg is acceleration due to gravityt is time
Rates of Change
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Ex: An object is dropped from a 200' foot tower. 1) How far does it fall in 2 seconds?2) What is its position in 3 seconds?3) When does it hit the ground?
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Rates of Change
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70 An object is dropped from a 100m tower. How far did the object fall in 1 second?
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Rates of Change
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71 An object is dropped from a 100m tower. What is the objects position relative to the top of the tower at 2 seconds?
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Rates of Change
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72 An object is dropped from a 100m tower. What is the objects position relative to the ground at 3 seconds? Te
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Rates of Change
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73 An object is dropped from a 100m tower. When does the object hit the ground? Te
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74 An object is dropped from a 50m tower on Mars,where the gravitational force is 3.69 m/sec2. When does the object hit the ground?
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Rates of Change
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Average velocity = average rate of change = slope of the secantie. (t1, s1) and (t2, s2)
Displacement= net change in position
t
s
The displacement from t1 to t2 is 0.
Rates of Change
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Instantaneous velocity (velocity) = instantaneous rate of change= slope of the tangent
ie. the derivative of your position function
Rates of Change
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A rocket launches upward at 160 ft/sec. Its height after t seconds is s= 160t -16t 2
a) What is the rockets height at 3 seconds?b) When does the rocket hit the ground?c) What is rockets average velocity between t=7 & t=8?d) What is the rockets velocity at t=5? Does this make sense?
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Rates of Change
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Speed is the absolute value of velocity.
Velocity shows magnitude and direction.
Speed shows only magnitude.
In the last example, the rocket had a velocity of -80 ft/s between t=7 and t=8. Its speed for that same interval is 80 ft/s.
Rates of Change
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75 A toy rocket is launch at an initial velocity of 64 ft/sec. Its position after t seconds is s(t)=64t -16t2. Find when its velocity is zero.
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Rates of Change
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76 A toy rocket is launch at an initial velocity of 64 ft/sec. Its position after t seconds is s(t)=64t -16t2. What is the average velocity from t=1 to t=3?
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Rates of Change
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77 A toy rocket is launch at an initial velocity of 64 ft/sec. Its position after t seconds is s(t)=64t -16t2. Find its speed at t=4.
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Rates of Change
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Acceleration is how fast velocity is changing.
Since velocity is how fast position is changing or the derivative of position, acceleration is the derivative of velocity or the second derivative of position.
Rates of Change
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Look back at our last example, what was our acceleration? (Take the second derivative to find out.)
What is the rockets acceleration at any given time?
The rocket slows as it reaches its max and then speeds up back to Earth. Look at v(t) and a(t) on number lines.
Rates of Change
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v(t)t
a(t) ---------------------------------------------------------------------------------
t
0++++++++++++++++++++++0--------------------------------------0
When velocity and acceleration have opposite signs the object slows down. When they have the same sign the object speeds up.
Rates of Change
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78 A toy rocket is launched at an initial velocity of 64 ft/sec. Its position after t seconds is s(t)=64t -16t2. What is the rockets acceleration at t=1?
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Rates of Change
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79 A toy rocket is launch at an initial velocity of 64 ft/sec. Its position after t seconds is s(t)=64t -16t2. For how many seconds is the rocket slowing down?
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