1 AP Calculus
Derivatives
20151103
www.njctl.org
2 Table of Contents
Rate of Change Slope of a Curve (Instantaneous ROC) Derivative Rules: Power, Constant, Sum/Difference Higher Order Derivatives Derivatives of Trig Functions Derivative Rules: Product & Quotient Calculating Derivatives Using Tables Equations of Tangent & Normal Lines Derivatives of Logs & e Chain Rule Derivatives of Inverse Functions Continuity vs. Differentiability Derivatives of Piecewise & Abs. Value Functions Implicit Differentiation
3 Why are Derivatives Important?
First, let's discuss the importance of Derivatives: Why do we need them? The idea of this slide is to remind students about how much they have learned about slopes in other math classes. Their exposure in previous a) What is the slope of the following function? math courses has likely been limited to finding slopes of lines only. So when b) What is the slope of the line prompted to find the slope of y=x2 this graphed at right? should lead to good discussion among
Teacher Notes students. Some students may argue that "there is no slope" others may realize "the slope is different at each point". c) Now, what about the slope of ???
4 Derivatives Exploration Lead students through an exploration by having them graph y=x2 (or any curve of their choice) and have them zoom in slowly by changing their window settings Exploration into the idea of being locally linear... little by little. You want the students to see that eventually their curve starts to resemble a line. The realization should be that this foreign concept of Derivatives will Click here to go to the lab titled "Derivatives allow them to be able to find the slopes of Exploration: y = x2" curves in particular places, due to the fact Teacher Notes that they are locally linear. URL for Lab: http://njctl.org/courses/math/ apcalculusab/derivatives/xsquared explorationlab/
5 Rate of Change
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6 Road Trip!
Consider the following scenario:
You and your friends take a road trip and leave When students answer 60 mph, ask the at 1:00pm, drive 240 miles, and arrive at 5:00pm. following: How fast were you driving? • How did you arrive at the answer? • Are you driving 60mph the entire time? • What does 60mph represent? (they should come up with the words average velocity) *they may say speed, which is valid at this point. Teacher Notes • How would you calculate how fast you were going at 2:37pm?
7 Position vs. Time
Now, consider the following position vs. time graph: Students can often grasp the concept of derivatives when you relate it to something they are familiar with, such as velocity.
Discuss with students: • What does the orange line represent? (average velocity over the entire interval)
Teacher Notes • What does each green segment represent? (instantaneous velocity at t1 and t2) position
t1 time t2 t3 t0
8 Recap
We will discuss more about average and instantaneous velocity in the next unit, but hopefully it allowed you to see the difference in calculating slopes at a specific point, rather than over a period of time.
9 SECANT vs. TANGENT
A secant line connects 2 b points on a curve. The slope y2 of this line is also known as the Average Rate of Change. a y1 A tangent line touches one point on a curve and is x1 x2 known as the Instantaneous Rate of Change.
10 Slope of a Secant Line
How would you calculate the slope of the secant line? Answer y2 b
a y1
x1 x2
11 Slope of a Secant Line Allow students to discuss what they think, What happens to the slope of the secant line as the point b eventually listening for the conclusion that the moves closer to the point a? secant line resembles the tangent line as those points get closer together.
b Encourage them to observe the fact that the y2 change in x, Δx, gets smaller (approaching 0) as the point b approaches a. a y1 In reference to the second question, students Teacher Notes should note that when b=a, using the x1 x2 traditional slope formula would result in .
What is the problem with the traditional slope formula when b=a?
12 Average Rate of Change
It's often useful to find the slope of a secant line, also known as the average rate of change, when using 2 distinct points.
Example: Find the average rate of change from x=2 to
x=4 if Answer
13 1 What is the average rate of change of the function on the interval from to ?
A 3 B 2
C 1 Answer D 1 E 3
14 2 What is the average rate of change of the function on the interval ?
A 14 B 0
C 56 Answer D 56 E 14
15 3 What is the average rate of change of the function on the interval ?
A B 0 C D E Answer
16 4 What is the average rate of change of the function on the interval ?
A
B 1 Answer
C 2
D
E 0
17 5 The wind chill is the temperature, in degrees Fahrenheit, a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v, in miles per hour. If the air temperature is 32oF, then the wind chill is given by and is valid for 5≤v≤60. (from the 2007 AP Exam) Find the average rate of change of W over the interval 5≤v≤60. CALCULATOR ALLOWED Answer Teacher Notes &
18 Slope of a Curve (Instantaneous Rate of Change)
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19 Recall: The Difference Quotient
This would be a good warm up, and reminder of the Recall from the previous unit, we used limits to calculate the difference quotient. instantaneous rate of change using the Difference Quotient. Feel free to work out this problem again, with students. Or simply slide the answer tab out for students to For example, given , we found an expression to be reminded of their previous work. represent the slope at any given point. & Answer Teacher Notes
20 Derivatives
The derivative of a function is a formula for the slope of the tangent line to that function at any point x. The process of taking derivatives is called differentiation. We now define the derivative of a function f (x) as
The derivative gives the instantaneous rate of change. In terms of a graph, the derivative gives the slope of the tangent line.
21 Derivatives Recall the Limits unit, when we discussed alternative representations for the difference quotient as well:
will result in an expression will result in an expression
*where a is constant
will result in a number
22 Notation
You may see many different notations for the derivative of a function. Although they look different and are read differently, they still refer to the same concept.
Notation How it's read
"f prime of x" "y prime"
"derivative of y with respect to x"
"derivative with respect to x of f(x)"
23 Formal Definition of a Derivative
In 1629, mathematician Fermat, was the one to discover that you could calculate the derivative of a function, or the slope a tangent line using the formula:
24 Example Using Fermat's notion of derivatives, we can either find an expression that represents the slope of a curve at any point, x, or if given an x value, we can substitute to find the slope at that instant. Emphasize to students that the expression they Example: a) Find the slope at any point, x, of the function solve for in part a represents the slope of the function b) Use that expression to find the slope of the curve at ANYWHERE and doesn't change.
In part b, depending on what value is substituted for x is what results in a different slope at various points.
a) & Answer Teacher Notes
25 6 Which expression represents if ?
A B C Answer D E
26 7 What is the slope of at x=1?
If you notice some students taking
A much longer than others, it may be that they didn't realize the function B was the same from the last slide. With a simple substitution into the correct derivative, they will have C Answer their slope. D
E
27 8 Find if
A
B Answer C D E
28 9 Find if
A If you notice some students taking much longer than others, it may be B that they didn't realize the function was the same from the last slide. With a simple substitution into the
C correct derivative, they will have
Answer their slope. D
E
29 Derivatives
As you may have noticed, derivatives have an important role in mathematics as they allow us to consider what the slope, or rate of change, is of functions other than lines. In the next unit, you will begin to apply the use of derivatives to real world scenarios, understanding how they are even more useful with things such as velocity, acceleration, and optimization, just to name a few.
30 Derivative Rules: Power, Constant & Sum/Difference Return to Table of Contents
31 Alternate Methods Depending on the function, calculating derivatives using Fermat's method with limits can be extremely time consuming. Can you imagine calculating the derivative of using that method?
Or what about ?
Fortunately, there are some "shortcuts" which make taking derivatives much easier!
The AP Exam will still test your knowledge of calculating derivatives using the formal definition (limits), so your energy was not wasted!
32 Exploration: Power Rule
This exploration is meant for students to Let's look back at a few of the derivatives you have calculated already. discover the Power Rule by recognizing patterns. As you show students these We found that: derivatives, allow students to first process quietly, and then encourage discussion in pairs or small groups. Finally, as a class • The derivative of is discuss what conclusions they have made. • The derivative of is There are usually some students who have already heard of the Power Rule, or have • The derivative of is Teacher Notes older siblings/friends that tell them, this is why the quiet reflection time is important for each student to have the opportunity to think.
What observations can you make? Do you notice any shortcuts for finding these derivatives?
33 The Power Rule
e.g.
*where c is a constant e.g.
34 The Constant Rule
All of these functions have the same Lead students in a discussion derivative. Their derivative is 0. about what the graphs of each of those functions look like. Hopefully, they will conclude that they are all equations of horizontal lines. Why do you think this is? Therefore, no matter where you are on the graph, the slope of any tangent line will be zero. Hence, the Think of the meaning of a derivative, Teacher Notes derivative is zero at any point, and how it applies to the graph of regardless of the xvalue. each of these functions.
where c is a constant
35 The Sum & Difference Rule
e.g. e.g.
36 Practice Take the derivatives of the following. Answer
37 Extra Steps Sometimes, it takes a little bit of manipulating of the function before Distribute then differentiate. applying the Power Rule. Here are 4 scenarios which require an extra step prior to differentiating: Teacher Notes
38 10 What is the derivative of ?
A B E C Answer D E
39 11
A B C Answer C
D
E
40 12 What is the derivative of 15?
A x D B 1 Answer C 14 D 0 E 15
41 13 Find if
A B D Answer C D E
42 14 Find y' if
A C C Answer
B D
43 15 Which expression represents the slope at any point on the curve ? Distribute!HINT
A B B Answer C D E
44 Derivatives at a Point If asked to find the derivative at a specific point, a question may ask... Find
Calculate What is the derivative at ?
Simply find the derivative first, and then substitute the given value for x.
Think... What would happen if you
substituted the xvalue first and Teacher Notes then tried to take the derivative?
45 16 What is the derivative of at ?
A 5 B 0 C 15 C D 5 Answer E 15
46 17 Find
A 36 B 144 A C 12 Answer D 72 E 24
47 18 What is the slope of the tangent line at if ?
A 6.5 B 6 C 0 E Answer D 3.5 E 4
48 19 Find y'(16) if
A C B E Answer
B D
49 Higher Order Derivatives
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50 Higher Order Derivatives
You may be wondering.... Can you find the derivative of a derivative!!??
The answer is... YES!
Finding the derivative of a derivative is called the 2nd derivative. Teacher Notes Furthermore, taking another derivative would be called the 3rd derivative. So on and so forth.
51 Notation
The notation for higher order derivatives is:
2nd derivative:
3rd derivative: Teacher Notes
4th derivative:
nth derivative:
52 Applications of Higher Order Derivatives
Finding 2nd , 3rd, and higher order derivatives have many practical uses in the real world. In the next unit, you will learn how these derivatives relate to an object's position, velocity, and acceleration. In addition, the 5th derivative is helpful in DNA analysis and population modeling.
53 Practice A good discussion question for students is to ask: Find the indicated derivative. How many derivatives must you take depending on the power of x until the derivative reaches 0? & Answer Teacher Notes
54 20 Find the 3rd derivative of
A B C Answer D E
55 21 Find if
A B C Answer D E
56 22 Find if
A
B Answer C
D
E
57 23 Find
A
B Answer C
D
E
58 24 Find
A
B
C Answer
D
E
59 25 Find
A B C D Answer E
60 Derivatives of Trig Functions The reason for placing trig derivatives prior to product & quotient rule is to allow for more of a variety of problems during these subsequent sections. Teacher Notes Return to Table of Contents
61 Derivatives of Trig Functions
So far, we have talked about taking derivatives of polynomials, however what about other functions that exist in mathematics?
Next, we will explore derivatives of trigonometric functions!
For example, if asked to take the derivative of , our previous rules would not apply.
62 Derivatives of Trig Functions Teacher Notes
63 Proof Let's take a moment to prove one of these derivatives... Answer Teacher Notes &
64 Derivatives of Inverse Trig Functions These derivatives can be very overwhelming for students. Again, encourage flashcards to aid students with memorization.
While these derivatives will be tested, they are not nearly as critical as the regular trig derivatives.
The inverse trig derivatives typically show up on the multiple choice portion of the exam. Teacher Notes
It is helpful to point out the relationship between positive and negative derivatives to aid them in memorization.
65 26 What is the derivative of ?
A B C
D Answer E F
66 27 What is the derivative of ?
A B C Answer D E F
67 28 What is the derivative of ?
A B C Answer D E F
68 29 What is the derivative of ?
A B
C Answer D E F
69 30 What is the derivative of ?
A B
C Answer D E F
70 31 What is the derivative of ?
A B C D Answer E F
71 32 What is the derivative of ?
A B C
D Answer E F
72 33 Find
It may be necessary to review trig calculations with your students at this time. There are many methods of teaching this, A 1 D including the unit circle, special right triangles, or an angle table. Whichever method students use, they must be efficient and accurate. They cannot rely on a B E 1 Answer calculator.
C F
73 34 Find
A D
B E Answer
C F
74 35 Find
A D
B E Answer
C F
75 36 Find
A D
B E Answer
C F
76 Derivative Rules: Product & Quotient
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77 Need for the Product Rule
Now... imagine trying to find the derivative of:
Using previous methods of multiplication/distribution, this would be extremely tedious and time consuming!
78 The Product Rule
Fortunately, an alternative method was discovered by the famous calculus mathematician, Gottfried Leibniz, known as the product rule. Many teachers find it helpful to have their students say the product rule aloud and repeat it multiple times, as it is practiced... for example:
"2nd times the derivative of the 1st plus 1st times the derivative of the 2nd." OR Let's take a look at how the product rule works... "1st times the derivative of the 2nd plus 2nd times the derivative of the 1st." Teacher Notes
Note: the order in which you add doesn't matter.
79 The Product Rule
Notice: You have previously calculated these derivatives by using the distributive property.
The problems above can also be viewed as the product of 2 functions. We can then apply the product rule.
80 The Product Rule
using the distributive property using the product rule It may be helpful for students to practice repeating the product rule. There are various ways to practice including:
Teacher Notes "f prime g plus f g prime" or "2nd times the derivative of the 1st plus 1st times the derivative of the 2nd." Notice: Both methods yield the same derivative!
81 Distribution vs. The Product Rule Why use the Product Rule if distribution works just fine? The complexity of the function will help you determine whether or not to distribute and use the power rule, versus using the product rule.
For example, with the previous function distributing is slightly faster than using the product rule; however, given the function, it may be easier to use the product rule than to try and distribute.
82 Practice
Finding the following derivatives using the product rule. Answer
83 37
A B
C Answer D
84 38
A B
C Answer D
85 39
A
B Answer
C
D
86 40 Find
A
B
C Answer
D
87 41
A
B Answer C
D
88 42
A B
C Answer D
89 43 Find Students may not realize they are supposed to be finding the 2nd A derivative. Therefore, using the trig derivative, followed by product B rule. Answer C D
90 44
True FALSE False Students can share/discuss the functions they use to disprove this statement. Teacher Notes
91 What About Rational Functions? So far, we have discussed how to take the derivatives of polynomials using the Power Rule, Sum and Difference Rule, and Constant Rule. We have also discussed how to differentiate trigonometric functions, as well as functions which are comprised as the product of two functions using the Product Rule.
Next, we will discuss how to approach derivatives of rational functions.
92 The Quotient Rule
Many teachers find it helpful to have their students say the quotient rule aloud and repeat it multiple times, as Notice, the problems above can be viewed as the quotient of 2 it is practiced... for example: functions. We can then apply the quotient rule. "Bottom times the derivative of the top minus top times the derivative of
Teacher Notes the bottom, over the bottom squared."
93 Example
Given: Find
f(x), or "top"
g(x), or "bottom" Answer
94 Example This is a good example to explain to students, again, that quotient rule is useful and effective, but in some scenarios it may not be the only way to differentiate (like they saw with distributing Given: Find or using product rule). On this example, they could use the quotient rule, or simplify the rational expression and then find the derivative using power rule.
Perhaps have them try both ways and compare. Clearly, in this particular example, simplifying is quicker than the quotient rule. Answer
95 Proof Now that you have seen the Quotient Rule in action, we can revisit one of the trig derivatives and walk through the proof. Answer
96 45 Differentiate
A Answer B
C
D
97 46 Find
A
B Answer
C
D
98 47 Find
A Notice: Numerator can be factored, simplifying the equation to y=x. B Answer
C
D
99 48 Find
A
B Answer
C
D
100 49 Differentiate
A
B Answer
C
D
101 50 Find the derivative of This example may confuse students, due to the fact that it contains a fraction. They may attempt to use quotient rule and get stuck.
A The easiest method would be to rewrite the expression and use the power rule to differentiate.
*Note: Quotient rule is still an acceptable method, B they could first combine the expression into a Answer rational expression using a common denominator,
Teacher Notes & or apply quotient rule to the second term only. C
D
102 Calculating Derivatives Using Tables
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103 Derivatives Using Tables
On the AP Exam, in addition to calculating derivatives on your own, you must also be able to use tabular data to find derivatives. These problems are not incredibly difficult, but can be distracting due to extraneous information.
104 Example Let's take a look at an example:
Let Calculate The functions f and g are differentiable for all real numbers. Using product rule... The table above gives values of the functions and their first
derivatives at selected values of x. Answer
105 Example
Let
Calculate Using power rule & quotient rule... The functions f and g are differentiable for all real numbers. The table above gives values of the functions and their first derivatives at selected values of x. Answer
106 Derivatives Using Tables Next is another type of question you may encounter on the AP Exam involving tabular data and derivatives.
Use the table at right to estimate Answer
107 The functions f and g are 51 differentiable for all real numbers. The table at right gives values of the functions and their first derivatives at selected values of x. Let Calculate A 2 C Answer B 10 C 1 D 2 E 30
108 The functions f and g are 52 differentiable for all real numbers. The table at right gives values of the functions and their first derivatives at selected values of x. Let Calculate A 93 A B 61 Answer C 75 D 95 E 0
109 53 Let Calculate
A 1.5 B 39 C 32 B Answer D 75 E 0
110 54 Use the table at right to estimate
A 2.4 B 3.1 C 32 E Answer D 0 E 4
111 55 Use the table at right to estimate
A 3.05 B 3.6 C 0 D Answer D 0.278 E 0.5
112 Equations of Tangent & Normal Lines
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113 Writing Equations of Lines
Recall from Algebra, that in order to write an equation of a line you either need 2 points, or a slope and a point. If we are asked to find the equation of a tangent line to a curve, our line will touch the curve at a particular point, therefore we will need a slope at that specific point.
Now that we are familiar with calculating derivatives (slopes) we can use our techniques to write these equations of tangent lines.
114 Equations of Tangent Lines Have a discussion with students First let's consider some basic linear functions... about the fact that a tangent line to any line is the SAME line!
Let them think about it on their own first, then discuss.
It may help to draw a sketch of a line
Teacher Notes versus a curve and show how there is only one tangent line to a line, but infinite tangent lines to a curve.
If asked to write the equation of the tangent line to each of these functions what do you notice?
115 Example Let's try an example: Write an equation for the tangent line to at x=2. Answer Teacher Notes &
116 Example Write an equation for the tangent line to at . Answer
117 Normal Lines In addition to finding equations of tangent lines, we also need to find equations of normal lines. Normal lines are defined as the lines which are perpendicular to the tangent line, at the same given point. y = x2
tangent line normal line at x = 1 at x = 1 Teacher Notes
How do you suppose we would calculate the slope of a normal line?
118 Example Let's try an example: Write an equation for the normal line to at x=2. Answer
119 Example
Example: Write an equation for the normal line to at . Answer
120 Example Example: Write an equation for the normal line to at .
In order to write an equation we need the point and slope. We can find the corresponding yvalue for our point by substituting our given xvalue into the equation. Answer Teacher Notes & Now, to find the slope, we can calculate the derivative, and substitute our xvalue.
121 56 Which of the following is the equation of the tangent line to at ?
A D
B E Answer
C F
122 57 Which of the following is the equation of the tangent line to at ?
A D
B E Answer
C F
123 58 Which of the following is the equation of the normal line to at ?
A D Answer B E
C F
124 59 Which of the following is the equation of the tangent line to at ?
A D Answer B E
C F
125 60 Which of the following is the equation of the normal line to at ?
A D
B E Answer
C F
126 61 Which of the following is the equation of the tangent line to at ?
A D
B E Answer
C F
127 Derivatives of Logs & e
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128 Exponential and Logarithmic Functions
The next set of functions we will look at are exponential and logarithmic functions, which have their own set of rules for differentiation.
129 Exponential Functions
First, let's consider the exponential function
While it appears that Power Rule may be an option, unfortunately it will not apply to this function, because the exponent is not a fixed number, and the base is not the variable.
130 Derivatives of Exponential Functions
By considering a particular value of a, , we are able to see the proof for the derivative of exponential functions.
Note: This proof is based on the fact that e, in the realm of calculus, is the unique number for which Teacher Notes
131 Derivatives of Exponential Functions
Technically, y=0 is also it's own derivative as well, but does not depend on another variable, so generally people say that is the only one.
Consider asking students if Teacher Notes cool! is the only nontrivial function whose they can think of y=0 before derivative is the same as the function! telling them.
132 Derivatives of Exponential Functions
At this point, we lack knowledge for the proof of , however, we can prove this derivative when we get to the section on Chain Rule.
133 Derivatives of Logarithmic Functions
Remind students that lnx follows the same rule that logax does, lnx is just a special case because lne=1. Teacher Notes
134 62
A D C B E Answer
C F
135 63
A D C B E Answer
C F
136 64
A D F B E Answer
C F
137 65 Find the derivative of
A D A B E Answer
C F
138 66
A D E B E Answer
C F
139 Chain Rule
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140 What About the Following? a) What type of function is this?
Students may reply, "a composite function" or "a Consider the following function: function within a function"
b) Would Power Rule or Product Rule be appropriate in finding the derivative of this function?
Let them discuss this notion and realize that the functions cosx and 2x+3 are not being multiplied, a) What type of function is this? Teacher Notes therefore Product Rule won't work. Also, they have only used Power Rule when a single variable is raised to a power, not another function. b) Would Power Rule or Product Rule be appropriate in finding the derivative of this function?
141 Chain Rule
We must apply a new rule when differentiating composite functions known as the Chain Rule.
If Teacher Notes
Then
142 Example Before starting, ask students to identify which Let's try the Chain Rule on a basic example. function is the "outer" function and which is the "inner" function. Find Hopefully, they will recognize that ( )5 is the outer function and x3+3 is the inner function. Answer Teacher Notes &
143 Applying Chain Rule Now let's take a look back at the original question and apply Chain Rule... take note of how many "layers" exist in this equation. Find Answer Teacher Notes &
144 Example
Given: Find Answer Teacher Notes &
145 67 Find
A
B
Answer B C
D
E
146 68 Find
A
B
C Answer
D
E
147 69 Find
A
B
Answer A
C
D
E
148 70 Find
A
B
C Answer
D
E
149 71 Find
A
B
Answer D C
D
E
150 72 Find
A
B
Answer A
C
D
151 73 Find
A
B
Answer B C
D
E
152 Derivatives of Inverse Functions
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153 Derivatives of Inverse Functions
We have already covered derivatives of inverse trig functions, but it is also necessary to calculate the derivatives of other inverse functions.
154 Inverse Functions Recall the definition of an inverse function..
We say that and are invertible if: Teacher Notes
Also, if and are invertible then:
155 Derivatives of Inverse Functions
Taking the derivative of inverse functions requires use of the If you are able to calculate the chain rule, as we can see below. inverse function, and then differentiate, that is just fine! The rule is necessary on functions where the inverse is impossible or Fact about inverse functions... very difficult to find. Teacher Notes Applying chain rule to derive...
Thus,
156 Be Careful with Notation Note: As you work through the following problems it is extremely important to pay close attention to notation as you work.
A common error is to forget and/or mix up the inverse sign and derivative sign.
Note the differences:
157 Example
Find the derivative of the inverse of Notes Answer & Teacher In addition, you may wish to show that differentiating the inverse function directly will yield the same answer as using the rule for inverse derivatives.
158 Example
If and find Answer Teacher Notes &
159 Example
Suppose and find
*This time, instead of the value for f' being given to us, we must calculate it, using the derivative function given Answer above.
160 74 Find the derivative of the inverse of
A
B Answer C
C
D
161 75 Find the derivative of the inverse of
A
B
Answer A
C
D
162 76 If and Find
A
B
Answer D
C
D
163 77 If and Find
A
B
Answer B
C
D
164 Continuity vs. Differentiability
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165 Definition of Continuity
In the previous Limits unit, we discussed what must be true for a function to be continuous:
Definition of Continuity 1) f(a) exists
2) exists
3)
Differentiability requires the same criterion, as well as a few others.
166 Differentiable Functions
In order for a function to be considered differentiable, it must contain: • No discontinuities • No vertical tangent lines • No Corners "sharp points" • No Cusps
167 Differentiability Implies Continuity If a function is differentiable, it is also continuous. However, the converse is not true.
Just because a function is continuous does not mean it is differentiable. Because there is not one single tangent line that can "balance" at x=0, it is not differentiable at this point. What does this mean??? Consider the function: Teacher Notes Another explanation: Imagine zooming in on the function, like we have previously done. The function must resemble a line ("locally linear") to Notice: If we were asked to find the be differentiable. derivative (slope) at x=0, there is a sharp corner. The slope quickly changes from 1 to 1 as you move closer to x=0. Therefore, this function is not differentiable at x=0.
168 A FUNCTION FAILS TO BE DIFFERENTIABLE IF... CORNER CUSP
DISCONTINUITY VERTICAL TANGENT
169 Types of Discontinuities:
removable removable jump infinite
essential
170 + no sharp points or vertical tangents...
171 78 Choose all values of x where f(x) is not differentiable.
A B C D Answer E F G
172 79 Choose all values of x where f(x) is not differentiable.
A B C D Answer E F G
173 80 Choose all values of x where f(x) is not differentiable.
A B C D Answer E F G
174 81 If f(x) is continuous on a given interval, it is also differentiable.
True False Answer
175 Derivatives of Piecewise & Abs. Value Functions
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176 Derivatives of Piecewise & Absolute Value Functions
Now that we've discussed the criterion for a function to be differentiable, we can look at how to find the derivatives of piecewise and absolute value functions, which often contain sharp corners, and discontinuities.
177 Derivatives of Piecewise & Absolute Value Functions When calculating derivatives of piecewise functions, the same rules apply for each piece; however, you must also consider the point in which the function switches from one portion to another.
For a piecewise function to be differentiable EVERYWHERE it must be: • Continuous at all points (equal limits from left and right) • Have equal slopes from left and right
178 Derivatives of Absolute Value Functions
Let's first consider the absolute value function...
Visually, we can see that this function is not differentiable at
x=0 due to the sharp corner. Teacher Notes Even if it is not differentiable at x=0, we can still find the derivative for the other portions of the graph.
notice: we do not include 0
179 Derivatives of Absolute Value Functions
It is apparent that every absolute value function will have a sharp point (thus, not being differentiable at that point). But again, we can still find the derivative, discluding the sharp point.
Example: Find the derivative of Note: We must first write our function as a piecewise. Teacher Notes
180 Derivatives of Piecewise Functions
Let's take a look at some additional piecewise functions.
Find Answer
Note: 2 is not included in the derivative because although it is continuous, the slopes do not match from the left and right.
181 Example
Is the following piecewise differentiable at ? Answer
Although the slopes are equivalent at the function is not continuous, there, thus not differentiable at .
182 Creating Continuity The method for solving this question is to create a system of equations, What values of k and m will make the function differentiable over because there are 2 unknown variables, k and m. the interval (0, 5)? First, we must ensure the function is continuous at x=3 (equal limits from right and left). To accomplish this, we can plug in 3 to each function and set them equal to each other to create our first equation for the system.
Secondly, we know the slopes must be equal, so taking the derivatives of each function, we can then plug 3 in again and set them equal to each other for our second equation. Teacher Notes & Answer
183 82
A f(x) is continuous at x=2 A
B f(x) is differentiable at x=2 Answer D C f(x) is not continuous at x=2 D f(x) is not differentiable at x=2
184 83
A f(x) is continuous at x=0 A B f(x) is differentiable at x=0 Answer B C f(x) is not continuous at x=0 D f(x) is not differentiable at x=0
185 84
A f(x) is continuous at x=1 A B f(x) is differentiable at x=1 Answer D C f(x) is not continuous at x=1 D f(x) is not differentiable at x=1
186 85
A f(x) is continuous at x=0 C B f(x) is differentiable at x=0 Answer D C f(x) is not continuous at x=0 D f(x) is not differentiable at x=0
187 86
A f(x) is continuous at x=5 A f(x) is differentiable at x=5 B Answer D C f(x) is not continuous at x=5 D f(x) is not differentiable at x=5
188 87 Which of the following is the correct derivative for the function ?
A D F Answer
B E
C F
189 88 Choose the correct values for k & m in order for f(x) to be differentiable on the interval (4,9).
D Answer A E F B F C G D H
190 Implicit Differentiation
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191 Explicit vs. Implicit Functions
Allow students to discuss the similarities and differences between the 2 functions. Compare/Contrast the following 2 functions: Often, students realize that one has the y variable isolated and the other does not, which is one basic understanding that helps students understand implicit Teacher Notes vs. functions.
192 Implicit Differentiation
So far, all of the derivatives we have taken have been with respect to the variable, x.
e.g.
"derivative of y with respect to x"
193 Implicit Differentiation
Mathematically, we are actually able to differentiate with respect to any variable, it just requires special attention and notation.
Example: Find y'(t).
"derivative of y with respect to t"
194 Implicit Differentiation
What happens if our function is in terms of x, but we are asked Students often get confused with the inclusion to find the derivative with respect to a different variable, t? of a dx/dt or any similar notation. A helpful explanation is to tell them that technically they have been doing this all along... Example: Find y'(t).
"derivative of y Teacher Notes with respect to t"
But, why are these needed? Due to the fact that we are differentiating with respect to a variable other than what is there, we must include a dx/dt.
195 Implicit Differentiation
When a function involves the variables y and x, and y is not isolated on one side of the equation, we must take additional steps in finding the derivative. Sometimes students tend to stop once they've differentiated across the entire equation. Remind them that the question is asking to find so Example: Find they must rearrange their equation until is isolated. Answer Teacher Notes &
196 Example
Given: Find 1. Differentiate both sides. Remember! You aren't finished until is isolated. 2. Collect all dy/dx to one side. 1. Differentiate both sides
2. Collect all dy/dx to one side Answer 3. Factor out dy/dx. 3. Factor out dy/dx 4. Solve for dy/dx. 4. Solve for dy/dx.
197 Practice Find Find
Find Find Answer
CHALLENGE! CHALLENGE! Find Find
198 Derivatives with Respect to t
Why am I being asked to find the derivative with respect to the variable, t, so often?
Often in Calculus, we are interested in seeing how things change with respect to TIME, hence taking the derivative (which shows us rate of change) with respect to the variable t.
This will become increasingly more apparent in the next unit when we study Related Rates.
199 89 Find
A
B C Answer
C
D
200 90 Find
A
B Answer
C
D
201 91 Find
A A
B Answer
C
D
202 92 Find
A C
B Answer
C
D
203 93 Find
A B
B Answer
C
D
204 Implicit Differentiation at a Point
Now that we have practiced using implicit differentiation, we can
extend the process to find the derivatives at specific points.
205 Example Find the slope of the tangent line to the circle given by: at the point Answer Plug in values for x and y:
206 Implicit vs. Explicit Differentiation
For this example, note the benefits of implicit differentiation vs. explicit differentiation. As an optional exercise, you may rework the example for the explicit function: which is the upper half of the graph.
Then, remember this must be done again if points on the lower half are also desired, given by:
207 Example, Continued As a further step in this example, we can now find the equation of the tangent line at the point, (3,4). Answer
208 Example
Example: Find the slope of the graph of at the point Answer Plug in point values:
209 94 Find the slope of the tangent line at the point (2, 4) for the equation: Answer A
B
C
D
210 95 Find the slope of the tangent line at x=3 for the equation: Note: Students may need prompting to substitute the xvalue into original function to find the corresponding yvalue to use in derivative.
A Answer
B
C
D
211 96 Find the slope of the tangent line at the point for the equation: Answer A
B
C
D
212 97 Find the equation of tangent line through point (1, 1) for the equation:
A Answer
B
C
D
213