Tag Archives: Difference Quotients

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

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Unit 2 contains topics rates of change, difference quotients, and the definition of the derivative (CED – 2019 p. 51 – 66). These topics account for about 10 – 12% of questions on the AB exam and 4 – 7% of the BC questions.

Topics 2.1 – 2.4: Introducing and Defining the Derivative 

Topic 2.1: Average and Instantaneous Rate of Change. The forward difference quotient is used to introduce the idea of rate of change over an interval and its limit as the length of the interval approaches zero is the instantaneous rate of change.

Topic 2.2: Defining the derivative and using derivative notation. The derivative is defined as the limit of the difference quotient from topic 1 and several new notations are introduced. The derivative is the slope of the tangent line at a point on the graph. Explain graphically, numerically, and analytically how the three representations relate to each other and the slope.

Topic 2.3 Estimating the derivative at a point.  Using tables and technology to approximate derivatives is used in this topic. The two resources in the sidebar will be helpful here.

Topic 2.4: Differentiability and Continuity. An important theorem is that differentiability implies continuity – everywhere a function is differentiable it is continuous.  Its converse is false – a function may be continuous at a point, but not differentiable there. A counterexample is the absolute value function, |x|, at x = 0.

One way that the definition of derivative is tested on recent exams which bothers some students is to ask a limit like

\displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{\tan \left( {\tfrac{\pi }{4}+x} \right)-\tan \left( {\tfrac{\pi }{4}} \right)}}{x}.

From the form of the limit students should realize this as the limit definition of the derivative. The h in the definition has been replaced by x. The function is tan(x) at the point where \displaystyle a=\tfrac{\pi }{4}. The limit is \displaystyle {{\sec }^{2}}\left( {\tfrac{\pi }{4}} \right)=2.

Topics 2.5 – 2.10: Differentiation Rules

The remaining topics in this chapter are the rules for calculating derivatives without using the definition. These rules should be memorized as students will be using them constantly. There will be additional rules in Unit 3 (Chain Rule, Implicit differentiation, higher order derivative) and for BC, Unit 9 (parametric and vector equations).

Topic 2.5: The Power Rule

Topic 2.6: Constant, sum, difference, and constant multiple rules

Topic 2.7: Derivatives of the cos(x), sin(x), ex, and ln(x). This is where you use the squeeze theorem.

Topic 2.8. The Product Rule

Topic 2.9: The Quotient Rule

Topic 2.10: Derivative of the other trigonometric functions

The rules can be tested directly by just asking for the derivative or its value at a point for a given function. Or they can be tested by requiring the students to use the rule of an general expression and then find the values from a table, or a graph. See 2019 AB 6(b)

The suggested number of 40 – 50 minute class periods is 13 – 14 for AB and 9 – 10  for BC. This includes time for testing etc. Topics 2.1, 2,2, and 2.3 kind of flow together, but are important enough that you should spend time on them so that students develop a good understanding of what a derivative is. Topics 2.5 thru 2.10 can be developed in 2 -3 days, but then time needs to be spent deciding which rule(s) to use and in practice using them. The sidebar resource in the CED on “Selecting Procedures for Derivative” may be helpful here.

Other post on these topics

DEFINITION OF THE DERIVATIVE

Local Linearity 1  The graphical manifestation of differentiability with pathological examples.

Local Linearity 2   Using local linearity to approximate the tangent line. A calculator exploration.

Discovering the Derivative   A graphing calculator exploration

The Derivative 1  Definition of the derivative

The Derivative 2   Calculators and difference quotients

Difference Quotients 1

Difference Quotients II

Tangents and Slopes

         Differentiability Implies Continuity

FINDING DERIVATIVES 

Why Radians?  Don’t do calculus without them

The Derivative Rules 1  Constants, sums and differences, powers.

The Derivative Rules 2  The Product rule

The Derivative Rules 3  The Quotient rule

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions 

2019 CED Unit 10 Infinite Sequences and Series

 

 

 

 

 

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Difference Quotients II

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The Symmetric Difference Quotient

In the last post we defined the Forward Difference Quotient (FDQ) and the Backward Difference Quotient (BDQ). The average of the FDQ and the BDQ is called the Symmetric Difference Quotient (SDQ):

\displaystyle \frac{f\left( x+h \right)-f\left( x-h \right)}{2h}

You may be forgiven if you think this might be a better expression to use to find the derivative. It has its advantages. In fact, this is the expression used in many calculators to compute the numerical value of the derivative at a point; in calculators it is called nDeriv. Usually, it works pretty well. But if you try to find the derivative of the absolute value of x at x = 0 it will tell you the derivative is 0, which is wrong. The absolute value function is not locally linear at the origin and has no derivative there.

What went wrong?  Read the expression above. The numerator is the difference of the function values at the same distance, h, on both sides of x. Since, for the absolute value function with x = 0, these values are the same, their difference is 0. The SDQ never looks at x = 0 and doesn’t realize there is no derivative there. Thus, the limit of the SDQ is not the derivative.

This problem does not occur with the definition of derivative, since for that limit to exist the limits as h approaches zero from both sides must be equal. For the absolute value function the limit from the left is –1 and the limit from the right is +1 and therefore there is no limit and no derivative there.

Since most functions we will consider are differentiable, most of the time the SDQ and nDeriv are okay to use.

Seeing Difference Quotients Converge

This is an activity to see difference quotients graphically. Use a graphing calculator or a graphing program on a computer. One with a slider feature is better although I’ll also tell you how to use a calculator without this feature.

  1. Enter the function you want to consider as Y1 in your calculator or give it a name if you are using a computer. This is so later you can change the function without having to re-enter the next three equations.
  2. Enter the FDQ as Y2 using Y1 as the function. See Figure 1 below.
  3. Enter the BDQ as Y3 again using Y1 as the function.
  4. Enter the SDQ as Y3 again using Y1 as the function.
  5. Either set up a slider for h or go to the home screen and store a value for h. In the latter case you will have to return to the home screen and change the values.

Now graph all four functions. As you change the values of h with the slider or from the home screen, you should see three similar graphs (the difference quotients) along with the first function you entered. As h approaches zero, the three similar graphs should come together (converge) on the graph of the derivative. See Figures 2 and 3 below.

Change the first function. Some good functions to try are y = x3 – 4x, y = x3/3, y = sin(x) and don’t forget y = |x|. Try guessing the equation of the derivative.

Figure 2 Shows y = x3/3 in Black with the three difference quotients, h is about 2.

Figure 3 shows the same graph with h almost 0; the three difference quotients, now almost on top of each other, are closing in on the derivative.

Here is a link to a Desmos demonstration of the three difference quotients

Difference Quotients I

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Difference Quotients & Definition of the Derivative

In the second posting on Local Linearity II, we saw that what we were doing, finding the slope to a nearby point, looked like this symbolically:

\displaystyle \frac{f\left( x+h \right)-f\left( x \right)}{h}

This expression is called the Forward Difference Quotient (FDQ). It kind of assumes that h > 0.

There is also the Backwards Difference Quotient (BDQ):

\displaystyle \frac{f\left( x \right)-f\left( x-h \right)}{h}=\frac{f\left( x-h \right)-f\left( x \right)}{-h}

The BDQ also kind of assumes that h > 0. If h < 0 then the FDQ becomes the BDQ and vice versa. So these are really the same thing. The limit (if it exists) as h approaches zero is the slope of the tangent line at whatever x is and this is important enough to have its own name. It is called the derivative of f at x with the notation (among others) {f}'\left( x \right) :

\displaystyle {f}'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( x+h \right)-f\left( x \right)}{h}

Since h must approach 0 from both sides, this expression incorporates the FDQ and the BDQ in one expression.

To emphasize that h is a “change in x” this limit is often written

\displaystyle {f}'\left( x \right)=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{f\left( x+\Delta x \right)-f\left( x \right)}{\Delta x}