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Slide 51 (Answer) / 213 Higher Order Derivatives Teacher Notes You may be wondering.... You may want to mention to Can you find the derivative of a derivative!!?? students that each derivative they take must be continuous in order The answer is... YES! to keep taking the next derivative. Continuity and differentiability will Finding the derivative of a derivative is called the 2 nd derivative. be discussed later, but it is helpful Furthermore, taking another derivative would be called the 3 rd to mention here. derivative. So on and so forth. [This object is a teacher notes pull tab] Slide 52 / 213 Slide 52 (Answer) / 213

Slide 53 / 213 Applications of Higher Order Derivatives Finding 2 nd , 3 rd , 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 5 th derivative is helpful in DNA analysis and population modeling. Slide 54 / 213 Slide 54 (Answer) / 213

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Slide 61 / 213 Derivatives of Trig Functions Return to Table of Contents Slide 61 (Answer) / 213 Teacher Notes 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 Return to subsequent sections. Table of [This object is a teacher notes pull tab] Contents Slide 62 / 213 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.

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The proofs of trig derivatives stem from 2 previously learned limits, Slide 64 (Answer) / 213 Proof Let's take a moment to prove one of these derivatives... you can revisit this idea or just mention that they are being used. If students are interested in looking at the proofs of all the trig derivatives, you can direct them to research them on their own time online. There are extensive proofs, simply by searching for trig derivatives. angle sum identity Slide 65 / 213 Slide 65 (Answer) / 213

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Slide 73 (Answer) / 213 Slide 74 / 213 34 Find A D B E C F Slide 74 (Answer) / 213 34 Find A D Answer D B E [This object is a pull tab] C F

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Slide 76 (Answer) / 213 36 Find A D Answer E B E [This object is a pull tab] C F Slide 77 / 213 Derivative Rules: Product & Quotient Return to Table of Contents Slide 78 / 213

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Slide 90 (Answer) / 213 Slide 91 / 213 44 True False Slide 91 (Answer) / 213 44 Teacher Notes True FALSE False Students can share/discuss the functions they use to disprove this statement. [This object is a teacher notes pull tab]

Slide 92 / 213 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. Slide 93 / 213 Slide 93 (Answer) / 213

Slide 94 / 213 Example Given: Find f(x), or "top" g(x), or "bottom" Slide 94 (Answer) / 213 Example Given: Find f(x), or "top" Answer g(x), or "bottom" [This object is a pull tab] Slide 95 / 213 Example Given: Find

Slide 95 (Answer) / 213 Slide 96 / 213 Proof Now that you have seen the Quotient Rule in action, we can revisit one of the trig derivatives and walk through the proof. Slide 96 (Answer) / 213

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Slide 103 / 213 Calculating Derivatives Using Tables Return to Table of Contents Slide 104 / 213 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. Slide 105 / 213

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Slide 114 / 213 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. Slide 115 / 213 Slide 115 (Answer) / 213

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Slide 117 (Answer) / 213 Slide 118 / 213 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 = x 2 tangent line normal line at x = 1 at x = 1 How do you suppose we would calculate the slope of a normal line? Slide 118 (Answer) / 213 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 = x 2 This question is meant to be in Teacher Notes general, for any normal line, not necessarily this specific curve. tangent line normal line at x = 1 Allow students to discuss how to at x = 1 come up with the slope of the normal line, hoping they make the connection that perpendicular lines have opposite reciprocal [This object is a teacher notes pull tab] slopes. How do you suppose we would calculate the slope of a normal line?

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Slide 128 / 213 Derivatives of Logs & e Return to Table of Contents Slide 129 / 213 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. Slide 130 / 213

Slide 131 / 213 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 Slide 131 (Answer) / 213 Slide 132 / 213 Derivatives of Exponential Functions cool! is the only nontrivial function whose derivative is the same as the function!

Slide 132 (Answer) / 213 Derivatives of Exponential Functions Technically, y=0 is also it's own derivative as well, but Teacher Notes does not depend on another variable, so generally people say that is the only one. Consider asking students if cool! is the only nontrivial function whose they can think of y=0 before derivative is the same as the function! telling them. [This object is a teacher notes pull tab] Slide 133 / 213 Slide 134 / 213

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Slide 153 / 213 Derivatives of Inverse Functions Return to Table of Contents Slide 154 / 213 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. Slide 155 / 213

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Slide 165 / 213 Continuity vs. Differentiability Return to Table of Contents Slide 166 / 213 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. Slide 167 / 213 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

Slide 168 / 213 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. What does this mean??? Consider the function: Notice: If we were asked to find the 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. Slide 168 (Answer) / 213 Another way to explain to this to students is to Differentiability Implies Continuity draw several tangent lines at x=0 and show that they all have different slopes. If a function is differentiable, it is also continuous. Teacher Notes 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: 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 [This object is a teacher notes pull tab] 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. Slide 169 / 213 A FUNCTION FAILS TO BE DIFFERENTIABLE IF... CORNER CUSP DISCONTINUITY VERTICAL TANGENT

Slide 170 / 213 Types of Discontinuities: removable removable jump infinite essential Slide 171 / 213 Slide 172 / 213

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Slide 174 / 213 Slide 174 (Answer) / 213 Slide 175 / 213 81 If f(x) is continuous on a given interval, it is also differentiable. True False