P alculus erivatives 2015 11 03 www.njctl.org www.njctl.org Table of ontents Rate of hange Slope of a urve (Instantaneous RO) erivative Rules: Power, onstant, Sum/ifference Higher Order erivatives erivatives of Trig Functions erivative Rules: Product & Quotient alculating erivatives Using Tables Equations of Tangent & Normal Lines erivatives of Logs & e hain Rule erivatives of Inverse Functions ontinuity vs. ifferentiability erivatives of Piecewise & bs. Value Functions Implicit ifferentiation Table Of ontents 1
Why are erivatives Important? First, let's discuss the importance of erivatives: Why do we need them? a) What is the slope of the following function? b) What is the slope of the line graphed at right? Teacher Notes c) Now, what about the slope of??? Introduction erivatives Exploration Exploration into the idea of being locally linear... lick here to go to the lab titled "erivatives Exploration: y = x 2 " Teacher Notes Introduction 2
Rate of hange Return to Table of ontents Rate of hange Road Trip! onsider the following scenario: You and your friends take a road trip and leave at 1:00pm, drive 240 miles, and arrive at 5:00pm. How fast were you driving? Teacher Notes Rate of hange 3
Position vs. Time Now, consider the following position vs. time graph: Teacher Notes position t 1 time t 2 t 3 t0 Rate of hange 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. Rate of hange 4
SENT vs. TNGENT secant line connects 2 b points on a curve. The slope y 2 of this line is also known as the verage Rate of hange. a y 1 tangent line touches one x 1 x 2 point on a curve and is known as the Instantaneous Rate of hange. Rate of hange Slope of a Secant Line How would you calculate the slope of the secant line? y 2 b y 1 a x 1 x 2 Rate of hange 5
Slope of a Secant Line What happens to the slope of the secant line as the point b moves closer to the point a? y 2 y 1 a b Teacher Notes x 1 x 2 What is the problem with the traditional slope formula when b=a? Rate of hange verage Rate of hange 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 Rate of hange 6
1 What is the average rate of change of the function on the interval from to? 3 2 1 1 E 3 Rate of hange 2 What is the average rate of change of the function on the interval? 14 0 56 56 E 14 Rate of hange 7
3 What is the average rate of change of the function on the interval? 0 E Rate of hange 4 What is the average rate of change of the function on the interval? 1 2 E 0 Rate of hange 8
5 The wind chill is the temperature, in degrees Fahrenheit, a h feels based on the air temperature, in degrees Fahrenheit, a wind velocity v, in miles per hour. If the air temperature of, is then 32 the wind chill is given by and is valid fo 5 v 60. (from the 2007 P Exam) Find the average rate of change W over of the interval v5 60. LULTOR LLOWE Teacher Notes & Rate of hange Slope of a urve (Instantaneous Rate of hange) Return to Table of ontents Slope of a urve 9
Recall: The ifference Quotient Recall from the previous unit, we used limits to calculate the instantaneous rate of change using the ifference Quotient. For example, given, we found an expression to represent the slope at any given point. Teacher Notes & Slope of a urve erivatives 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. Slope of a urve 10
erivatives 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 Slope of a urve Notation You may see many different notations for the derivative of a function. lthough 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)" Slope of a urve 11
Formal efinition of a erivative 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: Slope of a urve 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. Example: a) Find the slope at any point, x, of the function b) Use that expression to find the slope of the curve at Teacher Notes & Slope of a urve 12
6 Which expression represents if? E Slope of a urve 7 What is the slope of at x=1? E Slope of a urve 13
8 Find if E Slope of a urve 9 Find if E Slope of a urve 14
erivatives s 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. Slope of a urve erivative Rules: Power, onstant & Sum/ifference Return to Table of ontents erivative Rules 15
epending on the function, calculating derivatives using Fermat's method with limits can be extremely time consuming. an you imagine calculating the derivative of using that method? Or what about? lternate Methods Fortunately, there are some "shortcuts" which make taking derivatives much easier! The P Exam will still test your knowledge of calculating derivatives using the formal definition (limits), so your energy was not wasted! erivative Rules Exploration: Power Rule Let's look back at a few of the derivatives you have calculated already. We found that: The derivative of is The derivative of is The derivative of is Teacher Notes What observations can you make? o you notice any shortcuts for finding these derivatives? erivative Rules 16
The Power Rule e.g. e.g. *where c is a constant erivative Rules The onstant Rule ll of these functions have the same derivative. Their derivative is 0. Why do you think this is? Think of the meaning of a derivative, and how it applies to the graph of each of these functions. Teacher Notes where c is a constant erivative Rules 17
The Sum & ifference Rule e.g. e.g. erivative Rules Practice Take the derivatives of the following. erivative Rules 18
Extra Steps Sometimes, it takes a little bit of manipulating of the function before applying the Power Rule. Here are 4 scenarios which require an extra step prior to differentiating: Teacher Notes erivative Rules 10 What is the derivative of? E erivative Rules 19
11 E erivative Rules 12 What is the derivative of 15? x 1 14 0 E 15 erivative Rules 20
13 Find if E erivative Rules 14 Find y' if erivative Rules 21
15 Which expression represents the slope at any point on the curve? istribute! HINT E erivative Rules erivatives at a Point If asked to find the derivative at a specific point, a question may ask... Find alculate 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 x value first and then tried to take the derivative? Teacher Notes erivative Rules 22
16 What is the derivative of at? 5 0 15 5 E 15 erivative Rules 17 Find 36 144 12 72 E 24 erivative Rules 23
18 What is the slope of the tangent line at if? 6.5 6 0 3.5 E 4 erivative Rules 19 Find y'(16) if E erivative Rules 24
Higher Order erivatives Return to Table of ontents Higher Order erivatives Higher Order erivatives You may be wondering... an you find the derivative of a derivative!!?? The answer is... YES! Finding the derivative of a derivative is called the 2 nd derivative. Furthermore, taking another derivative would be called the 3 rd derivative. So on and so forth. Teacher Notes Higher Order erivatives 25
Notation The notation for higher order derivatives is: 2 nd derivative: 3 rd derivative: Teacher Notes 4th derivative: n th derivative: Higher Order erivatives pplications of Higher Order erivatives 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 N analysis and population modeling. Higher Order erivatives 26
Find the indicated derivative. Practice Teacher Notes & Higher Order erivatives 20 Find the 3rd derivative of E Higher Order erivatives 27
21 Find if E Higher Order erivatives 22 Find if E Higher Order erivatives 28
23 Find E Higher Order erivatives 24 Find E Higher Order erivatives 29
25 Find E Higher Order erivatives erivatives of Trig Functions Return to Table of ontents Teacher Notes erivatives of Trig Functions 30
erivatives 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. erivatives of Trig Functions erivatives of Trig Functions Teacher Notes erivatives of Trig Functions 31
Proof Let's take a moment to prove one of these derivatives... Teacher Notes & erivatives of Trig Functions erivatives of Inverse Trig Functions Teacher Notes erivatives of Trig Functions 32
26 What is the derivative of? E F erivatives of Trig Functions 27 What is the derivative of? E F erivatives of Trig Functions 33
28 What is the derivative of? E F erivatives of Trig Functions 29 What is the derivative of? E F erivatives of Trig Functions 34
30 What is the derivative of? E F erivatives of Trig Functions 31 What is the derivative of? E F erivatives of Trig Functions 35
32 What is the derivative of? E F erivatives of Trig Functions 33 Find 1 E 1 F erivatives of Trig Functions 36
34 Find E F erivatives of Trig Functions 35 Find E F erivatives of Trig Functions 37
36 Find E F erivatives of Trig Functions erivative Rules: Product & Quotient Return to Table of ontents erivative Rules 38
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! erivative Rules The Product Rule Fortunately, an alternative method was discovered by the famous calculus mathematician, Gottfried Leibniz, known as the product rule. Let's take a look at how the product rule works... Teacher Notes erivative Rules 39
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. erivative Rules The Product Rule using the distributive property using the product rule Teacher Notes erivative Rules 40
istribution 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. erivative Rules Practice Finding the following derivatives using the product rule. erivative Rules 41
37 erivative Rules 38 erivative Rules 42
39 erivative Rules 40 Find erivative Rules 43
41 erivative Rules 42 erivative Rules 44
43 Find erivative Rules 44 True False Teacher Notes erivative Rules 45
What bout Rational Functions? So far, we have discussed how to take the derivatives of polynomials using the Power Rule, Sum and ifference Rule, and onstant 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. erivative Rules The Quotient Rule Notice, the problems above can be viewed as the quotient of 2 functions. We can then apply the quotient rule. Teacher Notes erivative Rules 46
Example Given: Find f(x), or "top" g(x), or "bottom" erivative Rules Example Given: Find erivative Rules 47
Proof Now that you have seen the Quotient Rule in action, we can revisit one of the trig derivatives and walk through the proof. erivative Rules 45 ifferentiate erivative Rules 48
46 Find erivative Rules 47 Find erivative Rules 49
48 Find erivative Rules 49 ifferentiate erivative Rules 50
50 Find the derivative of Teacher Notes & erivative Rules alculating erivatives Using Tables Return to Table of ontents erivatives Using Tables 51
erivatives Using Tables On the P 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. erivatives Using Tables Let's take a look at an example: Example Let alculate 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. erivatives Using Tables 52
Example Let alculate 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. erivatives Using Tables erivatives Using Tables Next is another type of question you may encounter on the P Exam involving tabular data and derivatives. Use the table at right to estimate erivatives Using Tables 53
51 The functions f and g are differentiable for all real numbers. The table at right gives values of the functions and their first derivatives at selected values of x. Let alculate 2 10 1 2 E 30 erivatives Using Tables 52 The functions f and g are differentiable for all real numbers. The table at right gives values of the functions and their first derivatives at selected values of x. Let alculate 93 61 75 95 E 0 erivatives Using Tables 54
53 Let alculate 1.5 39 32 75 E 0 erivatives Using Tables 54 Use the table at right to estimate 2.4 3.1 32 0 E 4 erivatives Using Tables 55
55 Use the table at right to estimate 3.05 3.6 0 0.278 E 0.5 erivatives Using Tables Equations of Tangent & Normal Lines Return to Table of ontents Tangent & Normal Lines 56
Writing Equations of Lines Recall from lgebra, 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. Tangent & Normal Lines Equations of Tangent Lines First let's consider some basic linear functions... Teacher Notes If asked to write the equation of the tangent line to each of these functions what do you notice? Tangent & Normal Lines 57
Example Let's try an example: Write an equation for the tangent line to at x=2. Teacher Notes & Tangent & Normal Lines Example Write an equation for the tangent line to at. Tangent & Normal Lines 58
y = x 2 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. normal line at x = 1 tangent line at x = 1 Teacher Notes How do you suppose we would calculate the slope of a normal line? Tangent & Normal Lines Example Let's try an example: Write an equation for the normal line to at x=2. Tangent & Normal Lines 59
Example Example: Write an equation for the normal line to at. Tangent & Normal Lines Example: Write an equation for the normal line to at. Example Teacher Notes & Tangent & Normal Lines 60
56 Which of the following is the equation of the tangent line to at? E F Tangent & Normal Lines 57 Which of the following is the equation of the tangent line to at? E F Tangent & Normal Lines 61
58 Which of the following is the equation of the normal line to at? E F Tangent & Normal Lines 59 Which of the following is the equation of the tangent line to at? E F Tangent & Normal Lines 62
60 Which of the following is the equation of the normal line to at? E F Tangent & Normal Lines 61 Which of the following is the equation of the tangent line to at? E F Tangent & Normal Lines 63