Roberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 5. Graph sketching
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1 Roberto s Notes on Differential Calculus Chapter 8: Graphical analsis Section 5 Graph sketching What ou need to know alread: How to compute and interpret limits How to perform first and second derivative analses. What ou can learn here: How to gather and use information about a function in order to sketch an informative graph. Let me get one thing straight right off the bat. Knot on our finger The eact graph of a function can onl be obtained b plotting all its points. Since plotting can onl be done on finitel man points, while a graph has infinitel man points, ever graph of a function is approimate, even the ones made b the best computers. What changes is onl the degree of accurac of the graph. Strateg for a reasonable approach to graph sketching Since sketching the graph of a function can onl be done approimatel: Do not become obsessed with wanting to get all details and each absolutel correctl, BUT Ensure that all important features of the graph are identified, classified and clearl visible in the graph ou construct. And what should we consider as important features? Wh are ou saing this? I am confused! To help ou reach a balance in our search for a graph. The reflection I just offered leads to the following general strateg. I am glad ou asked! There are three tpes of important features, and b now ou have seen how to find them all. So, here are the three tpes, one at a time. Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page
2 Strateg for finding the location features of a function B focusing on the original function f ( ) and b using algebra and limits methods, we can obtain information on: Domain Intercepts Discontinuities Behaviour at infinit These features help us identif the location of certain ke details of the graph. Strateg for finding the up/down pattern of the function B focusing on the derivative f ( ) and b using algebra, limits and differentiation methods, we can obtain information on: Intervals of increase and decrease Etreme points And ou have probabl guessed what the third set of features includes. Strateg for finding the concavit pattern of the function B focusing on the second derivative f ( ) and b using algebra, limits and differentiation methods, we can obtain information on: Concavit Inflection points B using this information we can sketch a reasonabl informative graph in most cases, and certainl in all cases that ou will find in tests. But there is one more important strateg to consider and this deals with how to put together all the information ou find! Man students first collect all the information and then tr to visualize the whole graph in one fell swoop. Not a good idea! Instead, tr the following. Strateg for how to build the graph of a function from the information available To construct an informative sketch of the graph of a function f ( ), jot down each piece of information available on the Cartesian plane as soon as ou find it. Onl at the end connect all the pieces in a reasonable and consistent wa. Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page
3 That s too vague! That is because the specific details can var tremendousl from function to function. Here are some eamples for our perusal. Eample: f ( ) 4 4 We start with the function itself. The intercept is: 0 f 0 0 0, 0 ( ) ( ) The intercepts are: ( ) ( ) ( ) ( ) 0, 0, 0,, 0,, 0 We start placing these points on the graph: The first derivatives is: f '( ) It produces the following line graph: f + f ( ) Thus we have relative maima at (, 4) and at ( ) + 0, 4. Since the have the same coordinate and the function does not go an higher, the are also absolute maima. At the origin we have a minimum. We add this information to the graph, in the form of small wedges around the points, since we do not have information about the second derivative et. Being a polnomial, this function has no discontinuities, nor horizontal asmptotes. However, we look at the limits at infinit to figure out how the graph eits the visible window. lim 4 4 lim 4 4 ( ) ( ) We now add this information to the graph, even though it is rather sketch now. Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page
4 The second derivative is: f "( ) 8 4 It produces the following line graph: ( ) f f / / Therefore we have two inflection points at 0.8. We indicate these as well on the graph and complete it b joining all the points and using all the features. + The first derivative is 8 ( 8) (check it!) and hence its line graph is: f f ' + 4 Therefore, the critical value at information to the graph. 4 is a relative maimum. We add this Eample: 8 We can easil see that the onl intercept of this function occurs at the origin. Moreover, the law of the jungle tells us that there is a HA at 0 and the denominator tells us that is a vertical asmptote. We begin to place all these items on the graph. Actuall, b sneaking a peak at the calculator s graph, we can choose a suitable window for our graph: In fact, we can add more information: As the graph has no more -intersections, so it must approach the horizontal asmptote from above. Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page 4
5 The function is decreasing both to the right and to the left of the vertical asmptote. As the graph has no more -intersections, so it must approach the horizontal asmptote from above. We can add this information to the graph as well: The net eample shows ou a situation where not all features can be identified, but we can still get an informative graph, although incomplete work ma hide some important features. This is still ver rough and we can improve on it b adding second derivative information. Notice also that the scale is not right: the maimum is higher than where it should be, but if we place it in the proper position it is hardl visible! That is acceptable, since we are producing a sketch onl. The second derivative is 6 6 ( 8) (check it!) with line graph: Eample: ( ) f e This time we are going to cheat and have a peak at the graph from the calculator first! We know that an eponential is alwas positive, so we set the window accordingl: f + 6 f 0 8 Therefore the onl inflection point is at 6. At 0 we do have 0, but it does not change sign, so it is not an inflection point. The graph produced b the calculator is as shown here and is consistent with all the information we found, just a little smoother. Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page 5
6 It looks like we have two asmptotes and a maimum, with the middle section going where? It cannot go to below the ais, so does it hit it? And if so, where? And are there other interesting, but invisible features? Let s see. We start from the intercepts: 0 DNE 0 DNE Fine, no intercepts. The asmptotes, which are the onl two discontinuities, occur at the values for which the eponent becomes infinite, that is, when: 0 0, Let s check: ( ) lim lim lim e e e ( ) lim e lim e lim e So, this is an asmptote on the left side onl, while on the right it is a single point hole! The same happens at the other value: ( ) lim e lim e lim e ( ) lim e lim e lim e 0 0 Now we check for the horizontal asmptotes 0 lim lim e e e Yes, is a horizontal asmptote. If we had started from scratch, without looking at the calculator, we would sketch this information on the graph: Notice that I have changed the window to highlight the ke central features. The first derivative is f '( ) f e ( ) ( ) + + (check it!) with line graph: f 0.5 As epected, there is onl one critical value and we now know that the maimum is at.5. So at this point we can reconstruct a graph ver close to what our calculator gave us: What about the second derivative? It is a bit of a challenge to compute it: ou ma want to do it during a cold winter night b a roaring fire. But the Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page 6
7 concavities at this point do not add much, so it ma not be worth the effort. Remember that we are just sketching! However, if ou do succeed in obtaining, ou will get a pleasant surprise: it will reveal two inflection points in the middle section. close to the asmptotes! Not as pleasant as chocolate, but interesting, especiall to remind us that the calculator does not alwas provide all the information, or even accurate hints, eh? I can see that sketching some of these graphs can become an adventure! Ever graph has its own stor and its own surprises. Once ou get the hang of it, sketching a graph can be a ver fun activit, but onl if ou learn the ke methods, implement them carefull and get plent of eperience. Speaking of which, the Learning questions offer several good adventures for ou. Keep in mind that ou can use our calculator to check our work, but do not trust the calculator all the times: there ma be some hidden features and when there is a discrepanc or an unclear aspect, investigate it alwas! Summar When sketching the graph of a function, we are not aiming for perfection, but onl for an accurate representation of the main graphical features of such graph The main features we need to obtain are intercepts, discontinuities and behaviour at infinit (from the original function), up/down pattern and etreme points (from the first derivative) and concavit and inflection points (from the second derivative). Common errors to avoid When making a mistake in computation, do not force the features ou find to coeist, because the won t! Instead identif the inconsistenc and state the need for investigating the problem. Tring to force things will create graphs that are obviousl wrong, recognizing errors will show that ou are reflecting on what ou are finding. Learning questions for Section D 8-5 Review questions:. Describe the three strategies for collecting relevant information about the graph of a function.. Describe the strateg for using the information provided b calculus to sketch the graph of a function.. Eplain wh it is impossible to obtain the eact graph of a function. Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page 7
8 Memor questions:. What kind of information about the graph of a function is obtained b analzing the original function onl?. What kind of information about the graph of a function is obtained b analzing its first derivative?. What kind of information about the graph of a function is obtained b analzing its second derivative? 4. What graphical information is provided b the limits at infinit when the function has no horizontal asmptotes? Computation questions: In each of questions -8 a function and its first and second derivative are provided. Use them to identif all ke graphical features of the function and to sketch an informative graph. Compare our graph to that provided b a graphing calculator, but identif the location of the important features as eactl as ou can. Of course, ou ma want to compute those derivatives ourself, just to practice and to make sure that there are no errors! ( ) 4 ' ( 4 ) ( + + ) + ( + ) " 4 4( 6 + 8) ( 4 ) ( + )( + + ) ( + ) ( + 8) 8 ( + 8) + 4. ( ) + + ( ) Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page 8
9 5. e ' e e " 6. ln ln ln ' " (ln ) (ln ) 7. ln ' ln + " 8. ( ) ( + ) ' ( ) ( + ) " ( ) ( + ) " ( 8) / ' ( 8) / ( ) 4 0. ( + ) ' 4 ( + ) " 48 ( + ) 4. e ' ( ) e " ( 4 + ) e. + 4 ' + 4 ( + 4) 4 " ( + 4) ( ) / ( + ) 6 ( + ) 5 Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page 9
10 ( ) e ( ) ( ) 6 ( ) e ( + ) 4 4 ( ) 4 ( + ) e e ( ) e ( 4 + 7) e ln 8. ln 6ln 5 ' " ' ( ) ( + ) 5 ( ) 4 5 ( + ) ( ) + 8 ( ) 4 For each of the functions provided in questions -8, obtain all relevant information and use it to sketch an informative graph of the function f. ( ) + 6. f ( ) ( ) f ( ) 8. f ( ) 8 Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page 0
11 f 0. ( ) 5. cos, 4. sin sin. f ( ) tan + cos, 0 4. sin 5. ( ) 4 f e 6. ln ( ) 7. ( ) 8. f e 4 5( ) ( ) ( + ) 9. ( ) sinh (No need to analze the second derivative) Theor questions:. Which calculus method is ke in the analsis of asmptotes?. Mention two graphical features of a function that ma not be visible on the calculator. At an -value where DNE, how do we determine whether we have a vertical asmptote or a vertical tangent line? 4. Identif potential calculator errors that can mislead in identifing the graph of a function. 5. Identif three graphical features that can occur at an c if f( ) 0 when c and f( ) 0 when c. 6. Can the line tangent to a curve at a point cross the curve at that point? 7. Mention two graphical features that are obtained directl from the original function 8. What features of the function provide information on the window to use when sketching its graph? 9. Wh is it necessar to do al the calculus work to sketch the graph of a function, rather than simpl using the calculator s graph? 0. If ou graph a polnomial function and zoom in repeatedl on an one of its points, what shape will its graph take eventuall? Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page
12 Proof questions:. Construct the formula of a function that has vertical asmptote at, a single point hole at (, ), and a horizontal asmptote at 4. Templated questions:. Construct a reasonabl simple function and sketch its graph b using calculus methods What questions do ou have for our instructor? Differential Calculus Chapter 8: Graphical analsis Section 5: Graph sketching Page
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