Partial preview of the text
Download 4.5 summery curve sketching and more Lecture notes Calculus in PDF only on Docsity!
MATH 1180 4.5: Summary of Curve Sketching “As long as algebra and geometry traveled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforth marched on at a rapid pace toward perfection.” Joseph-Louis Lagrange In this section we'll be bringing together many topics from different courses different sections of this course: ° e Chapter 1 (Review): x and y intercepts, Symmetry, Domain and Range Chapter 2: Continuity, Vertical Asymptotes Chapter 3: Differentiability Chapter 4: Relative Extrema, Concavity, Points of Inflection, Horizontal Asymptotes, Infinite Limits at Infinity. Guidelines for Analyzing the Graph of a Function Determine the domain and range of the function Determine the intercepts, asymptotes, and symmetry of the graph Locate the x-values for which f(x) and f'"(x) are either zero or do not exist. Use the results to determine the relative extrema and points of inflection. Be Prepared to do problems like this without a calculator! (wth a Qreghiay calculator ) Values of x that will divide your intervals: Values for which f (x) is undefined Critical Numbers (f’ = 0 or f' is not defined) Possible points of infliction (f"" = 0 or f" is undefined ) MATH 1180 4.5: Summary of Curve Sketching Example: Analyze and sketch a gragh of fox) = 20% 3NXX-3) xintucepts: 2,-3 fo = 2 4X (x+2XX-2) Domain: &x| x##23 ~ : Joys (2-4) 4K -2 2-9) 2x , Ay = 1bx~ 4X 436K (x2- a (x2-4)? y = 20x y?-y4 0 Fe ee ctay-tg fe Cea 20-Ok el) x (x2-u) (Cx2-4)*) fro la (x2-4)%3 (x2. ~(x2-4)5 ans = Cory hk 2 = Lo] — = ae NCS IS\ [24 ? = J ea (Cortical +S = 20x wha ts fl) vad fined: £ (x)= 20X% 2 (x2-4)* — denam/natar: (x2- ¢{) =O (x2)? 0 =20x X=F2 |etechnicall y net ocitical ABs oe Mey ore outa eek Inflectian Points | win i> f"bx = na. vAdined? = 0(3x?+4) — No (eal (y2-u)3 > O=-20(3x* rH) e solutions! bt J" ts uadefived when x= +2, bit Tse rove Aot in domain. MATH 1180 4.5: Summary of Curve Sketching Example: Analyze and sketch a graph of the function below. Label any intercepts, relative extrema, points of inflection and asymptotes, both vertical and horizontal, if they exist. _cosx -—sinx 1+sinx ARMA CoSKX fix): CI+SInx)C sina) — (oSX (COS %) sing =| fix): ———e f@)= (l+sinx)? Clrsinx)> Sug = -sinx - sin? X- COS?X $i) Girt (\+sina)* So) = -SINK- (sin?x +052x) \+sinx)? ont o> (l+ stax) yw [ 2 Cw ae) (2) FG) = w \Y wy. (ltsinx)-O - Cl)cos This TS periodre Figs Gana) eos Ti peliod 27 +sin x )* ( A x) Domain: wha does @ +Sin x) =0 COSK , = aU+2k7r Le LE Tree Sina =~] X* “z , (1+ sinx) for Keg vt of domain! lat ecepts eZ) see First Derivative: we eaval 10 ) -1 Neu ez, a but! 0): Tesiny Zero. Uiefined BT js ovt of _ wh BF +2. 2 ) [ .,. 7 < 4 domain: Xa y+ Zk MATH 1180 4.5: Summary of Curve Sketching Qnd derivative: nen fon = cos % COSK=O whe “Clsine)? aS x - . X= S429. Undeined @ =r yo latecept: fey - os (0) < l+sin0 No exeme values? Inflection point @ 0) one porlod of = 2tr ee ee ee ae MATH 1180 4.5: Summary of Curve Sketching a 
