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4.5 An Algorithm for Curve Sketching
We learned a bunch of βpiecesβ to curve sketching, now letβs put it all together β¦ STEP 1: Determine the DOMAIN of the function and all discontinuities (π(π₯) is undefined) Polynomials are continuous and differentiable for βπ₯ββ. Some functions do have restrictions on their domains (i.e. π π₯ = π₯ is only defined forπ₯β₯0). Discontinuities can be infinite, removable or jump. Infinite: π π₯ = π₯+1 π₯β3 , π 3 is undefined (den=0 but numβ 0) β A VERTICAL ASYMPTOTE β Donβt forget to check whether π(π₯) approaches Β±β at these points. Removeable: π π₯ = (π₯+1)(π₯β3) π₯β3 , π 3 is undefined (den=0 and num=0) β A HOLE Jump: think piecewise functions
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4.5 An Algorithm for Curve Sketching
STEP 2: Determine any π₯ and π¦ intercept(s). STEP 3: Determine any horizontal or oblique asymptotes. Horizontal Asymptotes & End Behaviour Explore lim π₯β+β π(π₯) and lim π₯βββ π(π₯) to determine if there are any horizontal asymptotes. β Donβt forget to check whether π(π₯) approaches the asymptote from above or below in each direction. Oblique Asymptotes The occur for all rational functions in which the degree of the numerator is exactly one greater than that of the denominator. β Again check whether π(π₯) approaches the asymptote from above or below in each direction.
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4.5 An Algorithm for Curve Sketching
STEP 4: The First Derivative Test 1. Find all the CRITICAL POINTS, π, where π β² π =π or π β² π is undefined for π in the domain of π. It is important not to forget the points where πβ²(π) because these can produce local max/min. 2. Classify all critical points as local maximum, local mininum or neither β set up an interval chart and test a value in each interval to determine whether πβ²(π₯) is positive or negative and use this to classify .
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4.5 An Algorithm for Curve Sketching
STEP 5: The Second Derivative Test 1. Find all the points, π, where π " π =π or π " π does not exist. NOTE: this will automatically include points where π(π₯) or π β² π₯ is undefined. 2. Determine whether these are points of inflection. β set up an interval chart and test a value in each interval to determine whether π"(π₯) is positive or negative and use this to classify points of inflection. STEP 6: Graph the Function
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Sketch the following function
Practice: Sketch the following function π π₯ = 3π₯ 4 β8 π₯ 3 +6 π₯ 2 Step 1: π(π₯) is a polynomial function β π· π =β. Step 2: π(π₯) is a polynomial function β π π₯ is continuous for βπ₯ββ. Step 3: When π 0 =0βthe π¦βintercept occurs at 0,0 . π π₯ =0β0= 3π₯ 4 β8 π₯ 3 +6 π₯ 2 β0= π₯ 2 (3π₯ 2 β8π₯+6) π 2 β4ππ=64β4 18 =β8<0 β3 π₯ 2 β8π₯+6 has no real roots β there is only one π₯βintercept at (0,0) Step 4: π(π₯) is a polynomial function β there is no horizontal or oblique asymptote.
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Sketch the following function
Practice: Sketch the following function π π₯ = 3π₯ 4 β8 π₯ 3 +6 π₯ 2 Step 5: π(π₯) is a polynomial function β πβ²(π₯) is defined for βπ₯ββ. π β² π₯ =12 π₯ 3 β24 π₯ 2 +12π₯ πβ² π₯ =0β0=12 π₯ 3 β24 π₯ 2 +12π₯ β0= 12π₯ (π₯ 2 β2π₯+1) β0= 12π₯ (π₯β1) 2 βπ₯=0 and π₯=1 are critical points of π. Note: π 1 =1 Interval π<π π<π<π π>π Sign of πβ²(π₯) β + =β + + =+ π(π₯) decreasing increasing β΄ there is a local minimum at (0, 0).
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Sketch the following function
Practice: Sketch the following function π π₯ = 3π₯ 4 β8 π₯ 3 +6 π₯ 2 π β² π₯ =12 π₯ 3 β24 π₯ 2 +12π₯ Step 6: π(π₯) is a polynomial function β π"(π₯) is defined for βπ₯ββ. π" π₯ =36 π₯ 2 β48π₯+12 π" π₯ =0β0=36 π₯ 2 β48π₯+12 β0=3 π₯ 2 β4π₯+1 β0=(π₯β1)(3π₯β1) βπ₯=1 and π₯= 1 3 . Interval π< π π π π <π<π π>π Sign of π"(π₯) β β =+ β + =β + + =+ π(π₯) concave up concave down β΄ there are points of inflection at , and (1,1).
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In summary β¦ QUESTIONS: p #1, 2, 3a, 4, 5, 6
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