Test 2 Review - UH

1 Test 2 ReviewJiwen HeDepartment of Mathematics, University of jiwenhe/Math1431 Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20081 / 69 Online QuizzesAll current and previous quizzes in Math 1431 are now openeduntil November He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20082 / 69 Review for Test 2 Review for Test 2 by Prof. 8:00 - 10:00pm in 100 SECJ iwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20083 / 69 Good Sources of Practice ProblemsExamples from basic homework basic online quiz He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20084 / 69 Section Differentialsincrement: f=f(x+h) f(x)differential.

2 Df=f (x)h f dfin the sense that f dfhtends to 0 ash He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20085 / 69 Quiz 1 Quiz 1 Use differentials to estimate 26, by using your knowledge of None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20086 / 69 Section Newton-Raphson ApproximationNewton MethodLet the numbercbe a solution (root) of an equationf(x) = Newton-Raphson methodxn+1=xn f(xn)f (xn),n= 0,1, ,generates a sequence of approximationsx1,x2, ,xn, thatwill converge to the rootcJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20087 / 69 Quiz 2 Quiz 2 Use 1 iteration of Newton s method to estimate 26, startingfrom a guess of 5, by noting that 26 is a root ofx2 26 = None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20088 / 69 Section The Mean-Value TheoremTheoremIf f is differentiable on the open interval(a,b)and continuous onthe closed interval[a,b], then there is at least one number c in(a,b)for whichf (c) =f(b) f(a)b aor equivalentlyf(b) f(a) =f (c)

3 (b a).Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 20089 / 69 Quiz 3 Give the number of values in (0,2 ) where the MVThm is 0b. 1c. 2d. 3e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200810 / 69 Section Increasing and Decreasing FunctionsTheoremA function f is increasing on aninterval I iff is continuous andf (x)>0at all but finitely manyvalues in I .A function f is decreasing on aninterval I iff is continuous andf (x)<0at all but finitely manyvalues in I .Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200811 / 69 Examplef(x) =45x5 3x4 4x3+ 22x2 24x+ 6,f (x) = 4(x+ 2)(x 1)2(x 3)fis continuous increasing on ( , 2],decreasing on [ 2,3], and increasingon [3, ).

4 Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200812 / 69 Quiz 4 Assume the domain offis all real numbers. The graph off (x) isshown below. Give the number of intervals of increase 1b. 2c. 3d. 4e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200813 / 69 Quiz 5 Assume the domain offis all real numbers. The graph off (x) isshown below. Give the number of intervals of decrease 1b. 2c. 3d. 4e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200814 / 69 Section Local Extreme ValuesJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200815 / 69 Section Critical NumberJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200816 / 69 Quiz 6 Assume the domain offis all real numbers.

5 The graph off (x) isshown below. Give the number of critical values 2b. 3c. 4d. 5e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200817 / 69 Section First Derivative TestJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200818 / 69 Section Second Derivative TestJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200819 / 69 Quiz 7 Assume the domain offis all real numbers. The graph off (x) isshown below. Give the number of local minima 1b. 2c. 3d. 4e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200820 / 69 Quiz 8 Assume the domain offis all real numbers.

6 The graph off (x) isshown below. Give the number of local maxima 1b. 2c. 3d. 4e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200821 / 69 Section Absolute Max/Min offon [a,b]Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200822 / 69 Example: Abosolute Max/Min offon [a,b]f(x) =x 2 sinx,0 x 2 ,f (x) = 1 2 cosx,0 x 2 .f (x) = 0 atx= /3,5 continuous on [0,2 ].fis decreasing on [0, /3], increasingon [ /3,5 /3], and decreasing on[5 /3,2 ].Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200823 / 69 Section Absolute Max/Min offon [a, ) or ( ,b]Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200824 / 69 Quiz 9 Assume the domain offis all real numbers.

7 The graph off (x) isshown below. Give the number of absolute minima 1b. 2c. 3d. 4e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200825 / 69 Quiz 10 Assume the domain offis all real numbers. The graph off (x) isshown below. Give the number of absolute maxima 1b. 2c. 3d. 4e. None of theseJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200826 / 69 Section Some Max-Min ProblemsJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200827 / 69 Section Concavity and Points of InflectionDefinitionThe graph offisconcave uponIiff graph offisconcave downonIiff that join arcs ofopposite concavityarepoints He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28.

8 200828 / 69 ExampleDetermine the intervals on whichfincreases and the intervalson the intervals on which the graph offis concave upand the intervals on which the graph offis concave thex-coordinates of the points of He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200829 / 69 Section Second-Derivative TestTheoremIf f (x)>0for all x in I , then f increaseson I , and thegraph of f is concave f (x)<0for all x in I , then f decreaseson I , and thegraph of f is concave the point(c,f(c))is apoint of inflection, then eitherf (c) = 0orf (c)does not He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200830 / 69 ExampleDetermine concavity and find thepoints of inflection of the graph off(x) =x+ cosx,x [0,2 ].

9 F (x) = 1 sinx,f (x) = He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200831 / 69 Section Vertical AymptotesTypically, to locate the vertical asymptotes for a functionf,find the valuesx=cat whichfis discontinuousand determine the behavior vertical linex=cis a vertical asymptote forfif any one ofthe following conditions holdsf(x) or asx c+;f(x) or asx c ;f(x) or asx He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200832 / 69 Section Vertical Aymptotes: Rational FunctionThe linex= 4 is avertical asymptoteforf(x) =3x+ 6x2 2x 8=3(x+ 2)(x+ 2)(x 4).Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200833 / 69 Section Aymptotes: Rational Functionf(x) =xx 2 The linex= 2 is avertical liney= 1 is ahorizontal He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200834 / 69 Section Behavior of Rational Function asx LetR(x) =anxn+ +a1x+a0bkxk+ +b1x+b0be a rational function.

10 Thenifn<k,R(x) 0 asx ;ifn=k,R(x) anbnasx ;ifn>k,R(x) asx .Jiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200835 / 69 Section Aymptotes: Rational Functionf(x) =5 3x21 x2 The linesx= 1 arevertical liney= 3 is ahorizontal He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200836 / 69 Section Some Curve SketchingSketch the graph offStep 1: Domain offStep 2: InterceptsStep 3: Symmetry and PeriodicityStep 4: First Derivativef Step 5: Second Derivativef Step 6: Preliminary sketchStep 7: Sketch the graphJiwen He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200837 / 69 Problem 1 Use differentials to estimate He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200838 / 69 Problem 2 Use differentials to estimate sin He, University of HoustonMath 1431 Section 24076, Test 2 ReviewOctober 28, 200839 / 69 Problem 3A metal sphere with a radius of 10 cm is to be covered with a coating of silver.