Math1a Spring 2011Final ReviewMay 9, 2011Tuesday, May 10, 2011FunctionsTuesday, May 10, 2011Important Functionssincostanpolynomialsrational functionstrig polynomialexplogsqt = √arcsinarccosarctanAddSubtractComposeMultiplyDivideTuesday, May 10, 2011sin(x)xsin(0)=0Tuesday, May 10, 2011cos(x)xcos(0)=1Tuesday, May 10, 2011tan(x)xtan(0)=0Tuesday, May 10, 2011cot(x)xcot(π/2)=0Tuesday, May 10, 2011Trig FunctionsTuesday, May 10, 2011Problem 1sin(5 π/4)= ?cos(3 π/2)= ?tan(-π/6)= ?Tuesday, May 10, 2011exp(x)1xexp(a+b)=exp(a) exp(b)exp(0) = 1a = exp(b log(a))bTuesday, May 10, 2011log|x|1xlog(a b)=log(a) + log(b)log(1) = 0exp(log(a)) = aTuesday, May 10, 2011Problem 2exp(0)+exp(1)log(e) + log(1)elog(2) 3Tuesday, May 10, 2011xx2Tuesday, May 10, 2011xx3Tuesday, May 10, 20111/xxTuesday, May 10, 2011arcsin(x)xπ/21Tuesday, May 10, 2011arccos(x)xπ/2Tuesday, May 10, 2011arctan(x)xTuesday, May 10, 2011Problem 3arcsin(1)=?arccos(1)=?arctan(1) = ? Tuesday, May 10, 2011even odd or neithermonotoincally increasing, decreasingconcave up, concave downinflection points, roots, critical pointsThings to look out for:Tuesday, May 10, 2011LimitsTuesday, May 10, 2011Hopitals rule 0/0 88/HealingSimplifyHopital rulesMedicineHopitals rule Tuesday, May 10, 2011sin(3x)/x 2/x-2x/x(exp(4x)-1)/xtan(x)/x(cos(5x)-1)/x^2Problem 4: find the following limits for x->02Tuesday, May 10, 2011Problem 5What is the limit of x - 1x - 113as x goes to 1? Tuesday, May 10, 2011ContinuityTuesday, May 10, 2011Definitionaf(a)f(x) f(a) if x a Tuesday, May 10, 2011Discontinuity typesJumpInfinityOscillateTuesday, May 10, 2011Tuesday, May 10, 2011Most functions are nice and continuousTuesday, May 10, 2011Most functions are nice and continuousBe aware of the mean ones howeverTuesday, May 10, 2011Problem 6are the following functions continuous? cos(x) /log|x|on the real line? sin(sin(x))/xTuesday, May 10, 2011TheoremsTuesday, May 10, 2011Intermediate Value theoremRolle theoremMean value theoremFermat theoremWobbly Table theoremTheorems on ContinuityTuesday, May 10, 2011Intermediate valueTuesday, May 10, 2011RolleTuesday, May 10, 2011Mean ValueTuesday, May 10, 2011FermatTuesday, May 10, 2011The Wobbly Chair TheoremDocumenting moment of Discovery: Sunday, May 8, 2011 Tuesday, May 10, 2011DifferentiationTuesday, May 10, 2011f(x+h)-f(x)hlimh->0Differential quotientTuesday, May 10, 2011Matching Problemsff’Tuesday, May 10, 2011Matching Problemsff’Tuesday, May 10, 2011Product rule (f g)’ = f’g+fg’Addition rule (f+g)’ = f’+g’Chain rule f(g)’ = f’(g(x)) g’(x)Quotient rule (f/g)’ = (f’g-fg’)/gReciprocal rule (1/f)’ = -f’/f22Tuesday, May 10, 2011Related RatesTuesday, May 10, 2011Problem 7: an other ladder problemTuesday, May 10, 2011Find y’ if x=2,y=1 and x(t)=1+t xy(x + y) + y = 1022Find y’(t) at t=1.Tuesday, May 10, 2011Problem 8: Implicit differentiation Problem: y = x - x2 5Oscar Bolza1857-1942Bolza curveFind the slopeat x=1/2Tuesday, May 10, 2011ExtremizationTuesday, May 10, 2011Problem 9:Find the local and global maxima -12 x - 4 x + 3 x432on the interval [-2,3]Tuesday, May 10, 2011The function�2�1123�30�20�10102030Tuesday, May 10, 2011Problem 10:yx2x+y=10 Find maximal area.Tuesday, May 10, 2011IntegrationTuesday, May 10, 2011Substitution ∫ f(g(x)) g’(x) dx =∫f(u)duParts ∫ uv’dx = uv - ∫u’ v dxPartial Fraction (x-a)(x-b) (x-a) + (x-b)Trig Substitution =1 1 1∫√1-x 2dx=∫cos(u) du2 Integration TechniquesTuesday, May 10, 2011Problem 11: Integrate∫ (x+1) exp(x+2) dx3∫ √1-x dx2∫ 1/cos(x)dxu=tan(x/2)dx=2du/(1+u )sin(x)=2u/(1+u)cos(x)=(1-u )/(1+u )2222Tuesday, May 10, 2011Area ComputationTuesday, May 10, 2011a bfgTuesday, May 10, 2011Problem 12Find the area enclosed between sin(x) and sin (x)20.51.01.52.02.53.00.20.40.60.81.0Tuesday, May 10, 2011Volume ComputationTuesday, May 10, 2011Problem 13 Find volumef(z) = 1+z2-1<z<11-1cylindrical slicesand shellπ 56/15Tuesday, May 10, 2011Improper integrals“I love you from here to infinity!”Tuesday, May 10, 2011�1.0�0.50.00.51.02004006008001000�1.0�0.50.00.51.05101520f(x)=1/x 2f(x)=1/√|x|Problem 13: Which converges?∫f(x) dx-11Tuesday, May 10, 2011Book to winTuesday, May 10, 2011Yell the answer:(you have to give a short explanation)Tuesday, May 10, 2011No computers or Wolfram alpha!Tuesday, May 10, 20113No computers or Wolfram alpha!Tuesday, May 10, 201132No computers or Wolfram alpha!Tuesday, May 10, 201132 1No computers or Wolfram alpha!Tuesday, May 10, 201112∫e Problem 14: What is:-x /7xdx082Tuesday, May 10, 2011Tuesday, May 10, 2011ApplicationsTuesday, May 10, 2011Statistics: PDF and CDFBusiness: Marginal, total,average costPhysics: position, velocity acceleration, free fallMusic: hull function, scaleComputer science: curvature, ChaikinEconomics: Strawberry theoremStatistics: PDF and CDFGastronomy: wobbly tablePsychology: perception catastropheNumerics: Simpson, Newton mathodEntertainment: Math storiesTuesday, May 10, 2011Problem 15F(x) = -x log(x) 2Verify that g(x) = F(x)/xis extremal at F’=gTuesday, May 10, 2011Problem 16A car accelerates with a(t) = 1+sin(t)It is at x=0 at time t=0 and has zero velocity then. Where is it at time t=2π? Tuesday, May 10, 2011Tuesday, May 10, 2011Problem 17Find the curvature of log(cos(x)) at x=π/4�1.0�0.50.51.0�1.0�0.8�0.6�0.4�0.2Tuesday, May 10, 2011Last Problem 18f(x) = sin(1000 sin(x *x)) f(x) = sin(1000 x) sin(x)) f(x) = sin(10 sin(100x))) ABCWhich function is that?Tuesday, May 10, 2011What is Calculus?Tuesday, May 10, 2011What is Calculus?A course which fulfills a general education requirement.Tuesday, May 10, 2011What is Calculus?A course which fulfills a general education requirement.The dissection of lifeTuesday, May 10, 2011What is Calculus?A course which fulfills a general education requirement.The dissection of lifeThe language of the planetTuesday, May 10, 2011What is Calculus?A course which fulfills a general education requirement.The dissection of lifeThe language of the planetSee the world through the eyes of a genius (Newton)Tuesday, May 10, 2011What is Calculus?A course which fulfills a general education requirement.The dissection of lifeInteresting, tedious and not too badThe language of the planetSee the world through the eyes of a genius (Newton)Tuesday, May 10, 2011What is Calculus?A course which fulfills a general education requirement.The dissection of lifeInteresting,