MATH 16A, SUMMER 2008, REVIEW SHEET FOR FINAL EXAMBENJAMIN JOHNSONThe final exam will be held Thursday, August 14, from 8:10AM to 10:00AM in 3 Evans. Theexam will be cumulative, but will focus more on chapters 3, 4, 5, and 6 of the textbook.To do well on this exam you should be able to answer any simple question related to the essentialideas in calculus, including some material covered in chapters 0-2 (such as taking the derivative ofa function). You can assume that any material on the exam will be either recent material fromchapters 3-6; or material that has been covered extensively, used consistently since its introductionin the course, and appeared at least once on a previous quiz, midterm, or midterm review sheet.Consult your previous midterm review sheets, midterm solutions, and quiz solutions for highlights ofmaterial from earlier chapters. At least 60% (and probably more) of the exam will focus exclusivelyon material covered since the midterm exam. In relation to that material, you should be able to doat least each of the following:Chapter 3. TECHNIQUES OF DIFFERENTIATION3.1 The Product and Quotient Rules(1) State the product rule and the quotient rule.(2) Use the product rule or quotient rule to differentiate functions [such as f (x) =(x2+ 1)(x3+ 2x + 3) or g(x) =x2−13x2+2].3.2 The Chain Rule and the General Power Rule(1) Express a function as a composition of two functions. [For example, if h(x) =x+23x4, find f and g such that h = f ◦ g].(2) State the chain rule using either Newton’s notation or Leibniz’ notation.(3) Use the chain rule to compute the derivative of a function (such as y = (x2+ 1)2).3.3 Implicit Differentiation and Related Rates(1) Use implicit differentiation to find the slope of the tangent line to a curve (whenthe curve is not expressed as a function). [For example, find the slope of thetangent to the curve x2+ y2= 4 at the point (1, −√3)].(2) Use implicit differentiation to finddydx,dxdy,dydt, ordxdtgiven an equation relating xand y [such as y + 2x + xy = 38]. Answer a similar question if the letters aresomething other than x, y, and t.(3) Solve an application problem involving related rates. [For example, if a 10 footladder is sliding down a wall at a rate of 4 feet per second, how fast is the laddermoving along the ground when the ladder is 5 feet from the ground?]Chapter 4. THE EXPONENTIAL AND NATURAL LOGARITHM FUNCTIONS4.1 Exponential Functions(1) Use properties of exponents to simplify expressions involving exponents. [Forexample, write (24x· 2−x)12in the form 2kxfor some constant k.](2) Solve for x an equation in which x occurs as an exponent, [for example, 27 = 35x].4.2 The Exponential Function exDate: August 14, 2008.12 BENJAMIN JOHNSON(1) Give a numerical approximation to the number e that is accurate to within onedecimal place.(2) Compute the derivative of ex.(3) Sketch a graph of ex.4.3 Differentiation of Exponential Functions(1) Use the chain rule for exponential functions to compute the derivative of a function[such as f (x) = ex2+1].(2) Given real numbers k and C, determine all functions f satisfying f0(x) = kf (x)for all real numbers x, and f (0) = C.(3) Sketch a graph of ekxgiven a real number k.4.4 The Natural Logarithm Function(1) Simplify an expression involving natural logarithms, [for example, eln 4+2 ln 3].(2) Sketch a graph of the curve y = ln x.4.5 The Derivative of ln x(1) Compute the derivative of f(x) = ln x, or g(x) = ln |x|.(2) Use the chain rule with logarithmic functions to compute the derivative of afunction such as y = ln(3x2+ 4x + 2).4.6 Properties of the Natural Logarithm Function(1) Simplify expressions involving natural logarithms using properties of logarithms.[For example, write ln 5 + 2 ln 3 as a single logarithm].(2) Use the properties of natural logarithms to simplify an expression before takingthe derivative. [For example, to take the derivative of f (x) = ln[x(x + 1)(x +2)(x + 3)], you would want to simplify first using properties of logarithms].(3) Use logarithmic differentiation to compute the derivative of a function involvingmultiple instances of multiplication and division of simpler functions, [for example,g(x) = (x2+ 1)(x3− 3)(2x + 5)].Chapter 5. APPLICATIONS OF THE EXPONENTIAL AND NATURAL LOGARITHMFUNCTIONS5.1 Exponential Growth and Decay(1) Recognize the equivalence of a function f satisfying the differential equationf0(x) = kf (x) and f having the form f (x) = Cekxfor some constant C.(2) Determine the growth constant for an exponential growth function in an applica-tion problem, given either a differential equation, the doubling time, or any twovalues of the function.(3) Determine the decay constant for an exponential decay function given a differen-tial equation, the half-life, or any two points of the function.(4) Determine the half-life of an exponential decay function given the decay constant.(5) Answer any basic set of application questions regarding an exponential growthfunction or an exponential decay function as exemplified in the homework prob-lems from this section.5.2 Compount Interest(1) State the formula for calculating the amount of money in an account given aprinciple (initial) amount P , a continuously compounded interest rate r, and atime duration t.(2) Express the number e as a limit of an appropriate function (which does not involvee).(3) Find the amount of time it takes for an investment to double (or triple) given acontinuously compounding interest rate.MATH 16A, SUMMER 2008, REVIEW SHEET FOR FINAL EXAM 35.3 Applications of the Natural Logarithm Function to Economics(1) Compute the relative rate of change of a function usingf0(t)f(t)=ddxln f(t).(2) Answer an application problem involving relative rates of change.(3) Compute the elasticity of demand from a demand equation.(4) Answer an application question involving elasticity of demand.5.4 Further Exponential Models(1) Identify the learning curve function f (t) = M(1 − e−kt) with the differentialequation f0(t) = k(M − f (t))(2) Identify the logistic growth curve y =M1+B−M ktwith the differential equationy0= ky(M − y).(3) Answer any sequence of application questions involving the learning curve func-tion, the logistic growth curve, or any other function model in which variablesappear as an exponent.Chapter 6. THE DEFINITE INTEGRAL6.1 Antidifferentiation(1) Find an antiderivative for a variety or simple functions, [for example x2, ex, or1x].(2) Find a formula for a function f given its derivative