Economics 205 Quiz 1Joel Watson, Fall 20061. Consider an arbitrary function f :[0, 1] → [0, 1]. Define the sequence of real numbers{xn}∞n=1inductively by (i) x1= 0, and (ii) for each n ∈ P, xn+1= f(xn).(a) Given the assumptions made above, must it b e the case that {xn} converges? If so,explain why. If not, provide a counterexample.(b) Given the assumptions made ab ove, must it be the case that {xn} has a convergentsubsequence? If so, explain why. If not, provide a counterexample.(c) Suppose that f(x)=(x +1)/3. Write the first few elements of {xn} and calculatelimn→∞xn.2. Calculate the following limits. In the case in which one or both of the limits does notexist, state this.(a) limn→∞xn, where the sequence {xn} is defined by xn=(2n2+3)(4n+1)4n3for all n ∈ P(b) limx→1(x−1)2lnx13. For both cases below, write expressions for h0(x).(a) h(x) ≡ [f(x)]−2[1 − f(x)](b) h(x) ≡ f(3x + g ( x))4. Prove by induction: 1 + 2 + 3 + ...+ n = n(n +1)/2.5. Supp ose f : R → R is differentiable and satisfies f(0) = 0 and f0(0) = 3. Prove thatthere is a number σ>0 such that f(x) > 0 for every x ∈ (0,σ).2Economics 205 Quiz 2Joel Watson, Fall 20061. Consider the function f :(0, ∞) → R defined by f(x) ≡ x2− 8x +6lnx.(a) Calculate the first four derivatives of f evaluated at the point c =1.(b) Using your observation of the pattern that develops in part (a), write an expressionfor f(k)(1) (the kth derivative of f at c =1)fork ≥ 3.(c) Write the third degree Taylor polynomial of f, centered at the point c = 1, as a functionof x.(d) Suppose you want to use a Taylor polynomial to estimate the value f(2). Write theerror term of the nth degree Taylor polynomial, as a function of n and t,forx =2. (Centerthe polynomial at c = 1.) Find a conv enient upper bound on the absolute value of theerror term and use it to determine a value of n (as small as you can find) such that theerrorissmallerthan1/10.(e) Determine the critical points of f.(f) Determine the nature (local maximizer, minimizer, neither) of the critical points.12. Use matrix algreba to find the vector x∗that solves the following system of equations:Ax=y, where A =3 −112 , x =x1x2 ,andy =38 . Show your work.3. Calculate10xexdx.4. Consider the subset of R3defined as {x ∈ R3| x1x22− e2x3=2}. Find the plane tangentto this set at the point ˆx =(3, 1, 0). Represen t the plane as a subset of R3.5. Suppose f : R → R+and g : R → R+are both concave down, where R+denotes theset of nonnegative real numbers. Must it be the case that h(x) ≡ f(x)g(x)isalsoconcavedown? Prove or provide a counterexample if you can.2Economics 205 Quiz 3Joel Watson, Fall 20061. Consider the function f : R2×R → R defined by f(x1,x2; a) ≡ 11ax2+x1x2−ax21−x22.Think of a as a parameter in the problem of maximizing f(x1,x2; a) by choice of x.(a) Calculate the first- and second-derivative matrices of f with resp ect to x =(x1,x2).(b) Calculate the critical point(s) of f (with respect to x, as a function of a) and determinethe nature of each.(c) Calculate v0(3), where v(a) ≡ maxx∈R2f(x, a).2. The equation x2+ xyez= z − 6 defines a curved surface in R3. Find the equation ofthe tangent plane to this surface at the point (2, −5, 0).13. Consider the function F : R3→ R2defined by F (x, z1,z2)= xz1− z22z31z2− 16!. Considerwhether the identity F (x, z1,z2) ≡ 00!implicitly defines z = z1z2!as a function ofx in a neighborhood of ( x0,z01,z02)=(2, 2, 2). Is this the case? If so, write z = g(x) andcalculate Dg (2).4. Recall that a function f : Rn→ R is concave if, for all x, y ∈ Rnand every λ ∈ (0, 1),we have f(λx +(1− λ)y) ≥ λf(x)+(1− λ)f(y). Prove that if f is concave and x∗is alocal maximizer of f then x∗is a global maximizer of f.2Economics 205 Final ExaminationProf. Watson, Fall 2006You have three hours to complete this closed-book examination. You may use scratchpaper, but please write your final answers (including your complete arguments) on thesesheets.1. Consider the sequence {xn} that is defined inductively by x1=12and, for every n ∈ P,xn+1=1+xn2.(a) Write the first four numbers in this sequence.(b) Write xnas a function of n (not as a function of xn−1).(c) Does {xn} converge? If so, to what number?2. Consider the function f : R → R defined by f(x)=(x sin(1/x)ifx 6=00ifx =0. Formallywrite what it means for f to be continuous at x = 0. Show that f is continuous at x =0by finding an appropriate δ value for each ε.13. For each of the following matrices, say whether or not it is invertible (that is, whetherA−1exists) and, if so, find A−1.(a) A = 24510!(b) A = 27311!(c) A =120002−10 −123.4. Consider the function f :(0, ∞) → R that is given by f(x)=x2− ln x.(a) Write the second-degree Taylor polynomial for f centered at the point c =1.(a) Suppose you want to approximate f(2) to within 1/100 using a Taylor polynomialcentered at c = 1. Write the expression for the error term of the nth degree Taylorpolynomial and find a convenient value of n to achieve the desired bound.25. Consider the function f :(0, ∞) → R defined by f(x)=30+x2− 14x + 20 ln x.(a) Calculate f0(x) and f00(x).(b) Determine the critical points of f.(c) Determine the nature (lo cal maximizer, minimizer, or neither) of each critical point.(d) Calculate the tangent line to the graph of f at the point x =1.6. Let f : R2→ R be given by f(x, y)=xy +2x +5y − x2− y2. Solve max f(x, y)byfinding the global maximizer (x∗,y∗). Verify first- and second-order conditions.37. Consider the functions f : R3→ R2and g : R2→ R given byf(x1,x2,x3)= y1y2!= x2ex1− x1x23x1x2x3− x32!and g(y1,y2)=y1y2.(a) Compute Df(0, 2, 1). Note that your answer will be a 2 × 3 matrix.(b) Compute Dh(0, 2, 1), where h( x1,x2,x3) ≡ [g ◦ f](x1,x2,x3). Note that your answerwill be a 1 × 3 matrix.8. Consider the function f : R2→ R defined by f(x, y)=7xy − y3ex+1.(a) Does the identity f(x, y) = 0 define y as a function of x near the p oint ( x0,y0)=(0, 1)?If so, and letting g denote the implicit function (so we have y = g(x)), calculate g0(0).(b) Determine the plane tangent to the graph of f at the point (x0,y0,z0)=(0, 1, 0).49. Consider the function f : R → R defined by f(x)=x3− 9x2+15x + 74.(a) Solve max f(x) subject to x ∈ [2, 6]. Find the maximizer and note conditions youcheck.(b) Solve min f(x) subject to x ≥ 0. Find the minimizer and note conditions you check.10. Solve the problem max