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# Honors Calculus II MATH 1512

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This 0 page Class Notes was uploaded by Chelsea Nolan MD on Monday November 2, 2015. The Class Notes belongs to MATH 1512 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/233956/math-1512-georgia-institute-of-technology-main-campus in Mathematics (M) at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Math 1512 Honors Calculus Fall 2007 FINAL EXAM REVIEW QUESTIONS These questions are intended as a review for the nal exam There will be just one hard proof77 on the nal and it will be selected from questions 1 3 below There will also be one question on differential equations and it will be similar to one of questions 48 50 below H D 00 4 CH 00w lf1111n is a basis for R and T R a R is a one to one linear transformation prove that T11 Tvn is also a basis for R Let S e1 en be an orthonormal basis for a nite dirnensional Euclidean space V Prove that for every x E V we have x Zltzejgtej 7 H Let W be a subspace of a nite dirnensional Euclidean space V and let p denote the orthogonal projection of an element x E V onto W Prove that H96 WQH S H96 7 yll for all y E W with equality if and only if y De ne what it means for a set S of vectors in a linear space V to be linearly independent De ne what it means for a linear space V to be nite dimensional De ne what it means for a linear transformation T R a R to be diagonalz39zable De ne what it means for a sequence can of real numbers to converge Deterrnine all solutions to the system 5x2y7622u71 z7yziu72 H H H D H H H 00 Find an equation of the form y of bx c for the unique parabola passing through the points 0 0 1 1 4 8 Find a scalar parametric equation for the plane through the three points 273717 27 17 37 and 47374 Show that for any two vectors A B E R we have HA BHZ 711A 7 B112 4A B Find the reduced row echelon form and the rank of the matrix 3 0 0 0 A 1 2 71 1 1 i4 2 72 Find all real It for which the two vectors t 1t and tt 1 are linearly dependent ls S xyz E R3 z y z 71 a subspace of R3 Why or why not Let B denote the linear space of all real polynomials of degree at most 71 ls S f E B f 0 3f0 a subspace of P Find a basis for the subspace W of R4 spanned by the vectors 171717 7 272727217171r127072707 and determine the dimension of W Let P2 be the linear space of all polynomials of degree at most 2 Find the coordinates of t272t3 relative to the ordered basis 1 3t71 tzit for P2 Find an orthonormal basis for the subspace of R4 spanned by 1 1 00 0 1 1 1 777 and 10 0 1 Let W Q R4 be the linear span of 10 0 1 and 23 1 0 Find a basis for the orthogonal complement of W with respect to the standard inner product 2 O 21 22 2 2 2 2 C40 4 CT CT In the real linear space C071 with inner product given by ltfg f01fxgxdx7 nd the linear polynomial g Closest to the function fx em Let 2 2 1 A 3 1 7 b 0 0 1 0 Find the best approximate least squares solution to the overdeter mined system of equations Ax b Let T R2 a R3 be a linear transformation for which T17 0 101 and T01 71071 Determine the rank and nullity of T Find the inverse of the matrix 2 3 4 A 2 1 1 71 1 2 Compute the determinant of the matrix 1 1 0 A 0 1 1 1 2 3 Find all real or complex eigenvalues and eigenvectors of the matrix 0 0 1 0 0 0 0 1 A 7 1 0 0 0 0 1 0 0 ls the matrix 0 71 0 A 0 0 1 71 i3 3 diagonalizable Justify your answer I 00 K O H D 00 4 CT CT I Find a nonsingular matrix C and a diagonal matrix D for which D C lAC7 where 4 1 71 A 2 5 i2 1 1 2 If T is the linear transformation whose matrix with respect to the standard ordered basis of R2 is 1 7 2 3 7 nd the matrix of T with respect to the ordered basis b1 17 71b2 27 1 of R2 Let T be the linear transformation from R2 to R2 given by re ection around the line y 2x a Find the matrix of T with respect to the ordered basis b1 12b2 721 of R2 b Use part a and the change of basis formula to nd the matrix of T with respect to the standard ordered basis 61 62 of R2 Compute a closed form formula for Ak7 where k is a positive integer and 1 1 A 0 2 Show that a 0 Evaluate limTHOOC 2 Evaluate limmace Evaluate limnaoocos Evaluate 220 Does the series 221 5517 converge or diverge Justify your answer Does the series 221 converge or diverge Justify your answer 4 3 00 3 4 O 4 H 4 D 4 C40 4 4 45 4 CT 4 I 48 4 00 inkii Does the series Zk 71M ally7 or diverge Justify your answer converge absolutely7 converge condition Give an upper bound for the error when the partial surn 2 is i i i i 00 used to estimate the sum of the in nite series Zk W Expand g x3 sin2x2 as a power series in x Find the Taylor series for 6 centered around z 17 and prove that this series converges to 6 for all real x Use the Lagrange form of the remainder for Taylor series to estimate V83 to within 3 decimal places Find the radius of convergence and interval of convergence for the power k series 221 Ex Find the interval of convergence for the power series 7172z712 3z 713 i 4z 714 2 4 8 16 Find the sum of the in nite series 221 Find a simple closed forrn expression for the function fx given by the i i i i 00 k2 in nite series expansion f Zk Use power series to evaluate fol sin2d to within 2 decimal places Find a system of 3 linear rst order ODE7s which is equivalent to the third order ODE y 7 2y 3y 0 Calculate 6 4 when In H DOC OOH OHH Hintz What is A37

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