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# Solutions for Chapter 5.3: The Fundamental Theorem of Calculus

## Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition

ISBN: 9781133109631

Solutions for Chapter 5.3: The Fundamental Theorem of Calculus

Solutions for Chapter 5.3
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##### ISBN: 9781133109631

Since 85 problems in chapter 5.3: The Fundamental Theorem of Calculus have been answered, more than 25803 students have viewed full step-by-step solutions from this chapter. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Chapter 5.3: The Fundamental Theorem of Calculus includes 85 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• Column space C (A) =

space of all combinations of the columns of A.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Pseudoinverse A+ (Moore-Penrose inverse).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

• Reduced row echelon form R = rref(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Singular Value Decomposition

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

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