 3.2.1: 12 Use the given graph to estimate the value of each derivative. Th...
 3.2.2: 12 Use the given graph to estimate the value of each derivative. Th...
 3.2.3: Match the graph of each function in (a)(d) with the graph of its de...
 3.2.4: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.5: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.6: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.7: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.8: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.9: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.10: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.11: 411 Trace or copy the graph of the given function f. (Assume that t...
 3.2.12: Yeast population Shown is the graph of the population function Pstd...
 3.2.13: Tadpole weights The graph shows the average body weight W as a func...
 3.2.14: Ground reaction force in walking The graph shows the horizontal for...
 3.2.15: Marriage age The graph shows how the average age of first marriage ...
 3.2.16: 1618 Make a careful sketch of the graph of f and below it sketch th...
 3.2.17: 1618 Make a careful sketch of the graph of f and below it sketch th...
 3.2.18: 1618 Make a careful sketch of the graph of f and below it sketch th...
 3.2.19: Let fsxd x 2 . (a) Estimate the values of f9s0d, f9( 1 2), f9s1d, a...
 3.2.20: Let fsxd x 3 . (a) Estimate the values of f9s0d, f9( 1 2), f9s1d, f...
 3.2.21: 2131 Find the derivative of the function using the definition of a ...
 3.2.22: 2131 Find the derivative of the function using the definition of a ...
 3.2.23: 2131 Find the derivative of the function using the definition of a ...
 3.2.24: 2131 Find the derivative of the function using the definition of a ...
 3.2.25: 2131 Find the derivative of the function using the definition of a ...
 3.2.26: 2131 Find the derivative of the function using the definition of a ...
 3.2.27: 2131 Find the derivative of the function using the definition of a ...
 3.2.28: 2131 Find the derivative of the function using the definition of a ...
 3.2.29: 2131 Find the derivative of the function using the definition of a ...
 3.2.30: 2131 Find the derivative of the function using the definition of a ...
 3.2.31: 2131 Find the derivative of the function using the definition of a ...
 3.2.32: 3234 (a) Use the definition of the derivative to calculate f9. (b) ...
 3.2.33: 3234 (a) Use the definition of the derivative to calculate f9. (b) ...
 3.2.34: 3234 (a) Use the definition of the derivative to calculate f9. (b) ...
 3.2.35: Malarial parasites An experiment measured the number of malarial pa...
 3.2.36: Blood alcohol concentration Researchers measured the blood alcohol ...
 3.2.37: 3740 The graph of f is given. State, with reasons, the numbers at w...
 3.2.38: 3740 The graph of f is given. State, with reasons, the numbers at w...
 3.2.39: 3740 The graph of f is given. State, with reasons, the numbers at w...
 3.2.40: 3740 The graph of f is given. State, with reasons, the numbers at w...
 3.2.41: Graph the function fsxd x 1 s x  . Zoom in repeatedly, first towa...
 3.2.42: Zoom in toward the points (1, 0), (0, 1), and s21, 0d on the graph ...
 3.2.43: The figure shows the graphs of f, f9, and f 0. Identify each curve,...
 3.2.44: The figure shows graphs of f, f9, f 0, and f . Identify each curve...
 3.2.45: 4546 Use the definition of a derivative to find f9sxd and f 0sxd. T...
 3.2.46: 4546 Use the definition of a derivative to find f9sxd and f 0sxd. T...
 3.2.47: 4748 The graph of the derivative f9 of a function f is shown. (a) O...
 3.2.48: 4748 The graph of the derivative f9 of a function f is shown. (a) O...
 3.2.49: Recall that a function f is called even if fs2xd fsxd for all x in ...
 3.2.50: When you turn on a hotwater faucet, the temperature T of the water...
Solutions for Chapter 3.2: The Derivative as a Function
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 3.2: The Derivative as a Function
Get Full SolutionsBiocalculus: 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. Since 50 problems in chapter 3.2: The Derivative as a Function have been answered, more than 25081 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.2: The Derivative as a Function includes 50 full stepbystep solutions.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Outer product uv T
= column times row = rank one matrix.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.