 9.2.1: The temperature T (in C) at a location in the Northern Hemisphere d...
 9.2.2: At the beginning of this section we discussed the function I fsT, H...
 9.2.3: The windchill index W is the perceived temperature whenthe actual ...
 9.2.4: The wave heights h in the open sea depend on the speed v of the win...
 9.2.5: 56 Determine the signs of the partial derivatives for the function ...
 9.2.6: 56 Determine the signs of the partial derivatives for the function ...
 9.2.7: If fsx, yd 16 2 4x 2 2 y 2 , find fxs1, 2d and fys1, 2d and interpr...
 9.2.8: A contour map is given for a function f. Use it to estimate fxs2, 1...
 9.2.9: 932 Find the first partial derivatives of the function. fsx, yd y 5...
 9.2.10: 932 Find the first partial derivatives of the function. fsx, yd x 4...
 9.2.11: 932 Find the first partial derivatives of the function. fsx, td e2t...
 9.2.12: 932 Find the first partial derivatives of the function. fsx, td sx ...
 9.2.13: 932 Find the first partial derivatives of the function. z s2x 1 3yd10
 9.2.14: 932 Find the first partial derivatives of the function. z tan xy
 9.2.15: 932 Find the first partial derivatives of the function. fsx, yd x 2...
 9.2.16: 932 Find the first partial derivatives of the function. fsx, yd x y
 9.2.17: 932 Find the first partial derivatives of the function. w sin cos
 9.2.18: 932 Find the first partial derivatives of the function. w evu 1 v 2
 9.2.19: 932 Find the first partial derivatives of the function. fsr, sd r l...
 9.2.20: 932 Find the first partial derivatives of the function. fsx, td arc...
 9.2.21: 932 Find the first partial derivatives of the function. u tewyt
 9.2.22: 932 Find the first partial derivatives of the function. fsx, yd yxy...
 9.2.23: 932 Find the first partial derivatives of the function. fsx, y, zd ...
 9.2.24: 932 Find the first partial derivatives of the function. fsx, y, zd ...
 9.2.25: 932 Find the first partial derivatives of the function. w lnsx 1 2y...
 9.2.26: 932 Find the first partial derivatives of the function. w zexyz
 9.2.27: 932 Find the first partial derivatives of the function. u xe2t sin
 9.2.28: 932 Find the first partial derivatives of the function. u x yyz
 9.2.29: 932 Find the first partial derivatives of the function. fsx, y, z, ...
 9.2.30: 932 Find the first partial derivatives of the function. fsx, y, z, ...
 9.2.31: 932 Find the first partial derivatives of the function. u sx 21 1 x...
 9.2.32: 932 Find the first partial derivatives of the function. u sinsx1 1 ...
 9.2.33: 3336 Find the indicated partial derivative. fsx, yd ln(x 1 sx 2 1 y...
 9.2.34: 3336 Find the indicated partial derivative. fsx, yd arctans yyxd; f...
 9.2.35: 3336 Find the indicated partial derivative. fsx, y, zd yx 1 y 1 z; ...
 9.2.36: 3336 Find the indicated partial derivative. fsx, y, zd ssin2x 1 sin...
 9.2.37: 3740 Use implicit differentiation to find zyx and zyy. x 2 1 y 2 1 ...
 9.2.38: 3740 Use implicit differentiation to find zyx and zyy. yz lnsx 1 z
 9.2.39: 3740 Use implicit differentiation to find zyx and zyy. x 2 z arctan...
 9.2.40: 3740 Use implicit differentiation to find zyx and zyy. sinsxyzd x 1...
 9.2.41: Body surface area A model for the surface area of a human body is g...
 9.2.42: The windchill index is modeled by the function W 13.12 1 0.6215T 2...
 9.2.43: Blood flow One of Poiseuilles laws states that the resistance of bl...
 9.2.44: Antibiotic concentration If an antibiotic is administered to a pati...
 9.2.45: Flapping and gliding In the project on page 297 we expressed the po...
 9.2.46: Dialysis removes urea from a patients blood by diverting some blood...
 9.2.47: Lizard energy expenditure The average energy E (in kcal) needed for...
 9.2.48: Snake reversals and stripes In a study of the survivorship of juven...
 9.2.49: The van der Waals equation for n moles of a gas is SP 1 n2 a V 2 Ds...
 9.2.50: a) The gas law for a fixed mass m of an ideal gas at absolute tempe...
 9.2.51: 5156 Find all the second partial derivatives. fsx, yd x 3y 5 1 2x 4y
 9.2.52: 5156 Find all the second partial derivatives. fsx, yd sin2smx 1 nyd
 9.2.53: 5156 Find all the second partial derivatives. w su2 1 v 2
 9.2.54: 5156 Find all the second partial derivatives. v xyx 2 y
 9.2.55: 5156 Find all the second partial derivatives. z arctan x 1 y1 2 xy
 9.2.56: 5156 Find all the second partial derivatives. v exey
 9.2.57: 5758 Verify that the conclusion of Clairauts Theorem holds, that is...
 9.2.58: 5758 Verify that the conclusion of Clairauts Theorem holds, that is...
 9.2.59: 5964 Find the indicated partial derivative(s). fsx, yd 3xy 4 1 x 3y...
 9.2.60: 5964 Find the indicated partial derivative(s). fsx, td x 2e2ct; ftt...
 9.2.61: 5964 Find the indicated partial derivative(s). fsx, y, zd coss4x 1 ...
 9.2.62: 5964 Find the indicated partial derivative(s). fsr, s, td r lnsrs 2...
 9.2.63: 5964 Find the indicated partial derivative(s). u er sin ; 3ur 2
 9.2.64: 5964 Find the indicated partial derivative(s). u x a y bz c; 6ux y ...
 9.2.65: If fsx, y, zd xy 2 z 3 1 sec2 (xsz ), find fxzy. [Hint: Which order...
 9.2.66: If tsx, y, zd s1 1 xz 1 s1 2 xy , find txyz. [Hint: Use a different...
 9.2.67: Verify that the function u e22k2t sin kx is a solution of the heat ...
 9.2.68: Determine whether each of the following functions is a solution of ...
 9.2.69: Show that each of the following functions is a solution of the wave...
 9.2.70: Diffusion equation Verify that the function csx, td 1 s4Dt e2x2ys4D...
 9.2.71: If u xey 1 yex , show that 3 u x 3 1 3 u y 3 x 3 u xy 2 1 y 3 u x 2 y
 9.2.72: Show that the CobbDouglas production function P bL K satisfies the...
 9.2.73: You are told that there is a function f whose partial derivatives a...
Solutions for Chapter 9.2: Partial Derivatives
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 9.2: Partial Derivatives
Get Full SolutionsThis textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Since 73 problems in chapter 9.2: Partial Derivatives have been answered, more than 27182 students have viewed full stepbystep solutions from this chapter. Chapter 9.2: Partial Derivatives includes 73 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column space C (A) =
space of all combinations of the columns of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.