 11.3.11.3.1: Fill in the blanks. _______ is the study of the rates of change of ...
 11.3.11.3.2: Fill in the blanks. The _______ to the graph of a function at a poi...
 11.3.11.3.3: Fill in the blanks.A _______ is a line through the point of tangenc...
 11.3.11.3.4: Fill in the blanks.
 11.3.11.3.5: Fill in the blanks.
 11.3.11.3.6: In Exercises 512, use the limit process to find the slope of the gr...
 11.3.11.3.7: In Exercises 512, use the limit process to find the slope of the gr...
 11.3.11.3.8: In Exercises 512, use the limit process to find the slope of the gr...
 11.3.11.3.9: In Exercises 512, use the limit process to find the slope of the gr...
 11.3.11.3.10: In Exercises 512, use the limit process to find the slope of the gr...
 11.3.11.3.11: In Exercises 512, use the limit process to find the slope of the gr...
 11.3.11.3.12: In Exercises 512, use the limit process to find the slope of the gr...
 11.3.11.3.13: In Exercises 1318, find a formula for the slope of the graph of f a...
 11.3.11.3.14: In Exercises 1318, find a formula for the slope of the graph of f a...
 11.3.11.3.15: In Exercises 1318, find a formula for the slope of the graph of f a...
 11.3.11.3.16: In Exercises 1318, find a formula for the slope of the graph of f a...
 11.3.11.3.17: In Exercises 1318, find a formula for the slope of the graph of f a...
 11.3.11.3.18: In Exercises 1318, find a formula for the slope of the graph of f a...
 11.3.11.3.19: In Exercises 1924, use a graphing utility to graph the function and...
 11.3.11.3.20: In Exercises 1924, use a graphing utility to graph the function and...
 11.3.11.3.21: In Exercises 1924, use a graphing utility to graph the function and...
 11.3.11.3.22: In Exercises 1924, use a graphing utility to graph the function and...
 11.3.11.3.23: In Exercises 1924, use a graphing utility to graph the function and...
 11.3.11.3.24: In Exercises 1924, use a graphing utility to graph the function and...
 11.3.11.3.25: In Exercises 2538, find the derivative of the function.
 11.3.11.3.26: In Exercises 2538, find the derivative of the function.
 11.3.11.3.27: In Exercises 2538, find the derivative of the function.
 11.3.11.3.28: In Exercises 2538, find the derivative of the function.
 11.3.11.3.29: In Exercises 2538, find the derivative of the function.
 11.3.11.3.30: In Exercises 2538, find the derivative of the function.
 11.3.11.3.31: In Exercises 2538, find the derivative of the function.
 11.3.11.3.32: In Exercises 2538, find the derivative of the function.
 11.3.11.3.33: In Exercises 2538, find the derivative of the function.
 11.3.11.3.34: In Exercises 2538, find the derivative of the function.
 11.3.11.3.35: In Exercises 2538, find the derivative of the function.
 11.3.11.3.36: In Exercises 2538, find the derivative of the function.
 11.3.11.3.37: In Exercises 2538, find the derivative of the function.
 11.3.11.3.38: In Exercises 2538, find the derivative of the function.
 11.3.11.3.39: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.40: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.41: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.42: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.43: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.44: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.45: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.46: In Exercises 3946, (a) find the slope of the graph of f at the give...
 11.3.11.3.47: In Exercises 47 50, use a graphing utility to graph f over the inte...
 11.3.11.3.48: In Exercises 47 50, use a graphing utility to graph f over the inte...
 11.3.11.3.49: In Exercises 47 50, use a graphing utility to graph f over the inte...
 11.3.11.3.50: In Exercises 47 50, use a graphing utility to graph f over the inte...
 11.3.11.3.51: In Exercises 51 54, find the derivative of f. Use the derivative to...
 11.3.11.3.52: In Exercises 51 54, find the derivative of f. Use the derivative to...
 11.3.11.3.53: In Exercises 51 54, find the derivative of f. Use the derivative to...
 11.3.11.3.54: In Exercises 51 54, find the derivative of f. Use the derivative to...
 11.3.11.3.55: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.56: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.57: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.58: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.59: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.60: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.61: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.62: In Exercises 5562, use the function and its derivative to determine...
 11.3.11.3.63: Population The projected populations (in thousands) of New Jersey f...
 11.3.11.3.64: Market Research The data in the table shows the number N (in thousa...
 11.3.11.3.65: Rate of Change A spherical balloon is inflated. The volume is appro...
 11.3.11.3.66: Rate of Change An approximately spherical benign tumor is reducing ...
 11.3.11.3.67: Vertical Motion A water balloon is thrown upward from the top of an...
 11.3.11.3.68: Vertical Motion A Sacajawea dollar is dropped from the top of a 120...
 11.3.11.3.69: True or False? In Exercises 69 and 70, determine whether the statem...
 11.3.11.3.70: True or False? In Exercises 69 and 70, determine whether the statem...
 11.3.11.3.71: Library of Parent Functions In Exercises 7174, match the function w...
 11.3.11.3.72: Library of Parent Functions In Exercises 7174, match the function w...
 11.3.11.3.73: Library of Parent Functions In Exercises 7174, match the function w...
 11.3.11.3.74: Library of Parent Functions In Exercises 7174, match the function w...
 11.3.11.3.75: Think About It Sketch the graph of a function whose derivative is a...
 11.3.11.3.76: Think About It Sketch the graph of a function whose derivative is a...
 11.3.11.3.77: In Exercises 7780, sketch the graph of the rational function. As sk...
 11.3.11.3.78: In Exercises 7780, sketch the graph of the rational function. As sk...
 11.3.11.3.79: In Exercises 7780, sketch the graph of the rational function. As sk...
 11.3.11.3.80: In Exercises 7780, sketch the graph of the rational function. As sk...
 11.3.11.3.81: In Exercises 8184, find the cross product of the vectors.
 11.3.11.3.82: In Exercises 8184, find the cross product of the vectors.
 11.3.11.3.83: In Exercises 8184, find the cross product of the vectors.
 11.3.11.3.84: In Exercises 8184, find the cross product of the vectors.
Solutions for Chapter 11.3: The Tangent Line Problem
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 11.3: The Tangent Line Problem
Get Full SolutionsPrecalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Since 84 problems in chapter 11.3: The Tangent Line Problem have been answered, more than 45885 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 11.3: The Tangent Line Problem includes 84 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

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!

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.