 4.1.1: Indicate all critical points on the graph of f in Figure 4.12and de...
 4.1.2: Graph a function which has exactly one critical point, atx = 2, and...
 4.1.3: Graph a function with exactly two critical points, one ofwhich is a...
 4.1.4: In Exercises 48, use derivatives to find the critical points andinf...
 4.1.5: In Exercises 48, use derivatives to find the critical points andinf...
 4.1.6: In Exercises 48, use derivatives to find the critical points andinf...
 4.1.7: In Exercises 48, use derivatives to find the critical points andinf...
 4.1.8: In Exercises 48, use derivatives to find the critical points andinf...
 4.1.9: In Exercises 912, find all critical points and then use the firstde...
 4.1.10: In Exercises 912, find all critical points and then use the firstde...
 4.1.11: In Exercises 912, find all critical points and then use the firstde...
 4.1.12: In Exercises 912, find all critical points and then use the firstde...
 4.1.13: In Exercises 1314, find the critical points of the function andclas...
 4.1.14: In Exercises 1314, find the critical points of the function andclas...
 4.1.15: (a) Use a graph to estimate the xvalues of any criticalpoints and ...
 4.1.16: In Exercises 1619, the function f is defined for all x. Use thegrap...
 4.1.17: In Exercises 1619, the function f is defined for all x. Use thegrap...
 4.1.18: In Exercises 1619, the function f is defined for all x. Use thegrap...
 4.1.19: In Exercises 1619, the function f is defined for all x. Use thegrap...
 4.1.20: (a) Show that if a is a positive constant, then x = 0 isthe only cr...
 4.1.21: (a) If b is a positive constant and x > 0, find all criticalpoints ...
 4.1.22: (a) If a is a nonzero constant, find all critical points off(x) = a...
 4.1.23: If U and V are positive constants, find all critical pointsofF(t) =...
 4.1.24: Indicate on the graph of the derivative function f in Figure4.13 th...
 4.1.25: Indicate on the graph of the derivative f in Figure 4.14the xvalue...
 4.1.26: Indicate on the graph of the second derivative f in Figure4.15 the ...
 4.1.27: For 2730, sketch a possible graph of y = f(x),using the given infor...
 4.1.28: For 2730, sketch a possible graph of y = f(x),using the given infor...
 4.1.29: For 2730, sketch a possible graph of y = f(x),using the given infor...
 4.1.30: For 2730, sketch a possible graph of y = f(x),using the given infor...
 4.1.31: Suppose f has a continuous derivative whose values aregiven in the ...
 4.1.32: (a) The following table gives values of the differentiablefunction ...
 4.1.33: If water is flowing at a constant rate (i.e., constant volumeper un...
 4.1.34: If water is flowing at a constant rate (i.e., constant volumeper un...
 4.1.35: Find and classify the critical points of f(x) = x3(1x)4as local max...
 4.1.36: If m, n 2 are integers, find and classify the criticalpoints of f(x...
 4.1.37: The rabbit population on a small Pacific island is approximatedbyP ...
 4.1.38: Find values of a and b so that the function f(x) =x2 + ax + b has a...
 4.1.39: Find the value of a so that the function f(x) = xeax hasa critical ...
 4.1.40: Find constants a and b in the function f(x) = axebxsuch that f( 13 ...
 4.1.41: Graph f(x) = x + sin x, and determine where f is increasingmost rap...
 4.1.42: You might think the graph of f(x) = x2 + cos x shouldlook like a pa...
 4.1.43: 4344 show graphs of the three functions f, f, f.Identify which is w...
 4.1.44: 4344 show graphs of the three functions f, f, f.Identify which is w...
 4.1.45: 4546 show graphs of f, f, f. Each of thesethree functions is either...
 4.1.46: 4546 show graphs of f, f, f. Each of thesethree functions is either...
 4.1.47: Use the derivative formulas and algebra to find the intervalswhere ...
 4.1.48: Let f be a function with f(x) > 0 for all x. Set g = 1/f.(a) If f i...
 4.1.49: In 4950, the differentiable function f has x = 1 asits only zero an...
 4.1.50: In 4950, the differentiable function f has x = 1 asits only zero an...
 4.1.51: In 5152, the graph of f lies entirely above the xaxisand f(x) < 0 f...
 4.1.52: In 5152, the graph of f lies entirely above the xaxisand f(x) < 0 f...
 4.1.53: In 5354, explain what is wrong with the statement.An increasing fun...
 4.1.54: In 5354, explain what is wrong with the statement.For any function ...
 4.1.55: In 5557, give an example of:A function which has no critical points...
 4.1.56: In 5557, give an example of:A function, f, which has a critical poi...
 4.1.57: In 5557, give an example of:A function with local maxima and minima...
 4.1.58: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.59: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.60: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.61: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.62: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.63: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.64: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.65: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.66: Are the statements in 5866 true or false for a functionf whose doma...
 4.1.67: In 6772, give an example of a function f thatmakes the statement tr...
 4.1.68: In 6772, give an example of a function f thatmakes the statement tr...
 4.1.69: In 6772, give an example of a function f thatmakes the statement tr...
 4.1.70: In 6772, give an example of a function f thatmakes the statement tr...
 4.1.71: In 6772, give an example of a function f thatmakes the statement tr...
 4.1.72: In 6772, give an example of a function f thatmakes the statement tr...
 4.1.73: Given that f(x) is continuous everywhere and changesfrom negative t...
Solutions for Chapter 4.1: USING FIRST AND SECOND DERIVATIVES
Full solutions for Calculus: Single and Multivariable  6th Edition
ISBN: 9780470888612
Solutions for Chapter 4.1: USING FIRST AND SECOND DERIVATIVES
Get Full SolutionsCalculus: Single and Multivariable was written by Patricia and is associated to the ISBN: 9780470888612. This textbook survival guide was created for the textbook: Calculus: Single and Multivariable , edition: 6. Chapter 4.1: USING FIRST AND SECOND DERIVATIVES includes 73 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 73 problems in chapter 4.1: USING FIRST AND SECOND DERIVATIVES have been answered, more than 11072 students have viewed full stepbystep solutions from this chapter.

Absolute value of a real number
Denoted by a, represents the number a or the positive number a if a < 0.

Algebraic model
An equation that relates variable quantities associated with phenomena being studied

Data
Facts collected for statistical purposes (singular form is datum)

Distance (on a number line)
The distance between real numbers a and b, or a  b

Doubleblind experiment
A blind experiment in which the researcher gathering data from the subjects is not told which subjects have received which treatment

Elimination method
A method of solving a system of linear equations

Exponential regression
A procedure for fitting an exponential function to a set of data.

Instantaneous velocity
The instantaneous rate of change of a position function with respect to time, p. 737.

Lefthand limit of f at x a
The limit of ƒ as x approaches a from the left.

Limaçon
A graph of a polar equation r = a b sin u or r = a b cos u with a > 0 b > 0

Linear regression
A procedure for finding the straight line that is the best fit for the data

Position vector of the point (a, b)
The vector <a,b>.

Quadratic equation in x
An equation that can be written in the form ax 2 + bx + c = 01a ? 02

Reference angle
See Reference triangle

Right circular cone
The surface created when a line is rotated about a second line that intersects but is not perpendicular to the first line.

Singular matrix
A square matrix with zero determinant

Sum of complex numbers
(a + bi) + (c + di) = (a + c) + (b + d)i

Summation notation
The series a nk=1ak, where n is a natural number ( or ?) is in summation notation and is read "the sum of ak from k = 1 to n(or infinity).” k is the index of summation, and ak is the kth term of the series

Symmetric about the yaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

Translation
See Horizontal translation, Vertical translation.
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