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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

For Exercises determine whether each of the following statements is true or false, and explain why.If ƒ′(c) = 0, where c is a value in interval (a, b), then ƒ is a constant on the interval (a, b).
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).ƒ(x) = 8x2 + 6x + 5
For Exercises determine whether each of the following statements is true or false, and explain why.If an odd function has a y-intercept, it must pass through the origin.
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2). f(x) = x² + 7x³ x² 2
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).ƒ(x) = 3x2 - 4x + 8
For Exercises determine whether each of the following statements is true or false, and explain why.Every rational function has either a vertical or a horizontal asymptote.
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
For Exercises determine whether each of the following statements is true or false, and explain why.If ƒ″(c) = 0, the function does not have a relative maximum or minimum at c.
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).ƒ(x) = 4x4 - 3x3 - 2x2 + 6
For Exercises determine whether each of the following statements is true or false, and explain why.If ƒ″(c) = 0, the function has an inflection point at c.
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).ƒ(x) = 4x3 + 5x2 + 6x - 7
Describe how you would find the equation of the horizontal asymptote for the graph of f(x): = 3x² - 2x 2x² + 5
For Exercises determine whether each of the following statements is true or false, and explain why.If ƒ″(x) > 0 on an interval, the function is increasing on that interval.
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).ƒ(x) = 5x3 - 7x2 + 4x + 3
For Exercises determine whether each of the following statements is true or false, and explain why.The acceleration is the second derivative of the position function.
Determine whether each statement is true or false, and explain why.If ƒ′(c) = 0 and ƒ″(c) > 0, then by the second derivative test, ƒ(c) is a relative maximum.
For Exercises determine whether each of the following statements is true or false, and explain why.If f'(c) exists, f"(c) also exists.
Determine whether each statement is true or false, and explain why.If a function has an inflection point at x = c, then ƒ″(c) = 0.
For Exercises determine whether each of the following statements is true or false, and explain why.If f is continuous on (a, b), f'(x) < 0 on (a, c), and f'(x) > 0on (c, b), then f has a relative minimum at c.
If ƒ″(5) = 0, then ƒ must have an inflection point at x = 5.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.If ƒ″(x) > 0 on an interval, then the function is concave upward on the interval.
For Exercises determine whether each of the following statements is true or false, and explain why.If c is a critical number, then the function must have a relative maximum or minimum at c.
The first derivative test can be used to determine concavity.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.Acceleration is the instantaneous rate of change of velocity.
For Exercises determine whether each of the following statements is true or false, and explain why.If f'(x) > 0 on an interval, the function is positive on thatinterval.
Every function has a relative maximum and a relative minimum.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.Velocity may be positive or negative, but speed is always positive.
For Exercises determine whether each of the following statements is true or false, and explain why.A critical number c is a number in the domain of a function ffor which f'(c) = 0 or f'(c) does not exist.
Use the following information to answer Exercises. A certain drug is given to a patient every 12 hours. The steady state concentration function is given by Css(t) = 1.99De -0.14t - 1.62De-2.08t mcg/mL, where D is the size of the dose in milligrams.1. If a 500-mg dose is given every 12
If f(x) is an even function, then f'(x) is an even function.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.If a function is concave upward on an interval, it must be increasing on that interval.
Consider the graphs of the function y = √2x - 1 and the straight line y = x + k. Discuss the number of points of intersection versus the change in the value of k.
If ƒ′(x) < 0 for every x on an interval, then ƒ is negative on that interval.Determine whether each statement is true or false, and explain why.
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. f(x)* 2 + -2 + 2 4 + X
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.h(x) = 5 - x1/3
In each figure, the graph of the derivative of y = f(x) is shown. Find the locations of all relative extrema of f(x). -2 f'(x) 4 -4+ 4 X
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. f(x) |||| -2 2 + 2 X
A relative maximum is the greatest possible value of a function.Determine whether each statement is true or false, and explain why.
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. f(x) 2. 2/4 X
A critical number for ƒ must be in the domain of ƒ.Determine whether each statement is true or false, and explain why.
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. g(x) ↑ 2- -2 -2 2 X
Determine whether each statement is true or false, and explain why.If a function has a relative extremum, then it must occur at a critical number or at an endpoint.
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. f(x) |||| -2 2 2 X
The derivative in regions separated by critical numbers must alternate between positive and negative.Determine whether each statement is true or false, and explain why.
A function has relative extrema at all critical numbers of the function.Determine whether each statement is true or false, and explain why.
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. 14 H h(x) { -4 -2 -2 2 H 2 X
If ƒ is increasing on an interval, then ƒ′(x) > 0 for each x in that interval.Determine whether each statement is true or false, and explain why.
A function can have only one relative maximum and one relative minimum.Determine whether each statement is true or false, and explain why.
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. + g(x) -2 -2 2 X
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. g(x) ++ -2 -2 -2 T +++ 2 4 A X
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. g(x) H -2 -2 H 2 4, HA fa X
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. || | -4 f(x) -2 2 X
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. h(x)↑ A 1 |||| -4 -2 -2 2 x
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. h(x) -2 + ++ 4 2 X
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. h(x) -2 + 2/4 A X
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. --4 f(x) -2 -2 + 2 X
Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercises in the preceding section. f(x) ↑ + 111 + -2 -2. T 2 ++ X
For each of the exercises listed below, suppose that the function that is graphed is not f(x) but f′(x). Find the locations of all relative extrema, and tell whether each extremum is a relative maximum or minimum.Data from Exercise 6Find the locations and values of all relative extrema for the
For each of the exercises listed below, suppose that the function that is graphed is not ƒ′(x), but ƒ′(x). Find the open intervals where ƒ′(x) is (a) Increasing or (b) Decreasing.Exercise 5Find the open intervals where the functions graphed as follows are f(x) 王 -2 2 + 2 X
Find the open intervals where the functions graphed as follows are (a) Increasing, or (b) Decreasing. f(x) 1 -2 本 + 2 X
For each of the exercises listed below, suppose that the function that is graphed is not f(x) but f′(x). Find the locations of all relative extrema, and tell whether each extremum is a relative maximum or minimum.Data from Exercise 5Find the locations and values of all relative extrema for the
For each of the exercises listed below, suppose that the function that is graphed is not f(x) but f′(x). Find the locations of all relative extrema, and tell whether each extremum is a relative maximum or minimum.Data from Exercise 11Find the locations and values of all relative extrema for the
For each of the exercises listed below, suppose that the function that is graphed is not ƒ′(x) , but ƒ′(x). Find the open intervals where ƒ′(x) is (a) Increasing or (b) Decreasing.Exercise 11Find the open intervals where the functions graphed as follows are f(x) A 11 ET -2 + X 2
For each of the exercises listed below, suppose that the function that is graphed is not ƒ′(x), but ƒ′(x). Find the open intervals where ƒ′(x) is (a) Increasing or (b) Decreasing.Exercise 6Find the open intervals where the functions graphed as follows are – f(x) ↑ 2. -2 2/
For each of the exercises listed below, suppose that the function that is graphed is not ƒ′(x) , but ƒ′(x). Find the open intervals where ƒ′(x) is (a) Increasing or (b) Decreasing.Exercise 12Find the open intervals where the functions graphed as follows are + f(x) -2 本 ²f 2 X
For each of the exercises listed below, suppose that the function that is graphed is not f(x) but f′(x) . Find the locations of all relative extrema, and tell whether each extremum is a relative maximum or minimum.Exercise 12Data from Exercise 12Find the locations and values of all relative
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing. f(x) || 2/3 好 - x² - 24x - 4
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x2 - 10x + 33
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.y = 2.3 + 3.4x - 1.2x2
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing. 2 f(x) = ² x ² − x² x³ = - 3 x² - 4x + 2
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x2 + 8x + 5
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) 4 3 X3 - 21 x2 2 5x + 8
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.y = 1.1 - 0.3x - 0.3x2
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x3 + 6x2 + 9x - 8
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x3 + 3x2 - 24x + 2
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.g(x) = x3 + 3x2 + 3x + 4
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) = 3 x3 رانه + 3x – 4
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = 4x3 - 15x2 - 72x + 5
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.g(x) = -x3 + 6x2 - 12x - 3
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = 4x3 - 9x2 - 30x + 6
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = x4 + 4x3 + 4x2 + 1
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.ƒ(x) = 3x4 + 8x3 - 18x2 + 5
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x): = (5 - 9x)2/3 7 +1
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x4 - 18x2 - 4
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.y = -3x + 6
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing. f(x) = x + 3 x - 4
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing. 2 y = √x² + 1
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = x4 - 8x2 + 9
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing.y = 6x - 9
For each function in Exercises, find (a) The critical numbers;(b) The open intervals where the function is increasing; and(c) The open intervals where it is decreasing. y = -√9+ x²
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = 3 - (8 + 3x)2/3
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = 2x + 3x2/3
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) = x - - X
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.ƒ(x) = 3x5/3 - 15x2/3
Find the x-value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema. f(x) = x² + X

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