[SOLVED] Sketch the graphs of the following functions.

> Find two positive numbers, x and y, whose sum is 100 and whose product is as large as possible.

> Find the dimensions of the rectangular garden of greatest area that can be fenced off (all four sides) with 300 meters of fencing.

> A farmer has $1500 available to build an E-shaped fence along a straight river so as to create two identical rectangular pastures. (See Fig. 13.) The materials for the side parallel to the river cost $6 per foot, and the materials for the three sections

> A canvas wind shelter for the beach has a back, two square sides, and a top. Find the dimensions for which the volume will be 250 cubic feet and that requires the least possible amount of canvas.

> Find the dimensions of the closed rectangular box with square base and volume 8000 cubic centimeters that can be constructed with the least amount of material.

> A closed rectangular box with a square base and a volume of 12 cubic feet is to be constructed from two different types of materials. The top is made of a metal costing $2 per square foot and the remainder of wood costing $1 per square foot. Find the dim

> A rectangular garden of area 75 square feet is to be surrounded on three sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that minimize the cost of materials.

> Consider the problem of finding the dimensions of the rectangular garden of area 100 square meters for which the amount of fencing needed to surround the garden is as small as possible. (a) Draw a picture of a rectangle and select appropriate letters for

> Postal requirements specify that parcels must have length plus girth of at most 84 inches. Consider the problem of finding the dimensions of the square-ended rectangular package of greatest volume that is mailable. (a) Draw a square-ended rectangular box

> Find the slope of the tangent line to the curve y = (x2 - 15)6 at x = 4. Then write the equation of this tangent line.

> Figure 12(b) shows an open rectangular box with a square base. Consider the problem of finding the values of x and h for which the volume is 32 cubic feet and the total surface area of the box is minimal. (The surface area is the sum of the areas of the

> There are $320 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $6 per foot and the fencing for the other three sides costs $2 per foot. [See Fig. 12(a).] Consider the problem of finding the dimensi

> Find the positive values of x, y, and z that maximize Q = xyz, if x + y = 1 and y + z = 2. What is this maximum value?

> Find the positive values of x and y that minimize S = x + y if xy = 36, and find this minimum value.

> In Exercise 7, can there be a maximum for Q = x2 + y2 if x + y = 6? Justify your answer. Exercise 7: Find the minimum of Q = x2 + y2 if x + y = 6.

> Find the minimum of Q = x2 + y2 if x + y = 6.

> Find two positive numbers x and y that maximize Q = x2y if x + y = 2.

> Find the maximum of Q = xy if x + y = 2.

> Find the x-intercepts of the given function. y = 4 - 2x - x2

> Find the x-intercepts of the given function. y = 2x2 + 5x + 2

> Write the equation of the tangent line to the curve y = x3 + 3x - 8 at (2, 6).

> Find the x-intercepts of the given function. y = x2 + 5x + 5

> Find the x-intercepts of the given function. y = x2 - 3x + 1

> The canopy height (in meters) of the tropical bunch-grass elephant millet t days after mowing (for t Ú 32) is f (t) = -3.14 + .142t - .0016t2 + .0000079t3 - .0000000133t4. (Source: Crop Science.) (a) Graph f (t) in the window [32, 250] by [-1.2, 4.5]. (b

> In a medical experiment, the body weight of a baby rat in the control group after t days was f (t) = 4.96 + .48t + .17t2 - .0048t3 grams. (Source: Growth, Development and Aging.) (a) Graph f (t) in the window [0, 20] by [-12, 50]. (b) Approximately how m

> If f (a) = 0 and f (x) is decreasing at x = a, explain why f (x) must have a local maximum at x = a.

> If f (a) = 0 and f (x) is increasing at x = a, explain why f (x) must have a local minimum at x = a.

> Find the quadratic function f (x) = ax2 + bx + c that goes through (0, 1) and has a local minimum at (1,-1).

> Find the quadratic function f (x) = ax2 + bx + c that goes through (2, 0) and has a local maximum at (0, 1).

> Determine which function is the derivative of the other. f(x) g(x) y X

> Determine which function is the derivative of the other. f(x) |(x)b y

> Find the slope of the tangent line to the curve y = x3 + 3x – 8 at (2, 6).

> Sketch the graphs of the following functions for x > 0. y = 1/√x + x/2

> Sketch the graphs of the following functions for x > 0. y = 6√x - x

> Sketch the graphs of the following functions for x > 0. y = 1/x2 + x/4 – 5/4

> Sketch the graphs of the following functions for x > 0. y = 2/x +x/2 + 2

> Sketch the graphs of the following functions for x > 0. y = 12/x + 3x + 1

> Sketch the graphs of the following functions for x > 0. y = 9/x + x + 1

> Sketch the graphs of the following functions for x > 0. y = 2/x

> Sketch the graphs of the following functions for x > 0. y = 1/x + ¼ x

> Sketch the graphs of the following functions. f (x) = (x + 2)4 - 1

> Sketch the graphs of the following functions. f (x) = (x - 3)4

> Find the slope of the graph of y = f (x) at the designated point. f (x) = x10 + 1 + √(1 – x), (0, 2)

> Differentiate. y = 1/3x3

> Sketch the graphs of the following functions. f (x) = 3x4 - 6x2 + 3

> Sketch the graphs of the following functions. f (x) = x4 - 6x2

> Sketch the graphs of the following functions. f (x) = 1/3 x3 - 2x2

> Sketch the graphs of the following functions. f (x) = 1 - 3x + 3x2 - x3

> Sketch the graphs of the following functions. f (x) = -3x3 - 6x2 - 9x - 6

> Sketch the graphs of the following functions. f (x) = 2x3 + x - 2

> Sketch the graphs of the following functions. f (x) = 5 - 13x + 6x2 - x3

> Sketch the graphs of the following functions. f (x) = x3 + 2x2 + 4x

> Sketch the graphs of the following functions. f (x) = x3 + 3x + 1

> Find the slope of the graph of y = f (x) at the designated point. f (x) = 3x2 - 2x + 1, (1, 2)

> Sketch the graphs of the following functions. f (x) = -x3

> Sketch the graphs of the following functions. f (x) = x3 - 6x2 + 12x - 6

> Show that the function f (x) = -x3 + 2x2 - 6x + 3 is always decreasing.

> Show that the function f (x) = 1/3 x3 - 2x2 + 5x has no relative extreme points.

> Find the x-intercepts of the given function. y = 3x2 + 10x + 3

> Find the x-intercepts of the given function. y = 4x - 4x2 - 1

> The graph of the function has one relative maximum and one relative minimum point. Find these points using the first-derivative test. Use a variation chart as in Example 1. Variation Chart from Example 1: f (x) = x3 - 6x2 +1 Critical Values Interv

> The graph of the function has one relative maximum and one relative minimum point. Find these points using the first-derivative test. Use a variation chart as in Example 1. Variation Chart from Example 1: f (x) = x3 - 27x Critical Values Intervals

> Index-Fund Fees Suppose that the cost function in Exercise 45 is C(x) = -2.5x + 1, where x% is the index-fund fee. (The company has a fixed cost of $1 billion, and the cost decreases as a function of the index-fund fee.) Find the value of x that maximize

> When a mutual fund company charges a fee of 0.47% on its index funds, its assets in the fund are $41 billion. And when it charges a fee of 0.18%, its assets in the fund are $300 billion. (Source: The Boston Globe.) (a) Let x % denote the fee that the com

> Differentiate. y = (x – 1/x)-1

> The population (in millions) of the United States (excluding Alaska and Hawaii) t years after 1800 is given by the function f (t) in Fig. 18(a). The graphs of f â(t) and f ââ(t) are shown in Figs. 18(

> Consider the graph of g(x) in Fig. 17. (a) If g(x) is the first derivative of f (x), describe f (x) when x = 2? (b) If g(x) is the second derivative of f (x), describe f (x) when x = 2? Figure 17: y Figure 17 2 y = g(x) x

> Determine which function is the derivative of the other. x- (x)b = h (x)f=h fi

> Determine which function is the derivative of the other. fi y = f(x) y = g(x)

> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. f (x) = 5x2 + x - 3

> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. g(x) = x2 + 10x + 10

> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. g(x) = 3 + 4x - 2x2

> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. f (x) = 5 - 12x - 2x2

> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. f (x) = 1/4 x2 - 2x + 7

> Let a, b, c, d be fixed numbers with a ≠ 0, and let f (x) = ax3 + bx2 + cx + d. Is it possible for the graph of f (x) to have more than one inflection point? Explain your answer.

> Differentiate. f (x) = (√x/2 + 1)3/2

> Let a, b, c be fixed numbers with a ≠ 0, and let f (x) = ax2 + bx + c. Is it possible for the graph of f (x) to have an inflection point? Explain your answer.

> Sketch the following curves, indicating all relative extreme points and inflection points. y = x4 – 4/3 x3

> Sketch the following curves, indicating all relative extreme points and inflection points. y = 2x3 - 3x2 - 36x + 20

> Sketch the following curves, indicating all relative extreme points and inflection points. y = x4 + 1/3 x3 - 2x2 - x + 1

> Sketch the following curves, indicating all relative extreme points and inflection points. y = 1/3 x3 - x2 - 3x + 5

> Sketch the following curves, indicating all relative extreme points and inflection points. y = -x3 + 12x - 4

> Sketch the following curves, indicating all relative extreme points and inflection points. y = 1 + 3x2 - x3

> Sketch the following curves, indicating all relative extreme points and inflection points. y = x3 - 6x2 + 9x + 3

> Sketch the following curves, indicating all relative extreme points and inflection points. y = x3 - 3x + 2

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. ƒ(x) = {x³ + 2x² − 5x +

> Differentiate. y = π2x

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. f(x) = −x³ + 2x² - 12

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. 3 f(x) = 2x³ − 15x² + 36x - 24

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. 3 f(x) = -√√x³ + x² + 9x

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. f(x) = x³ + 9x - 2

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. f(x) = x³ - 12x

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. f(x) = x³ = x²

> The graph of the function has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. f (x) = x3 + 6x2 + 9x

> The graph of the function has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f (x) has a relative minimum point when a > 0 and a relativ

> Differentiate. y = x + 1 + √(x + 1)

> The graph of the function has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f (x) has a relative minimum point when a > 0 and a relativ

Question: Sketch the graphs of the following functions.