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mathematical applications for the management

Mathematical Applications For The Management, Life And Social Sciences 12th Edition Ronald J. Harshbarger, James J. Reynolds - Solutions

In Problem, find the indicated derivative.Find y(4) if y' = √4x - 1.
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→-4 x2 - 16/x + 4
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.R(x) = [x2(x2 + 3x)]4
Use the graph to perform the evaluations (a)–(f) and to answer (g)–(i). If no value exists, so indicate.(a) f (1)(b) lim x→6 f(x)(c) limx→3- f(x)(d) limx→-4 f(x)(e) limx→∞ f (x)(f) Estimate f'(4).(g) Find all x-values where f'(x) does not exist.(h) Find all x-values where
For each function in Problems, approximate f' (a) in the following ways.(a) Use f (a + h) - f (a)/h with h = 0.0001.(b) Graph the function on a graphing calculator. Then zoom in near the point until the graph appears straight, pick two points, and find the slope of the line you see.f'(3) for f (x)
In Problem, find the derivative of each function.w = 5u8/5 - 3u5/6 + u1/3 + 5
In Problem, find the indicated derivative.Find y(5) if d2y/dx2 = 3√3x + 2.
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→3 x2 - 9/x - 3
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.c(x) = [x3(x2 + 1)]-3
Given that the line y = 2/3x - 8 is tangent to the graph of y = f (x) at x = 6, find(a) f'(6).(b) f(6).(c) The instantaneous rate of change of f (x) with respect to x at x = 6.
In Problem, find the derivative of each function.f (x) = 5x-4/5 + 2x-4/3
In Problem, find the indicated derivative.Find f(6)(x) if f(4)(x) = x(x + 1)-1.
Each of Problem contains a function and its graph. For each problem, answer parts (a) and (b).(a) Use the graph to determine, as well as you can,(i) Vertical asymptotes. (ii) limx→∞ f (x).(iii) limx→-∞ f (x). (iv)
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→-5 x2 + 8x + 15/x2 + 5x
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only. 4 2х - 1 y = * + x ||
The graph shows the total revenue and total cost functions for a company. Use the graph to decide (and justify) at which of points A, B, and C(a) Profit is the greatest.(b) There is a loss.(c) Producing and selling another item will increase profit.(d) The next item sold will decrease profit. R(x)
In Problem, find the derivative of each function.f (x) = 6x-8/3 - x-2/3
At the indicated point, for each function in Problems, find(a) The slope of the tangent line.(b) The instantaneous rate of change of the function.y = (4x3 - 5x + 1)3 at (1, 0)
In Problem, find the indicated derivative.Find f(3)(x) if f'(x) = x2/x2 + 1.
Each of Problem contains a function and its graph. For each problem, answer parts (a) and (b).(a) Use the graph to determine, as well as you can,(i) Vertical asymptotes. (ii) lim x→∞ f (x).(iii) lim x→-∞ f (x). (iv) Horizontal
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→7 x2 - 8x + 7/x2 - 6x - 7
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only. 5 - 3 y 4 ||
In Problem, write the equation of the tangent line to the graph of the function at the indicated point. Check the reasonableness of your answer by graphing both the function and the tangent line. 3x – 2x – 1 y = at x = 1 4 - x?
In Problem, find the derivative of each function.g (x) = 3/x5 + 2/x4 + 6 3√x
In Problem, suppose thatWhat is limx→-1 f(x)? x² + 1 if x
If f(x) = 16x2 - x3, what is the rate of change of f'(x) at (1, 15)?
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→-1/4 x + 4x2/x2 - 1/16
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.g (x) = (8x4 + 3)2(x3 - 4x)3
In Problem, write the equation of the tangent line to the graph of the function at the indicated point. Check the reasonableness of your answer by graphing both the function and the tangent line. x² - 4x y = 2х - х3 at x = 2 %3D ||
In Problem, suppose thatWhat is limx→0 f(x), if it exists? x² + 1 if x
If y = 36x2 - 6x3 + x, what is the rate of change of y' at (1, 31)?
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→1/2 8x2 - 4x/x - 1/2
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.y = (3x3 - 4x)3(4x2 - 8)2
In Problem, suppose thatWhat is limx→1 f(x), if it exists? x² + 1 if x
In Problem, write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct.y = √3x2 - 2 at x = 3
In Problem, cost, revenue, and profit are in dollars and x is the number of units.Suppose that the total revenue function for a product is R(x) = 50x and that the total cost function isC (x) = 1900 + 30x + 0.01x2.(a) Find the profit from the production and sale of 500 units.(b) Find the marginal
In Problem, complete (a) and (b).(a) Use analytic methods to evaluate each limit.(b) What does the result from part (a) tell you about horizontal asymptotes?You can verify your conclusions by graphing the functions with a graphing calculator. lim x→-∞ x3 - 1/x3 + 4
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→10 3x2 - 30x/x2 - 100
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only. Vx + 5 f(x) 4 - x
In Problem, write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct. 1 at x = 2 (x³ – x)3
In Problem, cost, revenue, and profit are in dollars and x is the number of units.Suppose that the total revenue function is given byR (x) = 46xand that the total cost function is given byC(x) = 100 + 30x + 0.1x2(a) Find P(100).(b) Find the marginal profit function.(c) Find M̅P̅ at x = 100 and
In Problem, complete (a) and (b).(a) Use analytic methods to evaluate each limit.(b) What does the result from part (a) tell you about horizontal asymptotes?You can verify your conclusions by graphing the functions with a graphing calculator. 3x + 2 lim x2 – 4 X - 00
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→-6 2x2 - 72/3x2 + 18x
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only. V2x - — 1 g(x) 2х + 1
In Problem, write the equation of the tangent line to each curve at the indicated point. As a check, graph both the function and the tangent line.f (x) = 4x2 - 1/x at x = -1/2
In Problem, complete (a) and (b).(a) Use analytic methods to evaluate each limit.(b) What does the result from part (a) tell you about horizontal asymptotes?You can verify your conclusions by graphing the functions with a graphing calculator. 5x - 4x lim X -00 3x - 2
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→-2 x2 + 4x + 4/x2 + 3x + 2
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.y = x2 4√4x - 3
In Problem, write the equation of the tangent line to each curve at the indicated point. As a check, graph both the function and the tangent line.f (x) = x3/3 - 3/x3 at x = -1
In Problem, complete (a) and (b).(a) Use analytic methods to evaluate each limit.(b) What does the result from part (a) tell you about horizontal asymptotes?You can verify your conclusions by graphing the functions with a graphing calculator. 4x + 5x lim x - 4x X00
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→10 x2 - 8x - 20/x2 - 11x + 10
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.y = 3x 3√4x4 + 3
In Problem, complete (a) and (b).(a) Use analytic methods to evaluate each limit.(b) What does the result from part (a) tell you about horizontal asymptotes?You can verify your conclusions by graphing the functions with a graphing calculator. Зx2 + 5х lim бх + 1
In Problem, use properties of limits and algebraic methods to find the limits, if they exist. Į 10 – 2x if x 3
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.c(x) = 2x√x3 + 1
In Problem, complete the following.(a) Find the derivative of each function and check your work by graphing both your calculated derivative and the numerical derivative of the function.(b) Use your graph of the derivative to find points where the original function has horizontal tangent lines.(c)
In Problem, complete (a) and (b).(a) Use analytic methods to evaluate each limit.(b) What does the result from part (a) tell you about horizontal asymptotes?You can verify your conclusions by graphing the functions with a graphing calculator. 5x – 8 lim X - 00 4x + 5x
In Problem, use properties of limits and algebraic methods to find the limits, if they exist. 7x – 10 if x 5
Find the derivatives of the functions in Problem. Simplify and express the answer using positive exponents only.R(x) = x 3√3x3 + 2
In Problem, complete the following.(a) Find the derivative of each function and check your work by graphing both your calculated derivative and the numerical derivative of the function.(b) Use your graph of the derivative to find points where the original function has horizontal tangent lines.(c)
In Problem, find the coordinates of points where the graph of f (x) has horizontal tangents. As a check, graph f (x) and see whether the points you found look as though they have horizontal tangents.f (x) = x4 - 4x3 + 9
In Problem, use a graphing calculator to complete (a) and (b).(a) Graph each function using a window with 0 ≤ x ≤ 300 and -2 ≤ y ≤ 2. What does the graph indicate about limx→∞ f(x)?(b) Use the table feature with x-values larger than 10,000 to investigate limx→-∞ f (x).
In Problem, use properties of limits and algebraic methods to find the limits, if they exist. 4 if x < -1 lim f(x), where f(x) = 3x - x - 1 if x> -1 x-1
In Problem, complete the following.(a) Find the derivative of each function and check your work by graphing both your calculated derivative and the numerical derivative of the function.(b) Use your graph of the derivative to find points where the original function has horizontal tangent lines.(c)
In Problem, find the coordinates of points where the graph of f (x) has horizontal tangents. As a check, graph f (x) and see whether the points you found look as though they have horizontal tangents.f (x) = 3x5 - 5x3 + 2
Given the graph of y = f (x), determine for which x-values F, G, H, I, or J the function is(a) Continuous.(b) Differentiable.
For the functions in Problem, determine which are continuous. Identify discontinuities for those that are not continuous. x* - 3 if x s< 1 if x>1 y = 2х — 3 |
In Problem, use a graphing calculator to complete (a) and (b).(a) Graph each function using a window with 0 ≤ x ≤ 300 and -2 ≤ y ≤ 2. What does the graph indicate about limx→∞ f(x)?(b) Use the table feature with x-values larger than 10,000 to investigate limx→-∞ f (x).
In Problem, use properties of limits and algebraic methods to find the limits, if they exist. x - 4 if x < 2 x - 3 lim f(x), where f(x) x2 3 - x2 if x> 2 ||
The total physical output P of workers is a function of the number of workers, x. The function P = f (x) is called the physical productivity function. Suppose that the physical productivity of x construction workers is given byP = 10(3x + 1)3 - 10Find the marginal physical productivity, dP/dx.
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→2 x2 + 6x + 9/x - 2
In Problem,(a) Find f'(x).(b) Graph both f (x) and f'(x) with a graphing calculator.(c) Identify the x-values where f'(x) = 0, f'(x) > 0, and f'(x) < 0.(d) Identify x-values where f (x) has a maximum point or a minimum point, where f (x) is increasing, and where f (x) is decreasing.f
In Problem,(a) Find the slope of the tangent to the graph of f (x) at any point(b) Find the slope of the tangent at the given point(c) Write the equation of the line tangent to the graph of f (x) at the given point, and(d) Graph both f (x) and its tangent line (use a graphing utility).f (x) = x2 +
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→5 x2 - 6x + 8/x - 5
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→-1 x2 + 5x + 6/x + 1
For Problem, let
In Problem, complete the following.(a) Find the derivative of each function and evaluate it at the given x-value.(b) Use the numerical derivative feature of a graphing calculator to check your evaluation from part (a).f (x) = 2x3 + 5x - π4 + 8 at x = 2
In Problem,(a) Find the slope of the tangent to the graph of f (x) at any point(b) Find the slope of the tangent at the given point(c) Write the equation of the line tangent to the graph of f (x) at the given point, and(d) Graph both f (x) and its tangent line (use a graphing utility).f (x) =
In Problem, use properties of limits and algebraic methods to find the limits, if they exist.limx→3 x2 + 2x - 3/x - 3
For Problem, let
In Problem,(a) Find the slope of the tangent to the graph of f (x) at any point(b) Find the slope of the tangent at the given point(c) Write the equation of the line tangent to the graph of f (x) at the given point, and(d) Graph both f (x) and its tangent line (use a graphing utility).f (x) = 5x3 +
In Problems, use properties of limits and algebraic methods to find the limits, if they exist.limh→0 (x + h)3 - x3/h
In Problem, complete the following.(a) Find the derivative of each function and evaluate it at the given x-value.(b) Use the numerical derivative feature of a graphing calculator to check your evaluation from part (a).h(x) = 10/x3 - 10/5√x2 + x2 + 1 at x = -1
In Problems, use properties of limits and algebraic methods to find the limits, if they exist.limx→0 2(x + h)2 - 2x2/h
In Problem, complete the following.(a) Find the derivative of each function and evaluate it at the given x-value.(b) Use the numerical derivative feature of a graphing calculator to check your evaluation from part (a).g (x) = 5/x10 + 4/4√ x3 + x5 - 4 at x = 1
Suppose that the demand for a product is defined by the equationwhere p is the price and q is the quantity demanded.(a) Is this function discontinuous at any value of q? What value?(b) Because q represents quantity, we know that q ≥ 0. Is this function continuous for q ≥ 0? 200,000 2 (q + 1)? ||
For each function in Problems, do the following.(a) Find f'(x).(b) Graph both f (x) and f'(x) using a graphing calculator.(c) Use the graph of f'(x) to identify x-values wheref'(x) = 0, f'(x) > 0, and f'(x) < 0.(d) Use the graph of f (x) to identify x-values where f (x) has a maximum or
The following table shows the amount spent per person for health care in the United States for selected years from 2002 and projected to 2024. These data can be modeled bywhere x is the number of years after 2000 and y is per capita expenditures for health care.(a) Find the instantaneous rate of
For each function in Problems, do the following.(a) Find f'(x).(b) Graph both f (x) and f'(x) using a graphing calculator.(c) Use the graph of f9(x) to identify x-values wheref'(x) = 0, f'(x) > 0, and f'(x) < 0.(d) Use the graph of f (x) to identify x-values where f (x) has a maximum or
Suppose that the cost C of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given byDescribe any discontinuities for C(p). Explain what each discontinuity means. 8100p C(p) 100 - Р
For each function in Problems, do the following.(a) Find f'(x).(b) Graph both f (x) and f'(x) using a graphing calculator.(c) Use the graph of f'(x) to identify x-values where f'(x) = 0, f'(x) > 0, and f'(x) < 0.(d) Use the graph of f (x) to identify x-values where f (x) has a maximum or
The description of body-heat loss due to convection involves a coefficient of convection, Kc, which depends on wind velocity according to the following equation.Kc = 4√4v + 1Find the rate of change of the coefficient with respect to the wind velocity.
In Problems, use the table feature of a graphing calculator to predict each limit. Check your work by using either a graphical or an algebraic approach. S2 + x - x if x 7 lim f(x), where f(x) →7 13 – 9x
(a) If limx→2+ f(x) = 5, limx→2- f (x) = 5, and f(2) = 0, find limx→2f (x), if it exists. Explain your conclusions.(b) If limx→0+ f (x) = 3, limx→0- f(x) = 0, and f(0) = 0, find limx→0 f (x), if it exists. Explain your conclusions.
Suppose that the number of calories of heat required to raise 1 g of water (or ice) from 240°C to x°C is given by(a) What can be said about the continuity of the function f (x)?(b) What happens to water at 0°C that accounts for the behavior of the function at 0°C? 2x + 20 if -40
If limx→3 f(x) = 4 and limx→3 g (x) = -2, find(a) limx→3 [ f(x) + g(x)](b) limx→3 [f (x) - g (x)](c) limx→3 [f (x) · g (x)](d) limx→3 g(x)/f (x)
The following table shows the total national expenditures for health (in billions of dollars) for selected years from 2002 and projected to 2024. (These data include expenditures for medical research and medical facilities construction.)Assume that these data can be modeled with the functionA(t) =
If limx→-2 f(x) = 6 and limx→-2 g (x) = 3, find(a) limx→-2 [5 f(x) - 4g (x)](b) limx→-2 [g (x)]2(c) limx→3 [4 - xf(x)](d) limx→-2 [f(x)/g(x)]
If limx→2 [f(x) + g(x)] = 5 and limx→2 g (x) = 11, find(a) limx→2 f (x)(b) limx→2 {[ f (x)]2 - [g (x)]2}(c) limx→2 3g(x)/f(x) - g(x)
The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070 (actual and projected).Assume that the GDP can be modeled with the functionG(t) = 213(0.2t + 5)3 - 5020(0.2t + 5)2 + 8810t + 104,000where G(t) is in billions of dollars and t is the

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