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mathematics
calculus

Questions and Answers of Calculus

If an IRA is a variable-rate investment for 20 years at rate r percent per year, compounded monthly, then the future value S that accumulates from an initial investment of $1000 isWhat is the rate
The concentration C of a substance in the body depends on the quantity of the substance Q and the volume V through which it is distributed. For a static substance, this is given by C = Q/V For a
The table shows the total national expenditures for health (both actual and projected, in billions of dollars) for the years from 2001 to 2018. (These data include expenditures for medical research
Energy use per dollar of GDP indexed to 1980 means that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. The following data show the energy use per dollar of GDP,
The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070 (actual and projected).Assume the GDP can be modeled with the function G(t) = 212.9(0.2t
The figure shows the percent of the U.S. population with diabetes (diagnosed and undiagnosed) for selected years from 2010 and projections to 2050. Assume this percent can be modeled byy = 6.97(0.5x
Find the derivatives of the functions. Simplify and express the answer using positive exponents only. 1. f(x) = π4 2. f(x) = 1/4 3. g(x) = 4/x4 4. t = x4/4 5. g(x) = 5x3 + 4/3
1. The derivative of each function.(a) F1(x) = 3(x4 + 1)5/5(b) F2(x) = 3/5(x4 + 1)5(c) F3(x) = (3x4 + 1)5/5(d) F4(x) = 3/(5x4 + 1)52. (a) G1(x) = 2(x3 - 5)3/3(b) G2(x) = (2x3 - 5)3/3(c) G3(x) =
Suppose that the revenue function for a certain product is given by R(x) = 15(2x + 1)-1 + 30x - 15 where x is in thousands of units and R is in thousands of dollars. (a) Find the marginal revenue
Suppose that the revenue in dollars from the sale of x campers is given by R(x) = 60,000x + 40,000(10 + x)-1 - 4000 (a) Find the marginal revenue when 10 units are sold. (b) How is revenue changing
Suppose that the production of x items of a new line of products is given by x = 200[(t + 10) - 400(t + 40)-1] where t is the number of weeks the line has been in production. Find the rate of
1. If the national consumption function is given by C(y) = 2(y + 1)1/2 + 0.4y + 4 find the marginal propensity to consume, dC/dy. 2. Suppose that the demand function for q units of an appliance
When squares of side x inches are cut from the corners of a 12-inch-square piece of cardboard, an open-top box can be formed by folding up the sides. The volume of this box is given by V = x(12 -
Suppose that sales (in thousands of dollars) are directly related to an advertising campaign according to S = 1 + 3t - 9 / (t + 3)2 where t is the number of weeks of the campaign. (a) Find the rate
An inferior product with an extensive advertising campaign does well when it is released, but sales decline as people discontinue use of the product. If the sales S (in thousands of dollars) after t
An excellent film with a very small advertising budget must depend largely on word-of-mouth advertising. If attendance at the film after t weeks is given by A = 100t / (t + 10)2 what is the rate of
The dollars spent per person per year for health care (projected to 2018) are shown in the table. These data can be modeled bywhere x is the number of years past 1990 and y is the per capita
Find the second derivative. 1. f(x) = 2x10 - 18x5 - 12x3 + 4 2. y = 6x5 - 3x4 + 12x2
Find the indicated derivative.1. If y = x5 - x1/2, find d2y/dx2.2. If y = x4 + x1/3, find d2y/dx2.3. If f(x) = √x + 1, find f'"(x).
Use the numerical derivative feature of a graphing calculator to approximate the given second derivatives. 1. f"(3) for f(x) = x3 - 27/x 2. f"(-1) for f(x) = x2/4 - 4/x2 3. f"(21) for f(x) = √x2 + 4
Do the following for each function f (x). (a) Find f'(x) and f"(x). (b) Graph f(x), f'(x), and f"(x) with a graphing utility. (c) Identify x-values where f"(x) = 0, f"(x) > 0, and f"(x) < 0. (d)
A particle travels as a function of time according to the formula s = 100 + 10t + 0.01t3 where s is in meters and t is in seconds. Find the acceleration of the particle when t =2.
1. Acceleration If the formula describing the distance s (in feet) an object travels as a function of time (in seconds) is s = 100 + 160t - 16t2 what is the acceleration of the object when t = 4? 2.
Suppose that the revenue (in dollars) from the sale of a product is given by R = 70x + 0.5x2 - 0.001x3 where x is the number of units sold. How fast is the marginal revenue M̅R̅ changing when x =
When medicine is administered, reaction (measured in change of blood pressure or temperature) can be modeled by R = m2 (c/2 - m/3) where c is a positive constant and m is the amount of medicine
The amount of photosynthesis that takes place in a certain plant depends on the intensity of light x according to the equation f(x) = 145x2 - 30x3 (a) Find the rate of change of photosynthesis with
The revenue (in thousands of dollars) from the sale of a product is R = 15x + 30(4x + 1)-1 - 30 where x is the number of units sold. (a) At what rate is the marginal revenue M̅R̅ changing when the
The sales of a product S (in thousands of dollars) are given byS = 600x / x + 40where x is the advertising expenditure (in thousands of dollars).(a) Find the rate of change of sales with respect to
The daily sales S (in thousands of dollars) that are attributed to an advertising campaign are given bywhere t is the number of weeks the campaign runs. (a) Find the rate of change of sales at any
A product with a large advertising budget has its sales S (in millions of dollars) given bywhere t is the number of months the product has been on the market. (a) Find the rate of change of sales at
By using Social Security Administration data for selected years from 2012 and projected to 2050, the U.S. average annual wage, in thousands of dollars, can be modeled by W(t) = 0.0212t2.11 where t is
The makeup of various groups within the U.S. population may reshape the electorate in ways that could change political representation and policies. By using U.S. Department of Labor data from 1980
The economic dependency ratio is defined as the number of persons in the total population who are not in the workforce per 100 in the workforce. Since 1960, Baby Boomers in the workforce coupled with
The table gives the U.S. population to the nearest million for selected years from 1950 and projected to 2050.(a) Find a cubic function P(t) that models these data, where P is the U.S. population in
The median income f (x), in thousands of dollars, is a function of the age of workers ages 20-62, x, and can be modeled by f (x) = 0.000864x3 - 0.128x2 + 6.61x - 62.6 (a) Find the instantaneous rate
Find the third derivative.1. y = x5 - 16x3 + 122. y = 6x3 - 12x2 + 6x
1. (a) If the total revenue function for a product is R(x) = 4x, what is the marginal revenue function for that product? (b) What does this marginal revenue function tell us? 2. If the total revenue
Suppose that the cost function for a commodity isC(x) = 40 + x2 dollars(a) Find the marginal cost at x = 5 units and tell what this predicts about the cost of producing 1 additional unit.(b)
Suppose that the cost function for a commodity is C(x) = 300 + 6x + 1/20 x2 dollars (a) Find the marginal cost at x8 units and tell what this predicts about the cost of producing 1 additional
If the cost function for a commodity is C(x) = x3 - 4x2 + 30x + 20 dollars find the marginal cost at x = 4 units and tell what this predicts about the cost of producing 1 additional unit and 3
If the cost function for a commodity is C(x) = 1/90 x3 + 4x2 + 4x + 10 dollars find the marginal cost at x = 3 units and tell what this predicts about the cost of producing 1 additional unit and 2
If the cost function for a commodity is C(x) = 300 + 4x + x2 graph the marginal cost function.
If the cost function for a commodity is C(x) = x3 - 12x2 + 63x + 15 graph the marginal cost function.
The graph of a company's total cost function is shown. For each problem, use the graph to answer the following questions.(a) Will the 101st item or the 501st item cost more to produce? Explain.(b)
Cost, revenue, and profit are in dollars and x is the number of units. 1. If the total profit function is P(x) = 5x - 25, find the marginal profit. What does this mean? 2. If the total profit
Suppose that the total revenue function for a product is R(x) = 50x and that the total cost function is C(x) = 1900 + 30x + 0.01x2. (a) Find the profit from the production and sale of 500 units. (b)
Suppose that the total revenue function is given by R (x) = 46x and that the total cost function is given by C(x) = 100 + 30x + 1/10 x2 (a) Find P(100). (b) Find the marginal profit function. (c)
The graphs of a company's total revenue function and total cost function are shown. For each problem, use the graph to answer the following questions.(a) From the sale of 100 items, 400 items, and
Suppose that the total revenue function for a commodity is R = 36x - 0.01x2. (a) Find R(100) and tell what it represents. (b) Find the marginal revenue function. (c) Find the marginal revenue at x =
The graph of a company's profit function is shown. For each problem, use the graph to answer the following questions about points A, B, and C.(a) Rank from smallest to largest the amounts of profit
(a) Graph the marginal profit function for the profit function P(x) = 30x - x2 - 200, where P(x) is in thousands of dollars and x is hundreds of units. (b) What level of production and sales will
(a) Graph the marginal profit function for the profit function P(x) = 16x - 0.1x2 - 100, where P(x) is in hundreds of dollars and x is hundreds of units. (b) What level of production and sales will
The price of a product in a competitive market is $300. If the cost per unit of producing the product is 160 + 0.1x dollars, where x is the number of units produced per month, how many units should
The cost per unit of producing a product is 60 + 0.2x dollars, where x represents the number of units produced per week. If the equilibrium price determined by a competitive market is $220, how many
If the daily cost per unit of producing a product by the Ace Company is 10 + 0.1x dollars, and if the price on the competitive market is $70, what is the maximum daily profit the Ace Company can
The Mary Ellen Candy Company produces chocolate Easter bunnies at a cost per unit of 0.40 + 0.005x dollars, where x is the number produced. If the price on the competitive market for a bunny this
Suppose that the total revenue function for a commodity is R(x) = 25x - 0.05x2. (a) Find R(50) and tell what it represents. (b) Find the marginal revenue function. (c) Find the marginal revenue at x
Suppose that demand for local cable TV service is given by p = 80 - 0.4x where p is the monthly price in dollars and x is the number of subscribers (in hundreds). (a) Find the total revenue as a
Suppose that in a monopoly market, the demand function for a product is given by p = 160 - 0.1x where x is the number of units and p is the price in dollars. (a) Find the total revenue from the sale
(a) Graph the marginal revenue function from Problem 3. (b) At what value of x will total revenue be maximized for Problem 3. (c) What is the maximum revenue? Total revenue is in dollars and x is the
(a) Graph the marginal revenue function from Problem 4. (b) Determine the number of units that must be sold to maximize total revenue. (c) What is the maximum revenue? Total revenue is in dollars and
Cost is in dollars and x is the number of units. Find the marginal cost functions for the given cost functions. 1. C(x) = 40 + 8x 2. C(x) = 200 + 16x 3. C(x) = 500 + 13x + x2 4. C(x) = 300 + 10x +
Use the graph of y = f(x) in Figure 9.40 to find the functional values and limits, if they exist.1. (a) f (- 2)2. (a) f(-1)3. (a) f(4)
If the cost function for a particular good is C(x) = 3x2 + 6x + 600, what is the (a) Marginal cost function? (b) Marginal cost if 30 units are produced? (c) Interpretation of your answer in part (b)?
If the total cost function for a commodity is C(x) = 400 + 5x + x3, what is the marginal cost when 4 units are produced, and what does it mean?
The total revenue function for a commodity is R = 40x - 0.02x2, with x representing the number of units. (a) Find the marginal revenue function. (b) At what level of production will marginal revenue
If the total revenue function for a product is given by R(x) = 60x and the total cost function is given by C = 200 + 10x + 0.1x2, what is the marginal profit at x = 10? What does the marginal profit
The total revenue function for a commodity is given by R = 80x - 0.04x2. (a) Find the marginal revenue function. (b) What is the marginal revenue at x = 100? (c) Interpret your answer in part (b).
If the revenue function for a product is R(x) = 60x2 / 2x + 1 find the marginal revenue.
A firm has monthly costs given by C = 45,000 + 100x + x3 where x is the number of units produced per month. The firm can sell its product in a competitive market for $4600 per unit. Find the marginal
A small business has weekly costs of C = 100 + 30x + x2/10 where x is the number of units produced each week. The competitive market price for this business's product is $46 per unit. Find the
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) The revenue from the next item will be least.(b) The
Use tables to investigate each limit. Check your result analytically or graphically.1.2.
Use the graph of y = f(x) in Figure 9.40 to answer the questions.1. Is f (x) continuous at (a) x = - 1? (b) x = 1? 2. Is f(x) continuous at (a) x = - 2? (b) x = 2?
1. Is f (x) continuous at x = 0?2. Is f (x) continuous at x = 1?3. Is f (x) continuous at x = -1?Suppose that
Determine which are continuous. Identify discontinuities for those that are not continuous.1. y = x2 + 25 / x - 52. y = x2 - 3x + 2/x - 23.
Determine which are continuous. Identify discontinuities for those that are not continuous.1. y = x2 + 25 / x - 52. y = x2 - 3x + 2/x - 23.Discuss.
Use the graphs to find(a) The points of discontinuity,1. 2.
Evaluate the limits, if they exist. Then state what each limit tells about any horizontal asymptotes.1.2.
1. Find the average rate of change off(x) = 2x4 - 3x + 7 over [- 1, 2]Decide whether the statements are true or false.2.gives the formula for the slope of the tangent and the instantaneous rate of
Use the definition of derivative to find f(x) for f (x) = 3x2 + 2x - 1.
Use the definition of derivative to find f(x) if f(x) = x - x2.
Explain which is greater: the average rate of change of f over [- 3, 0] or over [- 1, 0].Use the graph of y = f (x) in Figure 9.40 to answer the questions.
Use the graph of y = f (x) in Figure 9.40 to answer the questions.1. Is f(x) differentiable at (a) x = - 1? (b) x = 1? 2. Is f(x) differentiable at (a) x = - 2? (b) x = 2?
Let f(x) = 3√4x / (3x2 - 10)2. Approximate f'(2) (a) By using the numerical derivative feature of a graphing calculator, and (b) By evaluating f(2 + h) - f(2)/h with h = 0.00001.
Use the given table of values for g (x) to(a) Find the average rate of change of g (x) over [2, 5].(b) Approximate g(4) as accurately as possible.
1. Estimate f'(4).2. Rank the following from smallest to largest and explain.A: f'(2)B: f'(6)C: the average rate of change of f (x) over [2, 10]Use the following graph of f (x) to complete.
1. If c = 4x5 - 6x3, find c'.2. If f(x) = 10x9 - 5x6 + 4x - 27 + 19, find f (x).
1. If p = q + √7, find dp / dq. 2. If y = √x, find y'. 3. If f (z) = 3√24, find f'(z). 4. If v(x) = 4/ 3√x, find v'(x).
1. If y = 1/x - 1/√x, find y'. 2. If f(x) = 3/2x2 - 3√x + 45, find f'(x).
Write the equation of the line tangent to the graph of y = 3x5 - 6 at x = 1.
Write the equation of the line tangent to the curve y = 3x3 - 2x at the point where x = 2.
(a) Find all x-values where the slope of the tangent equals zero,(b) Find points (x, y) where the slope of the tangent equals zero,(c) Use a graphing utility to graph the function and label the
1. If f(x) = (3x - 1) (x2 - 4x), find f'(x).2. Find y' if y = (x4 + 3) (3x3 + 1).3. If p = 5q3/2q3 + 1, find dp/dq.
1. If y = (x3 - 4x2)3, find y'. 2. If y = (5x6 + 6x4 + 5)6, find y'. 3. If y = (2x4 - 9)9, find dy/dx.
Find each limit, if it exists.2. 3. 4.
Find g'(x) if g(x) = 1/ √x3 - 4x.
Find f'(x) if f(x) = x2(2x4 + 5)8.
Find S' if S = (3x + 1)2 / x2 - 4.
Find dy/dx if y = [3x + 1) (2x3 - 1)]12.

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