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calculus with applications

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

A restaurant has an annual demand for 900 bottles of a California wine. It costs $1 to store 1 bottle for 1 year, and it costs $5 to place a reorder.(a) Find the optimum number of bottles per order.(b) How many times a year should the wine be ordered?
A farmer is constructing a rectangular pen with one additional fence across its width. Find the maximum area that can be enclosed with 2400 m of fencing.
Repeat Exercise 15, given that 80 units are produced daily and the rate of change of production is 12 units per day.Exercise 15Given the revenue and cost functions R = 50x - 0.4x2 and C = 5x + 15 (in dollars), where x is the daily production (and sales), find the following when 40 units are
What is elasticity of demand (in words; no mathematical symbols allowed)? Why is the derivative used to describe elasticity?
Find the absolute extrema if they exist, as well as all values of x where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.ƒ(x) = x3 - 3x2 - 24x + 5; [-3, 6]
Find dy/dx by implicit differentiation for the following.(xy)4/3 + x1/3 = y6 + 1
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.√17.02
A bookstore has an annual demand for 100,000 copies of a best-selling book. It costs $0.50 to store 1 copy for 1 year, and it costs $60 to place an order.(a) Find the optimum number of copies per order.(b) How many times a year should the copies be ordered?
Given the revenue and cost functions R = 50x - 0.4x2 and C = 5x + 15 (in dollars), where x is the daily production (and sales), find the following when 40 units are produced daily and the rate of change of production is 10 units per day.(a) The rate of change of revenue with respect to time(b) The
When solving applied extrema problems, why is it necessary to check the endpoints of the domain?
Find the absolute extrema if they exist, as well as all values of x where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.ƒ(x) = x3 - 6x2 + 9x - 8; [0, 5]
A manufacturer has found that the cost C and revenue R (in dollars) in one month are related by the equationFind the rate of change of revenue with respect to time when the cost is changing by $15 per month and the monthly revenue is $25,000. C = R² 450,000 + 12,000.
Find dy/dx by implicit differentiation for the following.x4y3 + 4x3/2 = 6y3/2 + 5
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.√0.99
A campground owner has 1400 m of fencing. He wants to enclose a rectangular field bordering a river, with no fencing needed along the river. (See the sketch.) Let x represent the width of the field.(a) Write an expression for the length of the field.(b) Find the area of the field (area = length *
A manufacturer has a steady annual demand for 13,950 cases of sugar. It costs $9 to store 1 case for 1 year, $31 in setup cost to produce each batch, and $16 to produce each case.(a) Find the number of cases per batch that should be produced.(b) Find the number of batches of sugar that should be
Find the dimensions of the rectangular field of maximum area that can be made from 300 m of fencing material. (This fence has four sides.) What is the maximum area?
Find the absolute extrema if they exist, and all values of x where they occur on the given intervals.ƒ(x) = -2x3 - 2x2 + 2x - 1; [-3, 1]
Can a relative extremum be an absolute extremum? Is a relative extremum necessarily an absolute extremum?
Find dy/dx by implicit differentiation for the following.4√x - 8√y = 6y3/2
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.√23
Suppose 100,000 lamps are to be manufactured annually. It costs $1 to store a lamp for 1 year, and it costs $500 to set up the factory to produce a batch of lamps.(a) Find the number of lamps to produce in each batch.(b) Find the number of batches of lamps that should be manufactured annually.
A manufacturer of handcrafted wine racks has determined that the cost to produce x units per month is given by C = 0.2x2 + 10,000. How fast is cost per month changing when production is changing at the rate of 12 units per month and the production level is 80 units?
The sale of compact discs of “lesser” performers is very sensitive to price. If a CD manufacturer charges p(x) dollars per CD, wherethen x thousand CDs will be sold.(a) Find an expression for the total revenue from the sale of x thousand CDs.(b) Find the value of x and the number of CDs that
Find the absolute extrema if they exist, and all values of x where they occur on the given intervals.ƒ(x) = x3 + 2x2 - 15x + 3; [-4, 2]
Assume x and y are functions of t. Evaluate dy/dt for each of the following. y ln x + xe = 1; dx dt = 5, x = 1, y = 0
Find dy/dx by implicit differentiation for the following.2√x + 4√y = 5y
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.√145
Find the locations of any absolute extrema for the functions with graphs as follows. XI f(x) + 0 X₂ X3 X
Find dy/dx by implicit differentiation for the following. 2y² 5 + x 5 - x
Suppose the demand function is of the form q = Cp-k, where C and k are positive constants.Find the elasticity E.If 0If k > 1, what does your answer from part (a) say about how prices should be set to maximize the revenue?If k = 1, what does your answer from part (a) tell you about setting prices
For Exercises, find dy for the given values of x and Δx. y = 6x 3 2x + 1 - x = 3, Δ.x = -0.04
Find the absolute extrema if they exist, and all values of x where they occur on the given intervals.ƒ(x) = 4x3 - 9x2 - 3; [-1, 2]
If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city, where(a) Find an expression for the total revenue from the sale of x thousand candy bars.(b) Find the value of x and the number of candy bars that lead to maximum revenue.(c) Find the
Assume x and y are functions of t. Evaluate dy/dt for each of the following. xe 2 In 2 + In x; = - dx dt = 6, x = 2, y = 0
Find the locations of any absolute extrema for the functions with graphs as follows. X1 f(x). X2 Mol X3 X4 X
Find dy/dx by implicit differentiation for the following. 3x 2-у 2+ 2 +у
Suppose that a demand function is linear-that is, q = m - np for 0 ≤ p ≤ m/n, where m and n are positive constants. Show that E = 1 at the midpoint of the demand curve on the interval 0 ≤ p ≤ m/n; that is, at p = m/(2n).
For Exercises, find dy for the given values of x and Δx. y = 2x - 5 x + 1 x = 2, Ax = -0.03
Find the absolute extrema if they exist, and all values of x where they occur on the given intervals.ƒ(x) = -x3 + 6x2 + 1; [-1, 6]
Assume x and y are functions of t. Evaluate dy/dt for each of the following. 13 y³ - 4x² .3 x³ + 2y 44 31 dx dt 5, x = -3, y = -2
Find the locations of any absolute extrema for the functions with graphs as follows. g(x) ↑ 0 X₁ X
Assume x and y are functions of t. Evaluate dy/dt for each of the following. 好 + y x-y = 9; dx · = 2,x = 4, y = 2 | dt
What must be true about the demand function if E = 0?
For Exercises, find dy for the given values of x and Δx. - y = V4x – 1; r = 5, Δx = 0.08 =
In Exercises, determine the average cost function C(x) = C(x)/x. To find where the average cost is smallest, first calculate C'(x), the derivative of the average cost function. Then use a graphing calculator to find where the derivative is 0. Check your work by finding the minimum from the graph of
Find the locations of any absolute extrema for the functions with graphs as follows. g(x) + 0 X
When the change in x is small, the differential of y is approximately the change in y.Determine whether each of the following statements is true or false, and explain why.
In Exercises, determine the average cost function C(x) = C(x)/x. To find where the average cost is smallest, first calculate C'(x), the derivative of the average cost function. Then use a graphing calculator to find where the derivative is 0. Check your work by finding the minimum from the graph of
Find dy/dx by implicit differentiation for the following. 3x3 - 8y2 = 10y
A Giffen good is a product for which the demand function is increasing. Economists debate whether such goods actually exist. What is true about the elasticity of a Giffen good?
A differential is a real number.Determine whether each of the following statements is true or false, and explain why.
Assume x and y are functions of t. Evaluate dy/dt for each of the following. 4x³ − 6xy² + 3y² = 228; dx dt = 3, x = -3, y = 4
Find dy/dx by implicit differentiation for the following.5x3 = 3y2 + 4y
For Exercises, find dy for the given values of x and Δx.y = √3x + 2 x = 4, Δx = 0.15
Describe elasticity of demand in your own words.
In Exercises, use the steps shown to find nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of the indicated expression.x + y = 105 and xy2 is maximized.(a) Solve x + y = 180 for y.(b) Substitute the result from part (a) into P = xy, the equation for the
Assume x and y are functions of t. Evaluate dy/dt for each of the following. 2xy - 5x + 3y³ = -51; dx dt = -6, x = 3, y = -2
Find the locations of any absolute extrema for the functions with graphs as follows. X₁ f(x) 0 X
Assume x and y are functions of t. Evaluate dy/dt for each of the following. 8y³ + x² = 1; dx dt = 2, x = 3, y = −1
In a related rates problem, all derivatives are with respect to time.Determine whether each of the following statements is true or false, and explain why.
Find dy/dx by implicit differentiation for the following.8x2 - 10xy + 3y2 = 26
Find the locations of any absolute extrema for the functions with graphs as follows. fox) + X2X3 이 X
For Exercises, find dy for the given values of x and Δx.y = x3 - 2x2 + 3; x = 1, Δx = -0.1
In Exercises, use the steps shown to find nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of the indicated expression.The sum of x and y is 140 and the sum of the squares of x and y is minimized.(a) Solve x + y = 180 for y.(b) Substitute the result from part
Implicit differentiation can be used to find dy/dx when x is defined in terms of y.Determine whether each of the following statements is true or false, and explain why.
Find dy/dx by implicit differentiation for the following.7x2 - 4y2 = 24
For Exercises, find dy for the given values of x and Δx.y = 4x3 - 3x; x = 3, Δx = 0.2
In the discussion of economic lot size, use the critical point theorem to show that √(2ƒM)/k is the economic lot size that minimizes total production costs.
In Exercises, use the steps shown in Exercise 5 to find nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of the indicated expression.x + y = 180 and the product P = xy is as large as possible.(a) Solve x + y = 180 for y.(b) Substitute the result from part (a)
For Exercises, find dy for the given values of x and Δx.y = 2x3 - 5x; x = -2, Δx = 0.1
Assume x and y are functions of t. Evaluate dy/dt for each of the following. y² - 8x³ 8x³ = -55; dx dt -4, x = 2, y = 3
Find the locations of any absolute extrema for the functions with graphs as follows. f(x) ↑ XI X2 X3 X
Determine whether each of the following statements is true or false, and explain why.Total revenue is maximized at the price where demand has unit elasticity.
Find dy/dx by implicit differentiation for the following.6x2 + 5y2 = 36
If the demand has unit elasticity at price p, then increasing the price will increase the total revenue.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.A function on an open interval cannot have both an absolute maximum and an absolute minimum.
Determine whether each statement is true or false, and explain why.If C and r are functions of time and C = 2πr, then dC/dt = 2π(dr/dt).
Determine whether each of the following statements is true or false, and explain why.Demand for a product is elastic if the elasticity is greater than 1.
Determine whether each statement is true or false, and explain why.A function can have only one absolute maximum, although it may occur at more than one value of x.
Determine whether each statement is true or false, and explain why.If q is a function of p and p = ln q, then dq/dp = q.
Differentials can be used to find relative error.Determine whether each statement is true or false, and explain why.
If the demand for premium unleaded gasoline is elastic, raising prices will decrease the total revenue.Determine whether each statement is true or false, and explain why.
For a function on an open interval, absolute extrema will occur at every critical number.Determine whether each statement is true or false, and explain why.
If x and y are functions of t, then dy/dt = 3 when y = 3x.Determine whether each statement is true or false, and explain why.
A continuous function on an open interval does not have an absolute maximum or minimum.Determine whether each of the following statements is true or false, and explain why.
A continuous function always has an absolute minimum and an absolute maximum.Determine whether each statement is true or false, and explain why.
If r is a function of t and 10 = r2 + t2, then 0 = 2r(dr/dt2) + 2t(dt/dr).Determine whether each statement is true or false, and explain why.
When using differentials to estimate the value of ƒ(x), the error increases as x increases.Determine whether each statement is true or false, and explain why.
If the quantity demanded falls 10% when the price rises 5%, then the demand is elastic.Determine whether each statement is true or false, and explain why.
If a function on a closed interval has only one critical number at x = c, then by the critical point theorem, the function has an absolute extrema at x = c.Determine whether each statement is true or false, and explain why.
A related rates problem is used to find the absolute extrema of a functionDetermine whether each statement is true or false, and explain why.
A continuous function on a closed interval has an absolute maximum and minimum.Determine whether each of the following statements is true or false, and explain why.
The absolute maximum of a function on a closed interval always occurs at a relative maximum.Determine whether each statement is true or false, and explain why.
If y is a function of x and y = xey, then dy/dx = ey + xey(dy/dx).Determine whether each statement is true or false, and explain why.
Differentials can be used to find the exact value of √65.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.If the elasticity of demand is greater than 1, then the demand is elastic.
1. Find Z′(m).2. Solve the equation Z' (m) = 0.As a practical matter, it is usually required that m be a wholenumber. If m does not come out to be a whole number, thenm+ and m-, the two whole numbers closest to m, must bechosen. Calculate both Z(m+t) and Z(m-); the smaller of the two provides the
For a continuous function on a closed interval, one can determine the absolute extrema by evaluating the function at the endpoints and at each critical number.Determine whether each statement is true or false, and explain why.
A related rates problem is used when we know the rate of change of one or more quantities, and we want to find the rate of change of another quantity.Determine whether each statement is true or false, and explain why

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