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mathematics
calculus with applications

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

Use the rules for derivatives to find the derivative of each function defined as follows. y = √2t¹ - 5 217
In physics, a major area of study is electricity, including such concepts as electric charge, electric force, and electric current. Two ideas that physicists use a great deal are electric potential and electric field. Electric potential is the same as voltage, such as for a battery. Electric field
Determine whether each statement is true or false, and explain why.If ƒ(x) = x2 and g(x) = ln x, then g[ƒ(x)] = ln x2.
Determine whether each of the following statements is true or false, and explain why.The derivative of π3 is 3π2.
The derivative of a product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why. Since a constant function does not change, its derivative is always 0.
Determine whether each statement is true or false, and explain why.The derivative of ƒ(t) = ln t is ƒ′(t) = 1/t.
Determine whether each statement is true or false, and explain why.For all functions ƒ(x) and g(x), it is always true that ƒ[g(x)] = g[ƒ(x)].
Determine whether each statement is true or false, and explain why.The derivative of ƒ(x) = x2(3x - 7) is ƒ′(x) = 2x • 3 = 6x.
Determine whether each statement is true or false, and explain why. The derivative of ƒ(x) = x4 is ƒ′(x) = 4x3.
Determine whether each statement is true or false, and explain why.The derivative of ƒ(x) = log x is ƒ′(x) = 1/x.
Determine whether each statement is true or false, and explain why.If ƒ(x) = (2x3 + 7x)5, then ƒ′(x) = 5(2x3 + 7x)4.
Determine whether each statement is true or false, and explain why.If ƒ(x) = (2x3 + 7x)5, then ƒ′(x) = 5(6x2 + 7)4.
The derivative of a quotient is the derivative of the numerator divided by the derivative of the denominator.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why. The derivative of ƒ(x) = 710 is ƒ′(x) = 10 • 79.
The derivative of ƒ(x) = ln(x2 + 3)4 is the same as the derivative of g(x) = 4 ln(x2 + 3).Determine whether each statement is true or false, and explain why.
Let f(x) = 5x2 - 2x and g (x) = 8x + 3. Find the following.ƒ[g(2)]
In Exercises, use the product rule to find the derivative of each function.y = (3x2 + 2)(2x - 1)
Determine whether each statement is true or false, and explain why. The derivative of a sum is the sum of the derivatives.
In Exercises, find the derivative of each function. y = 8x³ - 5x² X 12
Let f(x) = 5x2 - 2x and g (x) = 8x + 3. Find the following.ƒ[g(-5)]
In Exercises, use the product rule to find the derivative of each function.y = (5x2 - 1)(4x + 3)
Determine whether each statement is true or false, and explain why.The marginal cost function is approximately the cost of producing one more unit.
Use the rules for derivatives to find the derivative of each function defined as follows.y = 9x8/3
Find the derivative of each function.y = ln(-4x)
Let f(x) = 5x2 - 2x and g (x) = 8x + 3. Find the following.g[f(2)]
In Exercises, find the derivative of each function.y = 12x3 - 8x2 + 7x + 5
In Exercises, use the product rule to find the derivative of each function.y = (2x - 5)2
Find the derivative of each function.y = ln(8 - 3x)
In Exercises, find the derivative of each function. y = 3x4 - 6x³ + 8 + 5
Let f(x) = 5x2 - 2x and g (x) = 8x + 3. Find the following.g[ƒ(-5)]
Find the derivative of each function.y = ln(1 + x3)
Let f(x) = 5x2 - 2x and g (x) = 8x + 3. Find the following.ƒ[g(k)]
In Exercises, use the product rule to find the derivative of each function.k(t) = (t2 - 1)2
Find the derivative of each function.y = ln|4x2 - 9x|
In Exercises, find f[g(x) ] and g[f(x)]. X f(x) = −8x + 9; g(x) = - +4 5
Let f(x) = 5x2 - 2x and g (x) = 8x + 3. Find the following.g[ ƒ(5z)]
In Exercises, use the product rule to find the derivative of each function.g(t) = (3t2 + 2)2
Find the derivative of each function.y = ln|-8x3 + 2x|
Use the rules for derivatives to find the derivative of each function defined as follows.y = 5x3 - 7x2 - 9x + √5
In Exercises, use the product rule to find the derivative of each function.y = (2x - 3)(√x - 1)
In Exercises, find the derivative of each function.ƒ(x) = -2x1.5 + 12x0.5
Find the derivative of each function.y = ln√2x + 1
In Exercises, find f[g(x)] and g[f(x)]. f(x) = ×10 8 +7; g(x) = 6x - 1
In Exercises, use the product rule to find the derivative of each function.y = (x + 1)(√x + 2)
In Exercises, find the derivative of each function.ƒ(x) = 6x3.5 - 10x0.5
Find the derivative of each function.y = ln√x + 5
Use the rules for derivatives to find the derivative of each function defined as follows.y = 7x3 - 4x2 - 5x + √2
Use the information given in the figure to find the following values.(a) ƒ(1) (b) ƒ′(1)(c) The tangent line at ƒ(x) when x = 1 (-1, 1) -2 (1, 2) 1 f(x) X
A company that manufactures bicycles has determined that a new employee can assemble M(d) bicycles per day after d days of on-the-job training, where(a) Find the rate of change function for the number of bicycles assembled with respect to time.(b) Find and interpret M′(2) and M′(5).
In Exercises, find the equation of the tangent line to the graph of the given function at the given value of x.ƒ(x) = x(x2 - 4x + 5)4; x = 2
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms.ƒ(x) = ln[x2(ex + 1)]
Explain the concept of marginal cost. How does it relate to cost? How is it found?
Show that, for any constant k, d [f(x)] k || f'(x) k
Suppose that the average cost function is given by C(x) = C(x)/x, where x is the number of items produced. Show that the marginal average cost function is given by C'(x) = xC'(x) = C(x) 2 x²
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms. h(t) = In (t + 4)²] t
In Exercises, find all values of x for the given function where the tangent line is horizontal. f(x) = √x³ - 6x² + 9x + 1
The effect of a when graphing y = aƒ(x) was discussed. Now describe how this relates to the fact that Dx[aƒ(x)] = aƒ′(x).
Suppose that the demand function is given by p = D(q), where q is the quantity that consumers demand when the price is p. Show that the marginal revenue is given byR′(q) = D(q) + qD′(q).
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms.g(x) = ln[x5(x2 + 5)10]
In Exercises, find all values of x for the given function where the tangent line is horizontal. f(x) X (x² + 4)4
In Exercises, find the equation of the tangent line to the graph of the given function at the given value of x.ƒ(x) = x2√x4 - 12 ; x = 2
Find the derivative of each of the following functions by first rewriting the function using the rules of logarithms. r(x) = In (x² - 4) x³
Suppose that at the beginning of the year, a Vermont maple syrup distributor found that the demand for maple syrup, sold at $15 a quart, was 500 quarts each month. At that time, the price was going up at a rate of $0.50 a month, but despite this, the demand was going up at a rate of 30 quarts a
Use the differentiation feature on your graphing calculator to solve the problems (to 2 decimal places) below, where ƒ(x) is defined as follows:ƒ(x) = 1.25x3 + 0.01x2 - 2.9x + 1.(a) Find ƒ′(4).(b) Find all values of x where ƒ′(4) = 0.
Katie and Sarah are working on taking the derivative ofKatie uses the quotient rule to getSarah converts it into a product and uses the product rule and the general power rule:Explain the discrepancies between the two answers. Which procedure do you think is preferable? f(x) = 2x 3x + 4'
In Exercises, find the equation of the tangent line to the graph of the given function at the given point.y = ln(x2 + 2) at the point (1, ln 3)
A gasoline refinery found that the cost to produce 12,500 gallons of gasoline last month was $27,000. At that time, the cost was going up at a rate of $1200 per month, while the number of gallons of gasoline the refinery produced was going up at a rate of 350 gallons per month. At what rate was the
Margy and Nate are working on taking the derivative ofMargy uses the quotient rule and general power rule as follows: Nate rewrites the function and uses the general power rule as follows:Compare the two procedures. Which procedure do you think is preferable? f(x) = 2 (3x + 1)4
Assume that the cost, in dollars, of producing x gaming devices is given by C(x) = -0.04x2 + 80x + 75.(a) Find the marginal cost function, C′(x).(b) Find and interpret the marginal cost when x = 100 devices.(c) Find the
In Exercises, find the equation of the tangent line to the graph of the given function at the given point.y = ln(x + 5)2 at the point (-1, ln 16)
When a certain drug is injected into a muscle, the muscle responds by contracting. The amount of contraction, s (in millimeters), is related to the concentration of the drug, x (in milliliters), bywhere m and n are constants.(a) Find s′(x).(b) Find the rate of contraction when the concentration
Suppose the demand for a certain brand of vacuum cleaner is given bywhere p is the price in dollars. If the price, in terms of the cost c, is expressed as p(c) = 2c - 10, find the demand in terms of the cost. D(p) = -p² 100 + 500,
Suppose that the revenue, in dollars, from the sale of x book bags is given byR(x) = 48x - 0.3x2.(a) Find the marginal revenue function, R′(x).(b) Find and interpret the marginal revenue when x = 20 bags.(c) Find the actual revenue of the 21st bag. Compare it to your results from part (b).
The formula for the growth rate of a population in the presence of a quantity x of food was given asThis was referred to as Michaelis-Menten kinetics.(a) Find the rate of change of the growth rate with respect to the amount of food.(b) The quantity A in the formula for ƒ(x) represents the quantity
Why do we use the absolute value of x or of g(x) in the derivative formulas for the natural logarithm?
Assume that the price, in dollars, of a sound system is given bywhere q represents the demand for the product.(a) Find the revenue function, R(q), and the marginal revenue function, R′(q).(b) Find and interpret the marginal revenue when the demand is 10 systems. Suppose the cost of producing
Assume that the total revenue (in dollars) from the sale of x television sets is given byR(x) = 24(x2 + x)2/3.Find and interpret the following.(a) R(100) (b) R′(100)(c) R′(100)(d) Find the average revenue function and the marginal averagerevenue function.Find and interpret the
Using data collected by zoologist Reto Zach, the work done by a crow to break open a whelk (large marine snail) can be estimated by the functionwhere H is the height (in meters) of the whelk when it is dropped.(a) Find dW/dH.(b) One can show that the amount of work is minimized when dW/dH = 0. Find
Prove d/dx ln |ax| = d/dx ln |x| for any constant a.
Murrell’s formula for calculating the total amount of rest, in minutes, required after performing a particular type of work activity for 30 minutes is given by the formulawhere w is the work expended in kilocalories per minute, kcal/min.(a) A value of 5 for w indicates light work, such as riding
A friend concludes that because y = ln 6x and y = ln x have the same derivative, namely dy/dx = 1/x, these two functions must be the same. Explain why this is incorrect.
A sum of $1500 is deposited in an account with an interest rate of r percent per year, compounded daily. At the end of 5 years, the balance in the account is given byFind and interpret the rate of change of A with respect to r for the following interest rates.(a) 6% (b) 8% (c) 9% A
Assume that a demand equation is given by q = 5000 - 100p.(a) Find the revenue function, R(q), and the marginal revenue function, R′(q).(b) Find and interpret the marginal revenue for q = 500 units.(c) Find and interpret the marginal revenue for q = 1000 units.Suppose that the cost of producing q
Find the rate of change of the velocity with respect to the length of an organism if the typical velocity (in centimeters per second) of a marine organism of length l (in centimeters) is given by v = 2.69∫1.86.
An analyst has found that a company’s costs and revenues in dollars for one product are given by C(x) = 2x and R(x) = 6x - x2/1000, respectively, where x is the number of items produced.(a) Find the marginal cost function.(b) Find the marginal revenue function.(c) Using the fact that profit is
Suppose a demand function is given bywhere q is the demand for a product and p is the price per item in dollars. Find the rate of change in the demand for the product per unit change in price. P 30 (5 √²+1) 1. q = D(p) = 30(5
Economists have considered the output y of a manufacturing process as a function of the size of the labor force n using the function y = knp,where 0 < p <
A certain truck depreciates according to the formulawhere V is the value of the truck (in dollars), t is time measured in years, and t = 0 represents the time of purchase (in years). Find and interpret the rate at which the value of the truck is changing at the following times.(a) 2 years (b)
Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given byFind the rate at which the number of facts remembered is changing after the following numbers of hours.(a) 1 (b) 10 f(t) = 90t 99t90*
The average number of vehicles waiting in a line to enter a parking ramp can be modeled by the functionwhere x is a quantity between 0 and 1 known as the traffic intensity. Find the rate of change of the number of vehicles in line with respect to the traffic intensity for the following values of
U.S. postal rates have steadily increased since 1932. Using data depicted in the table in the next column for the years 1971–2020, the cost in cents to mail a single letter can be modeled using a quadratic formula as follows: C(t) = -0.00246t2 + 1.02t + 7.44 where t is the number of
Use a graphing calculator to sketch the graph of y = [ ƒ(x + h) - ƒ(x)]/h using ƒ(x) = ln |x| and h = 0.0001. Compare it with the graph of y = 1/x and discuss what you observe.
Some people have argued that the Pythagorean Theorem of Baseball is a better predictor of a team’s ability than its winning percentage:where W is a predictor of what fraction of games a team is expected to win, s is the number of runs scored, and a is the number of runs allowed.(a) The team with
Suppose the cost in dollars of manufacturing q items isgiven byC = 2000q + 3500,and the demand equation is given byIn terms of the demand q,(a) find an expression for the revenue R;(b) find an expression for the profit P;(c) find an expression for the marginal profit.(d) Determine the value of the
Use graphical differentiation to verify that d dx (In x) = 1 X
The total amount of money in circulation for the years 2000–2019 can be closely approximated by M(t) = 0.0322t3 + 1.71t2 + 18.2t + 595 where t represents the number of years since 2000 and M(t) is in billions of dollars. Find the derivative of M(t) and use it to find the rate of change
Suppose the population P of a certain species of fish depends on the number x (in hundreds) of a smaller fish that serves as its food supply, so that
Use the fact that d ln x/dx = 1/x, as well as the change-of-base theorem for logarithms, to prove that d loga x dx || 1 x In a
Insulation workers who were exposed to asbestos and employed before 1960 experienced an increased likelihood of lung cancer. If a group of insulation workers has a cumulative total of 100,000 years of work experience with their first date of employment t years ago, then the number of lung cancer

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