The initial size of a bacteria culture that grows exponentially was 10,000. After 1 day, there are 15,000 bacteria. (a) Find the growth constant if time is measured in days. (b) How long will it take for the culture to double in size?
> Differentiate the following functions. y = (1 + ex)(1 - ex)
> Compute the given derivatives with the help of formulas (1)–(4). (a) d/dx (2x) |x=1 (b) d/dx (2x) |x=-2
> Differentiate the following functions. y = 8ex(1 + 2ex)2
> Differentiate the following functions. y = (x2 + x + 1)ex
> Differentiate the following functions. y = xex
> Determine the growth constant k, then find all solutions of the given differential equation. y' = y/4
> Differentiate the following functions. y = (2x + 4 - 5ex)/4
> Differentiate the following functions. y = 3ex - 7x
> Find the slope–point form of the equation of the tangent line to the graph of ex at the point (a, ea).
> Differentiate the function. y = (x + 11 / x – 3)3
> Suppose that A = (a, b) is a point on the graph of ex. What is the slope of the graph of ex at the point A?
> Estimate the slope of ex at x = 0 by calculating the slope (eh – 1)/h of the secant line passing through the points (0, 1) and (h, eh). Take h = .01, .001, and .0001.
> Use the first and second derivative rules from Section 2.2 to show that the graph of y = ex has no relative extreme points and is always concave up.
> Find the point on the graph of f (x) = ex, where the tangent line is parallel to y = x.
> Show that d/dx (2.7x) |x=0 ≈ .99 by calculating the slope (2.7h – 1)/h of the secant line passing through the points (0, 1) and (h, 2.7h). Take h = .1, .01, and .001.
> Find an equation of the tangent line to the graph of f (x) = ex, where x = -1. (Use 1/e = .37.)
> The decay constant for the radioactive element cesium 137 is .023 when time is measured in years. Find its half-life.
> Solve each equation for x. 4ex(x2 + 1) = 0
> Solve each equation for x. ex(x2 - 1) = 0
> Solve each equation for x. e-x = 1
> Solve each equation for x. ex2-2x = e8
> Differentiate the function. y = 3/ (3√x + 1)
> Solve each equation for x. e1-x = e2
> Solve each equation for x. e5x = e20
> Write each expression in the form ekx for a suitable constant k. √(e-x*e7x), e-3x/e-4x
> Write each expression in the form ekx for a suitable constant k. (e4x*e6x)3/5, 1/e-2x
> Write each expression in the form ekx for a suitable constant k. (e5/e3)x, e4x+2*ex-2
> Ten grams of a radioactive substance with decay constant .04 is stored in a vault. Assume that time is measured in days, and let P(t) be the amount remaining at time t. (a) Give the formula for P(t) (b) Give the differential equation satisfied by P(t). (
> Show that d/dx (3x) |x=0 ≈ 1.1 by calculating the slope (3h – 1)/h of the secant line passing through the points (0, 1) and (h, 3h). Take h = .1, .01, and .001.
> In an expression of the form f (g (x)), f (x) is called the outer function and g (x) is called the inner function. Give a written description of the chain rule using the words inner and outer.
> Refer to Exercise 61. (a) What was the maximum value of the company during the first 6 months since it went public, and when was that maximum value attained? (b) Assuming that the value of one share will continue to increase at the rate that it did durin
> Refer to Exercise 61. (a) Find dx/dt |t=2.5 and dx/dt |t=4. Give an interpretation for these values. (b) Use the chain rule to find dW/dt |t=2.5 and dW/dt |t=4. Give an interpretation for these values. Exercise 61: After a computer software company went
> Refer to Exercise 61. Use the chain rule to find dW/dt |t=1.5 and dW/dt |t=3.5 . Give an interpretation for these values. Exercise 61: After a computer software company went public, the price of one share of its stock fluctuated according to the graph i
> Differentiate the function. y = 1 / (√x + 1)
> After a computer software company went public, the price of one share of its stock fluctuated according to the graph in Fig. 1(a). The total worth of the company depended on the value of one share and was estimated to be where x is the value of one sha
> Consider the functions of Exercise 59. Find d/dx g (f (x)) |x=1. Exercise 59: If f (x) and g (x) are differentiable functions, such that f (1) = 2, f ‘ (1) = 3, f ‘ (5) = 4, g(1) = 5, g ’ (1) = 6, g ‘ (2) = 7, and g ‘ (5) = 8, find d/dx f ( g (x)) |x=1.
> If f (x) and g (x) are differentiable functions, such that f (1) = 2, f ‘ (1) = 3, f ‘ (5) = 4, g(1) = 5, g ’ (1) = 6, g ‘ (2) = 7, and g ‘ (5) = 8, find d/dx f ( g (x)) |x=1.
> If f (x) and g (x) are differentiable functions, find g (x) if you know that f (x) = 1>x and d/dx f (g (x)) = (2x + 5)/(x2 + 5x – 4).
> A person is given an injection of 300 milligrams of penicillin at time t = 0. Let f (t) be the amount (in milligrams) of penicillin present in the person’s bloodstream t hours after the injection. Then, the amount of penicillin decays exponentially, and
> By trial and error, find a number of the form b = 2.(just one decimal place) with the property that the slope of the graph of y = bx at x = 0 is as close to 1 as possible.
> Determine the growth constant k, then find all solutions of the given differential equation. y' = 1.7y
> A population is growing exponentially with growth constant .05. In how many years will the current population triple?
> Determine the growth constant of a population that is growing at a rate proportional to its size, where the population triples in size every 10 years and time is measured in years.
> Determine the growth constant of a population that is growing at a rate proportional to its size, where the population doubles in size every 40 days and time is measured in days.
> The size of a certain insect population is given by P(t) = 300e0.01t, where t is measured in days. (a) How many insects were present initially? (b) Give a differential equation satisfied by P(t). (c) At what time will the initial population double? (d) A
> After t hours there are P(t) cells present in a culture, where P(t) = 5000e0.2t. (a) How many cells were present initially? (b) Give a differential equation satisfied by P(t). (c) When will the initial number of cells double? (d) When will 20,000 cells b
> Approximately 10,000 bacteria are placed in a culture. Let P(t) be the number of bacteria present in the culture after t hours, and suppose that P(t) satisfies the differential equation P (t) = .55P(t). (a) What is P(0)? (b) Find the formula for P(t). (c
> Let P(t) be the population (in millions) of a certain city t years after 2015, and suppose that P(t) satisfies the differential equation P (t) = .03P(t), P(0) = 4. (a) Use the differential equation to determine how fast the population is growing when it
> Differentiate the functions. y = (x2 – 1) / (x2 + 1)
> A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let P(t) denote the number of fruit flies t days later, and let k = .08 denote the growth constant. (a) Write a differential equation and initial condition tha
> A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let P(t) denote the number of fruit flies t days later, and let k = .08 denote the growth constant. (a) Write a differential equation and initial condition tha
> Determine the growth constant k, then find all solutions of the given differential equation. y' = .4y
> Let P(t) be the population (in millions) of a certain city t years after 2015, and suppose that P(t) satisfies the differential equation P (t) = .01P(t), P(0) = 2. (a) Find a formula for P(t). (b) What was the initial population, that is, the population
> Solve the given differential equation with initial condition. 5y = 3y’ , y(0) = 7
> Solve the given differential equation with initial condition. 6y’ = y, y(0) = 12
> Solve the given differential equation with initial condition. y’ -y/7 = 0, y(0) = 6
> Solve the given differential equation with initial condition. y’ - .6y = 0, y(0) = 5
> Solve the given differential equation with initial condition. y ’ = y, y(0) = 4
> Differentiate the functions. y = 1 / (x2 + x + 7)
> Solve the given differential equation with initial condition. y’ = 2y, y(0) = 2
> Solve the given differential equation with initial condition. y’ = 4y, y(0) = 0
> Solve the given differential equation with initial condition. y' = 3y, y(0) = 1
> Determine the growth constant k, then find all solutions of the given differential equation. 5y’ - 6y = 0
> Determine the growth constant k, then find all solutions of the given differential equation. y' = y
> Write expression in the form 2kx or 3kx, for a suitable constant k. 34x/32x, 25x+1 / (2 * 2-x), 9-x / 27-x/3
> Write expression in the form 2kx or 3kx, for a suitable constant k. 7-x * 14x, 2x/6x, 32x/18x
> Write expression in the form 2kx or 3kx, for a suitable constant k. 6x * 3-x, 15x/5x, 12x/22x
> Write expression in the form 2kx or 3kx, for a suitable constant k. (1/9)2x, (1/27)x/3, (1/16)-x/2
> Write expression in the form 2kx or 3kx, for a suitable constant k. (1/4)2x, (1/8)-3x, (1/81)x/2
> Differentiate the functions. y = x – 1 / x + 1
> Graph the function f (x) = 3x in the window [-1, 2] by [-1, 8], and estimate the slope of the graph at x = 0.
> Graph the function f (x) = 2x in the window [-1, 2] by [-1, 4], and estimate the slope of the graph at x = 0.
> The expression may be factored as shown. Find the missing factors. 57x/2 - 5x/2 = √5x( )
> The expression may be factored as shown. Find the missing factors. 3x/2 + 3-x/2 = 3-x/2( )
> The expression may be factored as shown. Find the missing factors. 5x+h + 5x = 5x( )
> Write expression in the form 2kx or 3kx, for a suitable constant k. 9-x/2, 84x/3, 27-2x/3
> The expression may be factored as shown. Find the missing factors. 2x+h - 2x = 2x( )
> The expression may be factored as shown. Find the missing factors. 52+h = 25( )
> The expression may be factored as shown. Find the missing factors. 23+h = 23( )
> Differentiate the functions. y = x7(3x4 + 12x - 1)2
> Differentiate the functions. y = (x + 1)(x3 + 5x + 2)
> From the following account balances of Platypus Pty Ltd as at 30 September 2019, prepare a statement of financial position in both the T-format and the narrative classified format. Cash at bank ………………………………………………………… $ 55 000 Accounts receivable (net) ……
> Advantage Tennis Coaching, a business owned by sole trader Nicholas Cash, had the following assets and liabilities as at the financial years ended 30 June 2019 and 30 June 2020. Required (a) What is the equity as at the end of the two financial years? (
> There are four financial statements: the statement of financial position, statement of profit or loss, statement of cash flows and statement of changes in equity. Describe the information conveyed by the statement of financial position relative to that c
> You are reviewing a statement of financial position and notice that goodwill appears on the statement. Relate this information to the entity’s past investing decisions. Discuss how goodwill is measured (1) at acquisition and (2) post-acquisition.
> The High Cloud Software Company wants to increase its asset base by recognising its customer list as an asset. Discuss whether this is permissible under accounting standards.
> List three essential characteristics necessary to consider an item either as an asset or a liability.
> An entity has total assets measured at $220 000 in the statement of financial position. The entity’s liabilities total $100 000, of which $60 000 is a bank loan. Calculate the entity’s net assets. Discuss the entity’s financing decision.
> The statement of financial position for Daffodil Pty Ltd reveals cash on hand of $8000, accounts receivable of $48 000, inventory measured at $50 000 and plant and equipment measured at $116 000. The liabilities of the entity are: accounts payable $28 00
> Discuss whether the following statements are true or false. (a) The terms ‘accounts payable and ‘creditors’ mean the same thing. (b) The statement of financial position is a financial statement that shows the financial performance of an entity as at a po
> Following the collapse of the Toys“R”Us franchise in the United States, Toys“R”Us Australia went into administration on 21 May 2018, leaving 700 staff and 44 stores nationwide with an uncertain future. Subsequent to the announcement of financial difficul
> Kookaburra Ltd is always running short of cash, despite growing sales volumes and its current assets exceeding its current liabilities. A review of its operations by a consultant finds that a considerable portion of the company’s inventory is obsolete st
> Despite Australia adopting IFRS issued by the IASB, the AASB has not adopted the International Financial Reporting Standard for Small and Medium-sized Entities (IFRS for SMEs). Instead, the AASB has introduced a differential reporting framework, referred
> The most recent financial statements for the Brisbane City Council are available at www.brisbane.qld.gov.au. Referring to these statements, address the following questions. (a) Discuss why the Council is preparing GPFS. (b) Summarise the measurement basi
> The following article extract refers to the ‘goodwill glob’ and implies that it produces data that lack serviceability. Clarke and Dean (2011, p. 63) outline that there will come a time when challenges are made to the data used in company financial state
> In a significant accounting-related court case, the Australian Securities and Investments Commission (ASIC) took the directors of Centro to court over the misclassification of liabilities. On 27 June 2011, the Federal Court found the directors had breach
> During the global financial crisis in July 2008, National Australia Bank (NAB) was exposed to collateralised debt obligations (CDO). Consequently, NAB wrote down $1 billion associated with these risky instruments, with its share price plummeting and shar
> There is an IFRS on fair value measurement. The objective of this standard is to define fair value and specify the framework for measuring fair value. Identify the characteristics of assets and liabilities that should be considered when determining fair
> As a trainee accountant, you have been asked to determine the monetary value that should be assigned to the inventory of sporting equipment on hand as at the end of the financial year for Outdoor Adventures Ltd. You are currently looking at the inventory
> Koala Furniture Ltd, an office furniture retailer and wholesaler, carries a particular brand of office chair. During the year ended 30 June, the following purchases occurred. August …………………………………………… 50 chairs at $45 October ………………………………………… 60 chairs at
