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mathematics
applied calculus
Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions
Verify that daily compounding is nearly the same as continuous compounding by graphing y = 100[1 + (.05/360)]360x, together with y = 100e0.05x in the window [0, 64] by [250, 2500]. The two graphs should appear the same on the screen. Approximately how far apart are they when x = 32? When x = 64?
A population is growing exponentially with growth constant .04. In how many years will the current population double?
A population is growing exponentially with growth constant .05. In how many years will the current population triple?
The function A(t) in Fig. 2(a) gives the balance in a savings account after t years with interest compounded continuously. Figure 2(b) shows the derivative of A(t).(a) What is the balance after 20 years?(b) How fast is the balance increasing after 20 years?(c) Use the answers to parts (a) and (b)
Consider the demand function q = 60,000e-0.5p from Check Your Understanding 5.3.(a) Determine the value of p for which the value of E( p) is 1. For what values of p is demand inelastic?(b) Graph the revenue function in the window [0, 4] by [-5000, 50,000], and determine where its maximum value
The curve in Fig. 1 shows the growth of money in a savings account with interest compounded continuously.(a) What is the balance after 20 years?(b) At what rate is the money growing after 20 years?(c) Use the answers to parts (a) and (b) to determine the interest rate. 1000 600 200 y slope =
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.
Let C(x) = 1000e0.02x. Determine and simplify the formula for Ec (x). Show that Ec (60) > 1, and interpret this result.
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.
Let C(x) = (1>10)x2 + 5x + 300. Show that Ec (50) < 1. (Hence, when 50 units are produced, a small relative increase in production results in an even smaller relative increase in total cost. Also, the average cost of producing 50 units is greater than the marginal cost at x = 50.)
Show that Ec is equal to the marginal cost divided by the average cost.
When a rod of molten steel with a temperature of 1800°F is placed in a large vat of water at temperature 60°F, the temperature of the rod after t seconds isThe graph of this function is shown in Fig. 2.(a) What is the temperature of the rod after 11 seconds?(b) At what rate is the temperature of
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) At what time will the population equal 1200?
A certain bacteria culture grows at a rate proportional to its size. If 10,000 bacteria grow at the rate of 500 bacteria per day, how fast is the culture growing when it reaches 15,000 bacteria?
Show that Ec(x) = x · C′(x)/C(x).
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 be present?
Ten thousand dollars is deposited in a money market fund paying 8% interest compounded continuously. How much interest will be earned during the second year of the investment?
A country that is the major supplier of a certain commodity wishes to improve its balance-of-trade position by lowering the price of the commodity. The demand function is q = 1000/p2.(a) Compute E( p).(b) Will the country succeed in raising its revenue? 24. Show that any demand function of the form
Investment A is currently worth $70,200 and is growing at the rate of 13% per year compounded continuously. Investment B is currently worth $60,000 and is growing at the rate of 14% per year compounded continuously. After how many years will the two investments have the same value?
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 reaches 5 million people.(b) Use the
A subway charges 65 cents per person and has 10,000 riders each day. The demand function for the subway is q = 2000√90 - p.(a) Is demand elastic or inelastic at p = 65?(b) Should the price of a ride be raised or lowered to increase the amount of money taken in by the subway?
If the present value of $1000 to be received in 5 years is $559.90, what rate of interest, compounded continuously, was used to compute this present value?
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?
Currently, 1800 people ride a certain commuter train each day and pay $4 for a ticket. The number of people q willing to ride the train at price p is q = 600(5 - √p). The railroad would like to increase its revenue.(a) Is demand elastic or inelastic at p = 4?(b) Should the price of a ticket be
In Exercises, determine the growth constant k, then find all solutions of the given differential equation.y′ - 6y = 0
If ln x = -1, write x using the exponential function.
In Figure 1.60, which shows the cost and revenue functions for a product, label each of the following: (a) Fixed costs (b) Break-even quantity (c) Quantities at which the company: (i) Makes a profit (ii) Loses moneyFigure 1.60 $ R с 9
Each of the following functions gives the amount of a substance present at time t. In each case, give the amount present initially (at t = 0), state whether the function represents exponential growth or decay, and give the percent growth or decay rate. (a) A = 100(1.07)t (b) A =
Find an equation for the line that passes through the given points.(0, 2) and (2, 3)
Find the following: (a) f(g(x)) (b) g(f(x)) (c) f(f(x))f(x) = 5x − 1 and g(x) = 3x + 2
Graph the function. What is the amplitude and period?y = 3 sin x
Determine whether or not the function is a power function. If it is a power function, write it in the form y = kxp and give the values of k and p. y = X
For Problems find k such that P = P0ekt has the given doubling time.10
Which graph in Figure 1.5 best matches each of the following stories? Write a story for the remaining graph.(a) I had just left home when I realized I had forgotten my books, so I went back to pick them up. (b) Things went fine until I had a flat tire. (c) I started out calmly but sped up
Decide whether the graph is concave up, concave down, or neither. X
The following functions give the populations of four towns with time t in years.(i) P = 600(1.12)t (ii) P = 1,000(1.03)t (iii) P = 200(1.08)t (iv) P = 900(0.90)t (a) Which town has the largest percent growth rate? What is the percent growth rate? (b) Which town has the
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 4*, (√3)*, ()*
Compute the given derivatives with the help of formulas (1)–(4).(a)(b) d 4 (2²¹) dx x=1
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 82x/3, 93x/2, 16-3x/4
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 27%, (√2)", ()*
Compute the given derivatives with the help of formulas (1)–(4).(a)(b) d dx (2¹) x=1/2
State as many laws of exponents as you can recall.
Compute the given derivatives with the help of formulas (1)–(4).(a)(b) d dx (ex) x=1
What is e?
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 9-x/2 84x/3, 27-2x/3
Compute the given derivatives with the help of formulas (1)–(4).(a)(b) d dx (1) - x=e
Write the differential equation satisfied by y = Cekt.
If ex = 5, write x in terms of the natural logarithm.
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 7/x(+³) *xç_(†) *xx (7)
State the properties that graphs of the form y = ekx have in common when k is positive and when k is negative.
If e-x = 3.2, write x in terms of the natural logarithm.
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 6x.3x, 최창 15* 12* 5x ⁹ 22x
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. (੬), (ਉਹ), (k)/2 16
Write each expression in the form ekx for a suitable constant k. (e)/5. 1 X ₂,2 \e
Write each expression in the form ekx for a suitable constant k. (e²).x, (1) X
What are the coordinates of the reflection of the point (a, b) across the line y = x?
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 7-*• 14*, 2x 32x 6x⁹ 18x
What is a logarithm?
If ln x = 4.5, write x using the exponential function.
Write each expression in the form ekx for a suitable constant k. X 2e4x+2.gr-2
Write each expression in the form ekx for a suitable constant k. 2x (+) ² 3 el-x. e³x-1
What is the x-intercept of the graph of the natural logarithm function?
Write each expression in the form ekx for a suitable constant k. (e4x. e6x)3/5 1 -2x e
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 34x 25x+1 9-x 32x³ 2.2 x 27-x/3
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 23x. 2-5x/2, 32x. ()2x/3
State the main features of the graph of y = ln x.
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 2x 3-5x 16x 6x⁹3-2x³8-x 외화
Write each expression in the form ekx for a suitable constant k. Vex.ex 7x -3x e -4.x e
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. 25x/4. (1)x, 3-2x. 35x/2
State the two key equations giving the relationships between ex and ln x.
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. (2-3x. 2-2x)2/5, (91/2.94)x/9
What is the difference between a natural logarithm and a common logarithm?
Give the formula that converts a function of the form bx to an exponential function with base e.
State the differentiation formula for each of the following functions(a) f (x) = ekx (b) f (x) = eg(x) (c) f (x) = ln g(x)
Which is larger, 2 ln 5 or 3 ln 3? Explain.
State the four algebraic properties of the natural logarithm function.
Which is larger, 1/2 ln 16 or 1/3 ln 27? Explain.
What differential equation is key to solving exponential growth and decay problems? State a result about the solution to this differential equation.
Consider the function f (x) = 5(1 - e-2x), x ≥ 0.(a) Show that f (x) is increasing and concave down for all x ≥ 0.(b) Explain why f(x) approaches 5 as x gets large.(c) Sketch the graph of f (x), x ≥ 0.
The atmospheric pressure P(x) (measured in inches of mercury) at height x miles above sea level satisfies the differential equation P′(x) = -.2P(x). Find the formula for P(x) if the atmospheric pressure at sea level is 29.92.
Determine the growth constant k, then find all solutions of the given differential equation.y′ = y
Let A(t) = 5000e0.04t be the balance in a savings account after t years.(a) How much money was originally deposited?(b) What is the interest rate?(c) How much money will be in the account after 10 years?(d) What differential equation is satisfied by y = A(t)?(e) Use the results of parts (c) and (d)
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.f (t) = t2 at t = 10 and t = 50
What is a growth constant? A decay constant?
Consider the function g(x) = 10 - 10e-0.1x, x ≥ 0.(a) Show that g(x) is increasing and concave down for x ≥ 0.(b) Explain why g(x) approaches 10 as x gets large.(c) Sketch the graph of g(x), x ≥ 0.
The herring gull population in North America has been doubling every 13 years since 1900. Give a differential equation satisfied by P(t), the population t years after 1900.
Determine the growth constant k, then find all solutions of the given differential equation.y′ = .4y
Let A(t) be the balance in a savings account after t years, and suppose that A(t) satisfies the differential equation A′(t) = .045A(t), A(0) = 3000.(a) How much money was originally deposited in the account?(b) What interest rate is being earned?(c) Find the formula for A(t).(d) What is the
A student learns a certain amount of material for some class. Let f (t) denote the percentage of the material that the student can recall t weeks later. The psychologist Hermann Ebbinghaus found that this percentage of retention can be modeled by a function of the form f (t) = (100 - a)e-λt + a,
A piece of charcoal found at Stonehenge contained 63% of the level of 14C found in living trees. Approximately how old is the charcoal?
Determine the growth constant k, then find all solutions of the given differential equation. y' = 1.7y
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.f (t) = t10 at t = 10 and t = 50
Determine the growth constant k, then find all solutions of the given differential equation. y' 4
What is meant by the half-life of a radioactive element?
If y = 2(1 - e-x), compute y′ and show that y′ = 2 - y.
Find the present value of $10,000 payable at the end of 5 years if money can be invested at 12% with interest compounded continuously.
Determine the growth constant k, then find all solutions of the given differential equation. y' I = 0 || =
Four thousand dollars is deposited in a savings account at 3.5% yearly interest compounded continuously.(a) What is the formula for A(t), the balance after t years?(b) What differential equation is satisfied by A(t), the balance after t years?(c) How much money will be in the account after 2
Find the logarithmic derivative and then determine the percentage rate of change of the functions at the points indicated.f (x) = e0.3x at x = 10 and x = 20
Explain how radiocarbon dating works.
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