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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
According to the 2010 census, the U.S. population was 309 million with an estimated growth rate of 0.8%/yr.a. Based on these figures, find the doubling time and project the population in 2050.b. Suppose the actual growth rate is just 0.2 percentage point lower than 0.8%/yr (0.6%). What are the
Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.The number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?
Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.Between 2005 and 2010, the average rate of inflation was about 3%/yr (as measured by the Consumer Price Index). If a cart of
Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.How long will it take an initial deposit of $1500 to increase in value to $2500 in a saving account with an APY of 3.1%? Assume
Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.The current population of a town is 50,000 and is growing exponentially. If the population is projected to be 55,000 in 10
Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.The population of Clark County, Nevada, was 2 million in 2013. Assuming an annual growth rate of 4.5%/yr, what will the county
Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.The population of a town with a 2010 population of 90,000 grows at a rate of 2.4%/yr. In what year will the population double
Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.f(t) = 2200 + 400t, g(t) = 400 • 2t/20
Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.f(t) = 100 + 10.5t, g(t) = 100et/10
Give two examples of processes that are modeled by exponential decay.
Give two examples of processes that are modeled by exponential growth.
How are the rate constant and the half-life related?
How are the rate constant and the doubling time related?
Explain the meaning of half-life.
Explain the meaning of doubling time.
Give two pieces of information that may be used to formulate an exponential growth or decay function.
In terms of relative growth rate, what is the defining property of exponential growth?
We will encounter the harmonic sumUse a left Riemann sum to approximate (with unit spacing between the grid points) to show that Use this factto conclude thatdoes not exist. 3 1/2 en+1 1
Assume that y > 0 is fixed and that x > 0. Show that d/dx (ln xy) = d/dx (ln x). Recall that if two functions have the same derivative, then they differ by an additive constant. Set x = 1 to evaluate the constant and prove that ln xy = ln x + ln y.
Use a left Riemann sum with at least n = 2 subintervals of equal length to approximateand show that ln 2 < 1. Use a right Riemann sum with n = 7 subintervals of equal length to approximate and show that ln 3 > 1. dt In 2 = .3 dt In 3 =
Use the following argument to show thata. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.b. Construct a rectangle over the interval [1, 2] with height 1/2. Explain why ln 2 > 1/2.c. Show
Use the inverse relations between ln x and ex (exp (x)), and the properties of ln x to prove the following properties.a. exp (0) = 1.b. exp (x - y) = exp(x)/exp(y).c. (exp (x))p = exp (px), p rational.
Differentiate ln x for x > 0 and differentiate ln (-x) for x < 0 to conclude that d/dx (ln |x|) = 1/x.
Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one-fourth the area of the square? Begin by showing that x and y must
Evaluate the following integrals. 16* 2x
Evaluate the following integrals. In 2 e3x – e-3x dx -3x e3x + e Зх
Evaluate the following integrals. sin (In x) 4x
Evaluate the following integrals. 2e 3ln.x dx х
Evaluate the following integrals. т 2sin x cos x dx
Evaluate the following integrals.∫ x210x3 dx
Evaluate the following integrals. 55x dx
Evaluate the following integrals.∫ 3-2x dx
Evaluate the following integrals.∫ 72x dx
Compute the following derivatives using the method of your choice.d/dx (cos (x2 sin x))
Compute the following derivatives using the method of your choice.d/dx (x(x10))
Compute the following derivatives using the method of your choice. dx
Compute the following derivatives using the method of your choice.d/dx (xe + ex)
Compute the following derivatives using the method of your choice.d/dx ((1/x)x)
Compute the following derivatives using the method of your choice.d/dx (xtan x)
Compute the following derivatives using the method of your choice.d/dx (e-10x2)
Compute the following derivatives using the method of your choice.d/dx (x2x)
What is the average value of f(x) = 1/x on the interval [1, p] for p > 1? What is the average value of f as p→∞?
Using calculus and accurate sketches, explain how the graphs of f(x) = xp ln x differ as x→ 0+ for p = 1/2, 1, and 2.
Consider the functiona. Are there numbers 0 < a < 1 such that b. Are there numbers a > 1 such that f(x) || •1+a Г. f(x) dx 1- a 0?
Use a calculator to approximate the following limits. Confirm your result with l’Hôpital’s Rule. In (1 + x) lim
Use a calculator to approximate the following limits. Confirm your result with l’Hôpital’s Rule. 2* – 1 lim
Use a calculator to approximate the following limits. Confirm your result with l’Hôpital’s Rule. lim (1 + 3h)2/h
Use a calculator to approximate the following limits. Confirm your result with l’Hôpital’s Rule. /h lim (1 + 2h)' h→0
Use the integral definition of the natural logarithm to prove that ln (x/y) = ln x - ln y.
Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.a. ln xy = ln x + ln y. b. ln 0 = 1.c. ln (x + y) = ln x + ln y. d. 2x = e2 ln x.e. The area under the curve y = 1/x and the x-axis on the interval [1, e] is 1.
Evaluate the derivatives of the following functions.Q(t) = t1/t
Evaluate the derivatives of the following functions.G(y) = ysin y
Evaluate the derivatives of the following functions.p(x) = x-ln x
Evaluate the derivatives of the following functions.H(x) = (x + 1)2x
Evaluate the derivatives of the following functions.h(t) = (sin t)√t
Evaluate the derivatives of the following functions.h(x) = 2(x2)
Evaluate the derivatives of the following functions.f(x) = xπ
Evaluate the derivatives of the following functions.f(x) = (2x)4x
Evaluate the following integrals. 4cot x dx sin?
Evaluate the following integrals.∫ x2 6x3 + 8 dx
Evaluate the following integrals. 1/2 10/P dp p² 1/3
Evaluate the following integrals. (1 + In x)x* dx
Evaluate the following integrals. 7/2 4sin x cos x dx Jo
Evaluate the following integrals. 10* dx -1
Evaluate the following integrals. In 3 et + e* x- dx -2x e2x – 2 + e In 2
Evaluate the following integrals. e* + e* dx e* — е х
Evaluate the following integrals. 2 z/2 dz 2 e12 + 1 -2
Evaluate the following integrals. Vĩ dx VI
Evaluate the following integrals. sin x sin. dx sec x
Evaluate the following integrals. 4 хе dx
Evaluate the following integrals. Include absolute values only when needed. 'y Inª (y² + 1) y? + 1 ấp.
Evaluate the following integrals. Include absolute values only when needed. e3 dx x In x In? (In x) e²
Evaluate the following integrals. Include absolute values only when needed. dx x In x In (In x)
Evaluate the following integrals. Include absolute values only when needed. e2r dx 4 + e2r
Evaluate the following integrals. Include absolute values only when needed. /2 sin x dx 1 + cos x
Evaluate the following integrals. Include absolute values only when needed. dx x In x х
Evaluate the following integrals. Include absolute values only when needed.∫ tan x dx
Evaluate the following integrals. Include absolute values only when needed. 2x - 1 dx
Evaluate the following derivatives.d/dx (ln3 (3x2 + 2))
Evaluate the following derivatives.d/dx ((ln 2x)-5)
Evaluate the following derivatives.d/dx (ln (cos2 x))
Evaluate the following derivatives.d/dx (sin (ln x))
Evaluate the following derivatives.d/dx (ln (ln x))
Evaluate the following derivatives. (x In. dx x=1
Evaluate d/dx (3x)
Express 3x, xπ, and xsin x using the base e.
What is the inverse function of ln x, and what are its domain and range?
Evaluate ∫ 4x dx
Give a geometrical interpretation of the function In x
What are the domain and range of ln x?
Archimedes’ principle says that the buoyant force exerted on an object that is (partially or totally) submerged in water is equal to the weight of the water displaced by the object (see figure). Let ρw = 1 g/cm3 = 1000 kg/m3 be the density of water and let ρ be the density of an object in
A large tank has a plastic window on one wall that is designed to withstand a force of 90,000 N. The square window is 2 m on a side, and its lower edge is 1 m from the bottom of the tank.a. If the tank is filled to a depth of 4 m, will the window withstand the resulting force?b. What is the maximum
Suppose a cylindrical glass with a diameter of 1/12 m and a height of 1/10 m is filled to the brim with a 400-Cal milkshake. If you have a straw that is 1.1 m long (so the top of the straw is 1 m above the top of the glass), do you burn off all the calories in the milkshake in drinking it? Assume
A square plate 1 m on a side is placed on a vertical wall 1 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then compute the force on each plate. surface surface 1 m
A plate shaped like an equilateral triangle 1 m on a side is placed on a vertical wall 1 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then compute the force on each plate. surface surface
A body of mass m is suspended by a rod of length L that pivots without friction (see figure). The mass is slowly lifted along a circular arc to a height h. a. Assuming that the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc
A 60-m-long, 9.4-mm-diameter rope hangs free from a ledge. The density of the rope is 55 g/m. How much work is needed to pull the entire rope to the ledge?
A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5 kg/m. a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the chain
A rigid body with a mass of 2 kg moves along a line due to a force that produces a position function x(t) = 4t2, where x is measured in meters and t is measured in seconds. Find the work done during the first 5 s in two ways.a. Note that x"(t) = 8; then use Newton’s second law (F = ma = mx"(t))
For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm/(x + R)2, where G = 6.7 × 10-11 N-m2/kg2 is the gravitational constant, M = 6 × 1024 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378 × 106 m is the radius of
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