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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

Find dy/dx.x + 2y/x – 3y = y1/2
Find dy/dx.ln (xy + 1) = 2xy3 + 4
Find dy/dx.ln (x + y) = 1 + x2 + y3
Find dy/dt . y 1+ √x 1-√x dx dt -4, x = 4
Find dy/dt.y = 8x3 – 7x2; dx/dt = 4, x = 2
Find dy/dt . x² + 5y x - 2y 2; dx dt = 1, x = 2, y = 0
Find dy/dt.y = 9 – 4x/3 + 2x; dx/dt = –1, x = –3
Find dy/dt. y 1 et² + 1' dx dt 3, x = 1
Find dy/dt.y = xe3x; dx/dt = –2, x = 1
Evaluate dy.y = 3x – 7/2x + 1; x = 2, Δx = 0.003
Evaluate dy.y = 8 – x2 + x3; x = –1, Δx = 0.02
Integration by substitution is related to what differentiation method? What type of integrand suggests using integration by substitution?
In your own words, describe what is meant by an integrand.
Use substitution to find each indefinite integral.∫4(2x + 3)4 dx
Find the following.∫6 dk
Find the following.∫9 dy
A reasonable assumption is that over time scarcity might drive up the price of oil and thus reduce consumption. Comment on the fact that the rate of oil consumption actually increased in 2002,
Find the following.∫(2z + 3) dz
Develop a spreadsheet that shows the time to exhaustion for various values of k.
Find the following.∫(3x – 5) dx
Go to the website WolframAlpha.com and enter “integrate.” Follow the instructions to find the time to exhaustion for various values of k. Discuss how the solution compares with the solutions
Find the following.∫(6t2 – 8t + 7) dt
Find the following.∫(5x2 – 6x + 3) dx
Find the following.∫(4z3 + 3z2 + 2z – 6)dz
Find the following.∫(16y3 + 9y2 – 6y + 3)dy
Find the following.∫(5√z + √2) dz
Find the following.∫(t1/4 + π1/4)dt
Find the following.∫5x(x2 – 8)dx
Find the following.∫x2(x4 + 4x + 3)dx
Find the following.∫(4√v – 3v3/2)dv
A sales manager presented the following results at a sales meeting.Find the total sales over the given period as follows.a. Plot these points. Connect the points with line segments.b. Use the
Find the following.∫(15x√x + 2√x) dx
Find the following.∫(10u3/2 – 14u5/2) du
In Exercises 19–40, find each indefinite integral.∫(2x + 3) dx
Find the following.(56t5/2 + 18t7/2)dt
In Exercises 19–40, find each indefinite integral.∫(5x – 1) dx
Find the following.∫(7/z2) dz
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = ln(x2 + 1)
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = x2 log |x|
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = –x/1 – x2
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = √x2 + 4
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = √2x2 + 9
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = 32x3/4
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = –6x1/3
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = 5e–x2
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = 0.5ex2
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = ln x/4x
Find ƒ″(x) for each function. Then find ƒ″(0) and ƒ″(2).f(x) = ln x + 1/x
Find the open intervals where f is increasing or decreasing.f(x) = 8xe–4x
Find the locations and values of all relative maxima and minima.f(x) = xex/x – 1
Find the locations and values of all relative maxima and minima.f(x) = ln (3x)/2x2
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = 2e–x2
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = x8/3 – 4x5/3
In Exercises 27–48, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.f(x) = x7/3 + 56x4/3
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave up or concave
In 2008, the price of gasoline in the United States spiked and then dropped. The average monthly price (in cents per gallon) of unleaded regular gasoline for 2008 can be approximated by the
The accompanying figure shows the product life cycle graph, with typical products marked on it. It illustrates the fact that a new product is often purchased at a faster and faster rate as people
Find any critical numbers for f in Exercises 57-64 and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f″(c) = 0 or f″(c)
Assume that the number of bacteria R(t) (in millions) present in a certain culture at time t (in hours) is given bya. At what time before 8 hours will the population be maximized?b. Find the maximum
Under the scenario that the fertility rate in the European Union (EU) remains at 1.8 until 2020, when it rises to replacement level, the predicted population (in millions) of the 15 member countries
The percent of concentration of a certain drug in the bloodstream x hours after the drug is administered is given byFor example, after 1 hour the concentration is given bya. Find the time at which
The next two exercises are a continuation of exercises first given in the section on Derivatives of Exponential Functions. Find the inflection point of the graph of each logistic function. This is
The next two exercises are a continuation of exercises first given in the section on Derivatives of Exponential Functions. Find the inflection point of the graph of each logistic function. This is
An autocatalytic chemical reaction is one in which the product being formed causes the rate of formation to increase. The rate of a certain autocatalytic reaction is given bywhere x is the quantity
Roger Clemens, ace pitcher for many major league teams, including the Boston Red Sox, is standing on top of the 37-ft-high “Green Monster” left-field wall in Boston’s Fenway Park, to which he
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.ln 0.98
The concentration of a certain drug in the bloodstream x hours after being administered is approximatelyUse the differential to approximate the changes in concentration for the following changes in
Every year, Corinna Paolucci sells 30,000 cases of her Famous Spaghetti Sauce. It costs her $1 per year in electricity to store a case, plus she must pay annual warehouse fees of $2 per case for the
The population of bacteria (in millions) in a certain culture x hours after an experimental nutrient is introduced into the culture isUse the differential to approximate the changes in population for
A man 6 ft tall is walking away from a lamp post at the rate of 50 ft per minute. When the man is 8 ft from the lamp post, his shadow is 10 ft long. Find the rate at which the length of the shadow is
The short-term demand for crude oil in the United States in 2008 can be approximated by q = f(p) = 2,431,129p–0.06, where p represents the price of crude oil in dollars per barrel and q represents
In 2008, the Valve Corporation, a software entertainment company, ran a holiday sale on its popular Steam software program. Using data collected from the sale, it is possible to estimate the demand
Let Find each derivative.a. du/dv b. dv/duc. Based on your answers to parts a and b, what do you notice about the relationship between du/dv and dv/du? √u + √2v + 1 = 5.
Find the absolute maximum and minimum of f(x) = 2x – 3x2/3(a) On the interval [–1, 0.5];(b) On the interval [0.5, 2]
The number of bank robberies in the United States for the years 2000–2009 is given in the following figure. Consider the closed interval [2000, 2009].a. Give all relative maxima and minima and when
Let eu2–v – v = 1. Find each derivative.a. du/dvb. dv/duc. Based on your answers to parts a and b, what do you notice about the relationship between du/dv and dv/du?
The number of bank burglaries (entry into or theft from a bank during nonbusiness hours) in the United States for the years 2000–2009 is given in the figure. Consider the closed interval
Answer the questions in Exercise 40 for two investments with respective rates of profitability P1'(t) = 130 + t2 hundred dollars per year and P2'(t) = 306 + 5t hundred dollars per year.Data from
Use the method of Lagrange multipliers to prove that of all isosceles triangles with a given perimeter, the equilateral triangle has the largest area.
Beverly estimates that when she is corresponding on a regular basis with x close friends and is working on y interesting projects, her total satisfaction is measured by the utility function U(x, y) =
A car traveling at 67 ft/sec decelerates at the constant rate of 23 ft/sec2 when the brakes are applied.a. Find the velocity v(t) of the car t seconds after the brakes are applied. Then find its
What is ƒbx dx for base (b > 0, b ≠ 1)?
Consider the functiona. Using a graphing calculator, try to find any local minima, or tell why finding a local minimum is difficult for this function.b. Find any local minima using the techniques of
On August 8, 2007, the power used in New York state (in thousands of megawatts) could be approximated by the functionwhere t is the number of hours since midnight, for 0 ≤ t ≤ 24. Find any
The demand equation for telephones at one store iswhere p is the price (in dollars) and q is the quantity of telephones sold per week. Find the values of q and p that maximize revenue. P = D(q) =
The annual unemployment rates of the U.S. civilian non institutional population for 1990–2009 are shown in the graph. Identify the years where relative extrema occur, and estimate the unemployment
a. Verify thatThis expression is called the relative rate of change. It expresses the rate of change of f relative to the size of f. Stephen B. Maurer denotes this expression by f̂ and notes that
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = x2–x2
On many calculators, graphs of rational functions produce lines at vertical asymptotes. For example, graphing y = (x – 1)/(x + 1) on the window [–4.9, 4.9] by [–4.9, 4.9] produces such a line
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.y = 1.1 – 0.3x – 0.3x2
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = 2/3 x3 – x2 – 24
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = 2/3 x3 – x2 – 4x + 2

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