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calculus

Questions and Answers of Calculus

Suppose that f is continuous and strictly increasing on [0,1] with f(0) = 0 and f(1) = 1, ifCalculate
In problems, show that f has an inverse by showing that it is strictly monotonic. (a) f(x) = -x5 - x3 (b) f(x) = x7 + x5 (c) f(θ) = cos θ, 0 ≤ θ ≤ π
Use your calculator to calculate each of the following (a) e3 (b) e2.1 (b) e√2 (d) ecos(ln4)
In problems, find Dxy (a) y = ex+2 (b) e-2lnx (c) lnecosx (d) ln e-2x-3
Use your knowledge of the graph of y = ex to sketch the graphs of (a) y = -ex (b) y = e-x
In Problems, first find the domain of the given function f and then find where it is increasing and decreasing, and also where it is concave upward and downward. Identify all extreme values and
In problems, find each integral.(a) ˆ« e3x+1 dx(b)(c)
The region bounded byy = 0, x = 0, and x = 1 is revolved about the y-axis. Find the volume of the resulting solid
Find the area of the region bounded by the curve y = e-x and the line through the points (0,1) and (1, 1/e)
Show that f(x) = x/ex-1 - ln (1-e-x) is decreasing for x > 0.
Stirling's formula says that for large n we can approximate n! = 1. 2.3...n by n! ≈ √2πn (n/e)n (a) Calculate 10! Exactly and then approximately using the above formula (b) Approximate 60!.
It will be shown later that for small x ex ≈ 1 + x + x2/2! + x3/3! + x4/4! Use this result to approximate e0.3 and compare your result with what you get by calculating it directly.
Find the length of the curve given parametrically by x = et sint, y = et cos t, 0 ≤ t ≤ π.
Let R be the region bounded by x = 0, y = ex, and the tangent line to y = ex that goes through the origin, Find (a) The area of R (b) The volume of the solid obtained when R is revolved about the
In problems, solve for x (a) log2 8 = x (b) log5 x = 2 (c) log4 x = 3/2 (d) logx 64 = 4 (e) log 2log9(x/3) = 1
In problems, use natural logarithms to solve each of the exponential equations. (a) 2x = 17 (b) 5x = 13 (c) 52s-3 = 4
In problems, find eth indicated derivative or integral.(a) Dx(62x)(b)(c) Dx log3 ex (b) Dx log10 (x3 + 9)
In problems, find dy/dx. You must distinguish amount problems of the type ax, xa, and xx.(a)(b) y = sin2x + 2sinx (c) y = x Ï€+1 + (Ï€+1)x
Let f(x) = πx and g(x) = xπ. Which is larger, f(e) or g(e)? f'(e) or g'(e)?
In problems, first find the domain of the given function f and then find where it is increasing and decreasing, and also where it is concave upward and downward. Identify all extreme values and
Sketch the graphs of log1/3 x and log3 x using the same coordinate axes.
The magnitude M of an earthquake on the Richter scale is M = 0.67 log10 (0.37E) + 1.46 Where E is the energy of the earthquake in kilowatt-hours find the energy of an earthquake of magnitude 7. Of
In the equally tempered scale to which keyed instruments have been tuned since the days of J.S. Bach (1685-1750), the frequencies of successive notes C, C#, D, D#, F, F#, G, G#, A, A#, B, C from a
Prove that log2 3 is irrational. Use proof by contradiction.
You suspect that the xy-data that you collect lie on either an exponential curve y = Abx or a power curve y = Cxd. To check, you plot ln y against x on one graph and ln y against ln x on another
Convince yourself that f(x) = (xx)x and g(x) = x((x)x) are not same function, then find f'(x) and g'(x).
Find limx→0 xx. Also find the coordinates of the minimum point for f(x) = xx on [0, 4]
Draw the graphs of y = x3 and y = 3x using the same axes and find all their intersection appoints.
Refer to problem 49. Draw the graphs of f and g using the same axes. Then draw the graphs of f' and g' using the same axes. In Problem 49 Convince yourself that f(x) = (xx)x and g(x) = x((x)x) are
Use loga x = (lnx) / (lna) to calculate each of the logarithms in Problems. (a) log5 12 (b) log7 (0.11) (c) log11 (8.12)1/5 (d) log10 (8.57)7
In problems, solve the given differential equation subject to the given condition, (a) dy/dt = -6y, y(0) = 4 (b) dy/dt = 6y, y(0) = 1 (c) dy/dt = 0.005y, y(10) = 2
Determine the proportionality constant k in dy/dt = ky for problem 9. Then use y = 4.5ekt to find the population after 100 years.
A population is growing at a rate proportional to its size. After 5 years, the population size was 164000. After 12 years, the population size was 235000. What was the original population size?
The mass of a tumor grows at a rate proportional to its size. The first measurement of its size was 4.0 grams. Four months later its mass was 6.76 grams. How large was the tumor six months before the
A radioactive substance has a half-life of 700 years. If there were 10 grams initially, how much would be left after 300 years?
If a radioactive substance loses 15% of its radioactivity in 2 days, what is its half-life?
An unknown amount of a radioactive substance is being studied. After two days, the mass is 15.231 grams. After eight days, the mass is 9.086 grams. How much was there initially? What is the half-life
Human hair from a grave in Africa proved to have only 51% of the carbon 14 of living tissue. When was the body buried?
An object is taken from an oven at 300oF and left to cool in a room at 75oF. If the temperature fell to 200oF in ½ hour, what will it be after 3 hours?
A thermometer registered -20oC outside and then was brought into a house where the temperature was 24oC. After 5 minutes it registered 0oC. When will it register 20oC?
An object initially at 26oC is placed in water having temperature 90oC. If the temperature of the object rises to 70oC in 5 min what be the temperature after 10 minutes?
A batch of brownies is taken from a 350oF oven and placed in a refrigerator at 40oF and left to cool. After 15minutes the brownies have cooled to 250oF. When will the temperature of brownies be
A dead body is found at 10p.m. to have temperature 82oF. One hour later the temperature was 76oF. The temperature of the room was constant 70oF. Assuming the temperature of the body was 98.6oF when
If $375 in put in the bank today, what will it be worth at the end of 2 years if interest is 3.5% and is compounded as specified? (a) Annually (b) Monthly (c) Daily (d) Continuously
Do problem 25 assuming that the interest rate is 4.6% In problem 25 If $375 in put in the bank today, what will it be worth at the end of 2 years if interest is 3.5% and is compounded as
How long does it take money to double in value for the specified interest rate? (a) 6% compounded monthly (b) 6% compounded continuously
It will be shown later for small x that ln (1+x) ≈ x. Use this fact to show that the doubling time for money invested at p percent compounded annually is about 70/p years.
Sketch the graph of the solution in Problem 34 when y0 = 6.4, L = 16, and k = 0.00186 limt→0 y = 16
Consider a country with a population of 10 million in 1985, a growth rate of 1.2% per year, and immigration from other countries of 60000 per year. Use the differential equation of problem 38 to
Show that the relative rate of change of any polynomial approaches zero as the independent variable approaches infinity.
Prove that is the relative rate of change is a positive constant then the function must represent exponential growth.
Prove that is the relative rate of change is a negative constant then the function must represent exponential decay.
Assume that (1) world population continues to grow exponentially with growth constant k = 0.0132, (2) it take s1/2 acre of land to supply food for one person, and (3) there are 13500000 square miles
Using the same axes, draw the graphs for 0 ≤ t ≤ 100 of the following two models for the growth of world population (a) Exponential growth: y = 6.4e0.0132t (b) Logistic growth: y =
A bacterial population grows at a rate proportional to its size. Initially, it is 10000, and after 10days it is 20000. What is the population after 25 days?
How long will it take the population of Problems 5 to triple? In problem 5 A bacterial population grows at a rate proportional to its size. Initially, it is 10000, and after 10days it is 20000. What
The population of the United States was 3.9 million in 1790 and 178 million in 1960. If the rate of growth is assumed proportional to the number present, what estimate would you give for the
The population of a certain country is growing a t 3.2% per year; that is if it is A at the beginning of the year, it is 1.032A at the end of that year. Assuming that if is 4.5 million now, what will
In problems, solve each differential equation (a) dy/dx + y = e-x (b) (x+1) dy/dx + y = x2 -1 (c) (1-x2) dy/dx + xy = ax, |x| < 1
A tank contains 20 gallons of a solution, with 10 pounds of chemical A in the solution. At a certain instant, we begin pouring in a solution containing the same chemical in a concentration of 2
A tank initially contains 200 gallons of brain, with 50 pounds of salt in solution. Brine containing 2 pounds of salt per gallon is entering the tank at the rate of 4 gallons per minute and is
A tank initially contains 120 gallons of pure water. Brine with 1 pound of salt per gallon flows into the tank at 4 gallons per minute, and the well-stirred solution runs out at 6 gallons per minute.
A tank initially contains 50 gallons of brine, with 30 pounds of salt in solution. Water runs into the tank at 3 gallons per minute and the well-stirred solution runs out at 2 gallons per minute. How
Find the current I as a function of time for the circuit of Figure 3 if the switch S is closed and I = 0 at t = 0
Find I as a function of time for the circuit of Figure 4 assuming that the switch is closed and I = 0 at t = 0
Mary bailed out of her plane at an altitude of 8000 feet fell freely for 15 seconds, and then opened her parachute. Assume that the drag coefficients are a = 0.10 for free fall and a = 1.6 with the
In problems, a slop field is given for a differential equation of the form y' = f(x,y). Use the slope field to sketch the solution that satisfies the given initial condition. In each case, find
In problems, use Euler's Method with h = 0.2 to approximate the solution over the indicated interval. (a) y' = 2y, y(0) = 3, [0,1] (b) y' = -y, y(0) = 2, [0,1]
Apply Euler's Method to the equation y ' = y, y(0) = 1 with an arbitrary step size h = 1/N where N is a positive integer. (a) Derive the relationship yn = y0 (1+h)n (b) Explain why yN is an
Suppose that the function f(x,y) depends only on x. The differential equation y' = f(x,y) can then be written as y' = f(x), y(x0) = 0 Explain how to apply Euler's Method to this differential
For problems, use the improve Euler method with h = 0.2 on the equation in Problems 11-12 compare your answer with obtained using Euler's Method. In problem 11 and 12 (a) y' = 2y, y(0) = 3, [0,1] (b)
Apply the Improved Euler Method to the equation y' = y, y(0) = 1, with h = 0.2, 0.1, 0.05, 0.01, 0.005 to approximate the solution on the interval [0,1]. Compute the error in approximating y(1) and
In problems, a slop field is given for a differential equation of the form y' = f(x,y). In both cases, every solution has the same oblique asymptote. Sketch the solution that satisfies the given
In problems, plot a slop field for each differential equation. Use the method of separation fo variables or an integrating factor to find a particular solution of different equation that satisfies
In problems, find the exact value without using a calculator. (a) arcos(√2/2) (b) arcsin(-√3/2) (b) sin-1(-√3/2) (d) sin-1(-√2/2)
In problems, approximate each value (a) sin-1(0.1113) (b) arcos (0.6341) (c) cos(arccot 3.212) (d) sec(arcos 0.5111)
In problems, express Î¸ in terms of x using the inverse trigonometric functions sin-1, cos-1, tan-1 and sec-1.(a)(b) (c) (d) (e)
In problems, find each value without using a calculator. (a) cos[2sin-1(-2/3)] (b) tan[2tan-1(1/3)] (c) sin[cos-1(3/5) + cos-1(5/13)] (d) cos[cos-1(4/5) + sin-1(12/13)]
In problems, show that each equation is an identity (a) tan(xin-1x) = x/√1-x2 (b) sin(tan-1x) = x/√1 + x2 (c) cos(2sin-1x) = 1 - 2x2 (d) tan(2tan-1x) = 2x/1-x2
Find each limit. (a) limx→∞ tan-1x (b) limx→-∞ tan-1x
Find each limit. (a) limx→∞ sec-1x (b) limx→-∞ sec-1x
Find each limit (a) Limx→1- sin-1 x (b) Limx→1+ sin-1 x
Does limx→1 sin-1 x exist? Explain.
Describe what happens to the slope of the tangent line to the graph of y = sin-1x at the point c if c approaches 1 from the left.
In problems, find dy/dx (a) y = ln(2 + sinx) (b) y = etanx (c) y = ln(secx + tanx) (d) y = -ln(csc x + cot x)
In problems, evaluate each integral. (a) ∫cos 3x dx (b) ∫xsin(x2) dx (c) ∫sin 2x cos 2x dx (d) ∫tanx dx = ∫sin x / cosx dx
A picture 5 feet in height is hung on a wall so that its bottom is 8 feet from the floor, as shown figure 9. A viewer with eye level at 5.4 feet stands b feet from the wall. Express Î¸,
By repeated use of the addition formula tan(x + y) = (tanx + tany) / (1 - tanx tany) Show that π/4 = 3tan-1 (1/4) + tan-1(5/99)
Draw the graph of y = π/2 - arcsinx. Make a conjecture. Prove it.
Draw the graph of y = sin (arcsinx) on [-1, 1], Then draw the graph of y = arsin (sinx) on [-2π, 2 π]. Explain the difference that you observe.
Show that ∫dx/√a2 - x2 = sin-1 x/a + C, a > 0 By writing a2 - x2 = a2 [1-(x/a)2] and making the substitution u = x/a
Show the result in problem 81 by differentiating the right side to get the integrand. In problem 81 ∫dx/√a2 - x2 = sin-1 x/a + C, a > 0 By writing a2 - x2 = a2 [1-(x/a)2] and making the
Show that ∫dx/a2 + x2 = 1/a tan-1 x/a + C, a ≠ 0
Show by differentiating the right side, thata > 0
Use the result of Problem 85 show thatWhy is this result expected?
The lower edge of a wall hanging, 10 feet in height, is 2 feet above the observer's eye level. Find the ideal distance b to stand from the wall for viewing the hanging; that is find b that maximizes
The structure steel work of a new office building is finished. Across the street, 60 feet from the ground floor or the freight elevator shaft in the building, a spectator is standing and watching the
An airplane is flying at a constant altitude of 2 miles and a constant speed of 600 miles per hour on a straight course that will take it directly over an observer on the ground. How fast is the
A revolving beacon light is located on an island and is 2 miles away from the nearest point P of the straight shoreline of the mainland. The beacon throws a spot of light that moves along the
A man on a dock is pulling in a rope attached to a rowboat at a rate of 5 feet per second. If the man's hands are 8 feet higher than the point where the rope is attached to the boat, how fast is the

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