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mathematics
calculus

Questions and Answers of Calculus

If M is an -dimensional manifold in Rn, with the usual orientation, show that dV = dx1^ . . . ^ dxn, so that the volume of M, as defined in this section, is the volume as defined in Chapter 3. (Note
Generalize Theorem 5-6 to the case of an oriented (n - 1) -dimensional manifold in Rn. The generalization is w Є A n-1(Mx) defined by
a. If f: |a, b|→ R is non-negative and the graph of f in the x,y -plane is revolved around the -axis in R3 to yield a surface M, show that the area of M is
If Ί: Rn → Rn is a norm preserving linear transformation and M is a k-dimensional manifold in Rn, show that M has the same volume as Ί(M).
a. If c: [0, 2π] x [-1, 1] → c: [0 , 2π] x [-1, 1] → R3 is defined by c (u,v) = (2 eos (u) + vsin (u/2) eos (u), 2sin (u) + vsin (u/2) sin (u), veos (u/2)).
If there is a nowhere-zero k-form on a k -dimensional manifold M, show that M is orientable
a. Show that this length is the least upper bound of lengths of inscribed broken lines.
Let f, g: [0, 1] → R3 be nonintersecting closed curves. Define the linking number l (f, g) of and g by (cf. Problem 4-34
Generalize the divergence theorem to the case of an -manifold with boundary in Rn.
Applying the generalized divergence theorem to the set M = {XЄrN: |x| < a} and , find the volume of Sn – 1 = {xЄRn: |x| = 1} in terms of the -dimensional volume of Bn = {x Є Rn:
Define F on R3 by F(x) = (0, 0, cx3)x and let M be a compact three-dimensional manifold-with-boundary with MC {x: x3
The graph of a function f is given. x(a) State the value of f (€“ 1).(b) Estimate the value of f (2)(c) For what values of x is f (x) = 2?(d) Estimate the values of x such that f (x) =
The graphs of f and t are given.(a) State the values of f (€“4) and g (3).(b) For what values of x is f (x) = g(x)?(c) Estimate the solution of the equation f (x) = €“ 1.(d) On
Figures 1, 11, and 12 were recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use
In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight water temperature is a function of time. Give three other
Determine whether the curve is the graph of a function of if it is, state the domain and range of the function.
The graph shown gives the weight of a certain person as a function of age. Describe in words how this person€™s weight varies over time. What do you think happened when this person was 30
The graph shown gives a salesman€™s distance from his home as a function of time on a certain day. Describe in words what the graph indicates about his travels on this day.
You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph
Sketch a rough graph of the number of hours of daylight as a function of the time of year.
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough
A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.
An airplane flies from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let x(t) be the
The number N (in thousands) of cellular phone subscribers in Malaysia is shown in the table. (Midyear estimates are given.)(a) Use the data to sketch a rough graph of N as a function of(b) Use your
Temperature readings T (in °F) were recorded every two hours from midnight to 2:00 P.M. in Dallas on June 2, 2001. The time t was measured in hours from midnight.(a) Use the readings to sketch a
If f (x) = 3x2 – x + 2, fine f (2), f (– 2), f (a), f (– a), f (a + 1), 2f (a), f (2a), f (a2), [f (a)] 2, and f (a + h).
A spherical balloon with radius r inches has volume V(r) = 4/3πr3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r +
Find f (2 + h), f (x + h), and f(x) + h) – f (x)/h, where h ≠ 0 21. f (x) = x – x2 22. f (x)= x / x + 1
Find the domain of the function.
Find the domain and range and sketch the graph of the function h(x) = √4 - x2.
Find the domain and sketch the graph of the function.
Find an expression for the function whose graph is the given curve.
47. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.48. A rectangle has area 16 m . Express the perimeter of the rectangle as a
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width of the window.
A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side at each corner and then folding up the sides as
A taxi company charges two dollars for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a function of the
(a) Sketch the graph of the tax rate R as a function of the income I.(b) How much tax is assessed on an income of $14,000 On $26,000?(c) Sketch the graph of the total assessed tax T as a function of
The functions in Example 10 and Exercises 54 and 55(a) Are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.
Graphs of f and are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.
(a) If the point (5, 3) is on the graph of an even function, what other point must also be on the graph?(b) If the point (5, 3) is on the graph of an odd function, what other point must also be on
A function f has domain [– 5, 5] and a portion of its graph is shown.(a) Complete the graph of f if it is known that f is even.(b) Complete the graph of f if it is known that f is odd.
Determine whether f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph.
Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
(a) y = x – 6 / x + 6 (b) y = x + x2/√x – 1 (c) y = 10x (d) y = x10 (e) y = 2t6 + t3 – π (f) y = cos θ + sin θ
Match each equation with its graph. Explain your choices. (Don€™t use a computer or graphing calculator.)
(a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family.(b) Find an equation for the family of linear functions such that f (2) = 1 and sketch
What do all members of the family of linear functions f(x) = 1 + m(x +3) have in common sketched several members of the family.
What do all members of the family of linear functions f(x) = c - x have in common Sketch several members of the family.
The manager of a weekend flea market knows from past experience that if he charges dollars for a rental space at the flea market, then the number of spaces he can rent is given by the equation y =
The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the linear function F = 9/5C + 32.(a) Sketch a graph of this function.(b) What is the slope of the graph and
Jason leaves Detroit at 2:00 P.M. and drives at a constant speed west along I-96. He passes Ann Arbor, 40 mi from Detroit, at 2:50 P.M.(a) Express the distance traveled in terms of the time
Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per
The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day.(a) Express the cost as a function of the number of chairs
At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lb/in2. Below the surface, the water pressure increases by 4.34 lb/in2 for every 10 ft of
The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi.(a) Express the monthly
For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices?
The table shows (lifetime) peptic ulcer rates (per 100 populations) for various family incomes as reported by the 1989 National Health Interview Survey.a) Make a scatter plot of these data and decide
Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures.(a) Make a scatter
The table gives the winning heights for the Olympic pole vault competitions in the 20th century.(a) Make a scatter plot and decide whether a linear model is appropriate.(b) Find and graph the
A study by the U. S. Office of Science and Technology in 1972 estimated the cost (in 1972 dollars) to reduce automobile emissions by certain percentages:Find a model that captures the
Use the data in the table to model the population of the world in the 20th century by a cubic function. Then use your model to estimate the population in the year 1925.
The table shows the mean (average) distances d of the planets from the Sun (taking the unit of measurement to be the distance from Earth to the Sun) and their periods T (time of revolution in
Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows.(a) Shift 3 units upward.(b) Shift 3 units downward.(c) Shift 3 units to the right.(d)
Explain how the following graphs are obtained from the graph of y = f(x).(a) y = 5 f (x) (b) y = f (x – 5)(c) y = - f (x) (d) y = - 5 f (x)(e) y = f (5x) (f) y = 5 f(x) – 3
The graph of v = f(x) is given. Match each equation with its graph and give reasons for your choices.(a) y = f (x - 4) (b) y = f (x) + 3(c) y = 1/3 f(x) (d) y = - f (x + 4)(e) y = 2f(x + 6)
The graph of f is given. Draw the graphs of the following functions.(a) y = f (x + 4) (b) y = f (x) + 4(c) y = 2f(x) (d) y = -½(x) + 3
The graph of f is given. Use it to graph the following functions.(a) y = f(2x) (b) y = f(1/2 x)(c) y = f(-x) (d) y = -f(-x)
The graph of y =√3x €“ x2 is given. Use transformations to create a function whose graph is as shown.
(a) How is the graph of y = 2 sin x related to the graph of y = sin x ? Use your answer and Figure 6 to sketch the graph of y = 2 sin x. (b) How is the graph of y = 1 + √x related to the
Graph the function, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.
The city of New Orleans is located at latitude 30oN. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. Use the fact that on
A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cepheid, the time between periods of maximum brightness is 5.4 days, the average
(a) How is the graph of y = f (| x |) related to the graph of f? (b) Sketch the graph of y = sin | x |. (c) Sketch the graph of y = √| x |.
Use the given graph of f to sketch the graph of y = 1/f (x).Which features of f are the most important in sketching y = 1/f(x)? Explain how they are used.
Use graphical addition to sketch the graph of f + g.
Find f + g, f €“ g, fg, and f/g and state their domains.
Use the graphs of f and g and the method of graphical addition to sketch the graph of f + g.
Find the functions f o g, g o f, f o f, and g o g and their domains.
Find f o g o h.
Express function in the form f o g.
Express the function in the form f o g o h.
Use the table to evaluate each expression.(a) f (g(1)) (b) g (f(1)) (c) f (f(1))(d) g (g(1)) (e) (g o f)(3) (f) (f o g) (6)
Use the given graphs of f and to evaluate each expression, or explain why it is undefined.(a) f(g(2)) (b) g (f(0)) (c) (f o g (0))(d) (g o f)(6) (e) (g o g)(- 2) (f) (f o f) (4)
Use the given graphs of f and to estimate the value of f (g(x)) for x = -5, -4, -3 ....5. Use these estimates to sketch a rough graph of f o g.
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.(a) Express the radius of this circle as a function of the time (in seconds).(b) If A is the area
An airplane is flying at a speed 350 mi/hf at an altitude of one mile and passes directly over a radar station at time t = 0. (a) Express the horizontal distance (in miles) that the plane has flown
The Heaviside function H is defined byIt is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on.(a) Sketch
The Heaviside function defined in Exercise 59 can also be used to define the ramp function y = ctH (t), which represents a gradual increase in voltage or current in a circuit.(a) Sketch the graph of
(a) If g(x) = 2x + 1 and h (x) = 4x2 + 4x + 7, find a function f such that f o g = h. (Think about what operations you would have to perform on the formula for to end up with the formula for h)(b) If
If f(x) = x + 4 and h(x) = 4x – 1 , find a function such that g o f = h.
Suppose g is an even function and let h = f o g. Is h always an even function?
Suppose t is an odd function and let h = f o g. Is h always an odd function? What if f is odd? What if is f even?
Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f (x) = x4 + 2.(a) [– 2, 2] by [– 2, 2] (b) [0, 4] by
Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f (x) = x2 + 7x + 6.(a) [– 5, 5] by [– 5, 5](b) [0,
Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f (x) = 10 + 25x – x3.(a) [– 4, 4] by [– 4, 4](b)
Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f (x) = √8x – x2. (a) [– 4, 4] by [– 4,
Determine an appropriate viewing rectangle for the given function and use it to draw the graph.
Graph the ellipse 4x2 + 2y2 = 1 by graphing the functions whose graphs are the upper and lower halves of the ellipse.
Graph the hyperbola y2 - 9x2 = 1 by graphing the functions whose graphs are the upper and lower branches of the hyperbola.

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