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mathematics
calculus

Questions and Answers of Calculus

The number N (in millions) of US cellular phone subscribers is shown in the table. (Midyear estimates are given.)(a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph
If f(x) = 3x2 - x + 2, find f(2), f(-2), f(a), f(-a), f(a + 1), 2f(a) f(2a), f(a2), [f(a)]2, and f(a + h).
Evaluate the difference quotient for the given function. Simplify your answer. F(x) = 4 + 3x - x2, f(3 + h) - f(3)/h
The graph of a function f is given.(a) State the value of f(1).(b) Estimate the value of f(-1).(c) For what values of x is f(x) = 1?(d) Estimate the value of such that f(x) = 0.(e) State the domain
Find the domain of the function. F(x) = x + 4 / x2 - 9
Find the domain and sketch the graph of the function. f(x) = 2 - 0.4 x
Figure 1 was 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 it to
Find an expression for the function whose graph is the given curve. The line segment joining the points (1, -3) and (5, 7)
A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides. Find a formula for the described function and state its domain.
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
In a certain state the maximum speed permitted on freeways is 65 ml/h and the minimum speed is 40 ml/h. The fine for violating these limits is $15 for every mile per hour above the maximum speed or
In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over
(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
Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. F(x) = x /x2 + 1
Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic
If the recommended adult dosage for a drug is D(in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c = 0.0417 D(a + 1). Suppose the dosage for an
The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the linear function F = 9/5 C + 32. (a) Sketch a graph of this function. (b) What is the slope of the graph
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
At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 Ib/in2. Below the surface, the water pressure increases by 4.34 Ib/in2 for every 10 ft of
For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices.(a)(b)
The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey. Income Ulcer rate (per 100
The table gives the winning heights for the men's Olympic pole vault competitions up to the year 2004.(a) Make a scatter plot and decide whether a linear model is appropriate. (b) Find and graph the
Many physical quantities are connected by inverse square laws, that is, by power functions of the form f(x) = km-2. In particular, the illumination of an object by a light source is inversely
The table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles.(a) Use a power function to model N as a function of
Match each equation with its graph. Explain your choices.(a) y = x2(b) y = x5(c) y = x8
(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) = c - x have in common? Sketch several members of the family.
Find an expression for a cubic function f if f(1) = 6 and f(-1) = f(0) f(2) = 0.
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
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. To check the
(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|.
Find (a) f + g, (b) f - g, (c) fg, (d) f/g and state their domains. F(x) = x3 + 2x2, g(x) = 3x2 - 1
The graph of y = f(x) is given. Match each equation with its graph and give reasons for your choices.(a) y = f(x) + 8(b) y = f(x + 8)(c) y = 1/3 f(x)(d) y = -f (x + 4)(e) y = 2 f(x + 6)
Find the functions (a) f o g, (b) g o f, (c) f o f, and (d) g o g and their domains. F(x) = x2 - 1 g(x) = 2x + 1
Find f o g o h. a. F(x) = 3x - 2, g(x) = sin x, h (x) = x2 b. f(x) = √x - 3, g(x) = x2, h(x) = x3 + 2
Express the function in the form f o g. a. F(x) = (2x + x2)4 b. F(x) = 3√x / 1 + 3√x c. v(t) = sec(t2) tan (t2)
Express the function in the form f o g oh. a. R(x) = √√x - 1 b. H(x) = sec4 (√x)
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)
Use the given graphs of f and g 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)
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
A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance between the lighthouse and the ship
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)
Let and be linear functions with equations f(x) = m1x + b1 and g(x) = m2 x + b2. Is f o g also a linear function? If so, what is the slope of its graph?
(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)
Suppose t is an even function and let h = f o g. Is h always an even function?
The graph of y = ˆšx - x2 is given. Use transformations to create a function whose graph is as shown.
Graph the function by hand, 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. a. y = 1/x
Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f(x) = √x3 - 5x2. (a) [-5, 5] by [-5, 5] (b) [0, 10] by
(a) Try to find an appropriate viewing rectangle for F(x) = (x -10)3 2-x. (b) Do you need more than one window? Why?
Graph the ellipse 4x2 + 2y2 = 1 by graphing the functions whose graphs are the upper and lower halves of the ellipse.
Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? y = 3x2 - 6x + 1, y = 0.23x - 2.25; [-1, 3] by [-2.5, 1.5]
Find all solutions of the equation correct to two decimal places. a. X4 - x = 1 b. tan x = √1 - x2
Use graphs to determine which of the functions and g(x) = x3 /10 is eventually larger (that is, larger when is very large).
For what values of is it true that |tan x - x| < 0.01 and -π/2 < x < π/2?
In this exercise we consider the family of root functions f(x) = ˆšx, where is a positive integer.(a) Graph the functions y = ˆšx, y = 4ˆšx, and y = 6ˆšx on
Determine an appropriate viewing rectangle for the given function and use it to draw the graph. F(x) = x2 - 36x + 32
Graph the function f(x) = x4 + cx2 + x for several values of c. How does the graph change when changes?
Graph the function y = xn2-x, x ≥ 0, for n = 1, 2, 3, 4, 5, and 6. How does the graph change as increases?
What happens to the graph of the equation y2 = cx3 + x2 as c varies?
The figure shows the graphs of y = sin 96x and y = sin 2x as displayed by a TI-83 graphing calculator. The first graph is inaccurate. Explain why the two graphs appear identical.Y - sin 96xY = sin 2x
1. Use the Law of Exponents to rewrite and simplify the expression. (a) 4-3 / 2-8 (b) 1/ 3√x4 2. (a) b8 (2)4 (b) (6y3)4 / 2y5
Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3. (a) y = 10x+2 (b) y =
Starting with the graph of y = ex, write the equation of the graph that results from (a) Shifting 2 units downward (b) Shifting 2 units to the right (c) Reflecting about the x-axis (d) Reflecting
Find the domain of each function. (a) f(x) = 1 - ex2 / 1 - e1-x1 (b) f(x) = 1 + x / ecos x
Find the exponential function f(x) = Cax whose graph is given.
If f(x) = 5x, show that
Suppose the graphs of f(x) = x2 and g(x) = 2* are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that, at a distance 2 ft to the right of the origin, the height of the graph
Compare the functions f(x) = x10 and g(x) = ex by graphing both f and in several viewing rectangles. When does the graph of finally surpass the graph of f?
Under ideal conditions a certain bacteria population is known to double every three hours. Suppose that there are initially 100 bacteria. (a) What is the size of the population after 15 hours? (b)
Use a graphing calculator with exponential regression capability to model the population of the world with the data from 1950 to 2010 in Table 1 on page 54. Use the model to estimate the population
If you graph the function f (x) = 1 - e1/x / 1 + e1x you'll see that f appears to be an odd function. Prove it.
(a) Write an equation that defines the exponential function with base a > 0. (b) What is the domain of this function? (c) If a ≠ 1, what is the range of this function? (d) Sketch the general shape
Graph the given functions on a common screen. How are these graphs related? y = 2*, y = e*, y = 5*, y = 20*
Assume that is a one-to-one function. (a) If f(6) = 17, what is f-1 (17)? (b) If f-1 (3), what is f(2)?
If g(x) = 3 + x + ex, find g-1 (4)
The formula C = 9/5(F - 32), where F ≥ -459.67, expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is
Find a formula for the inverse of the function. (a) F(x) = 1 + √2 + 3x (b) f(x) = e2x-1 (c) y = ln (x + 3)
Find an explicit formula for f-1 and use it to graph f-1, f and the line y = x on the same screen. To check your work, see whether the graphs of f and f-1 are reflections about the line. F(x) = x4 +
Use the given graph of f to sketch the graph of f-1.
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.(a)(b) (c) (d) f (x) = x2 - 2x
Let f(x) = √1 - x2, 0 ≤ x ≤ 1. (a) Find f-1. How is it related to f? (b) Identify the graph of and explain your answer to part (a).
(a) How is the logarithmic function defined? (b) What is the domain of this function? (c) What is the range of this function?
Find the exact value of each expression.1. (a) log5 125(b) log3 (1/27)2. (a) log2 6 - log2 15 + log2 20(b) log3 100 - log3 18 - log3 50
Express the given quantity as a single logarithm. (a) ln 5 + 5 ln 3 (b) 1/3 ln (x + 2)3 + 1/2 [ln x - ln (x2 + 3x + 2)2]
Use Formula 10 to graph the given functions on a common screen. How are these graphs related? y = log1.5 x, y = ln x, y = log10 x, y = log50x
Suppose that the graph of y = log2 x is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the
Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3. (a) y = log10 (x +
(a) What are the domain and range of f? (b) What is the -intercept of the graph of f? (c) Sketch the graph of f. F(x) = ln x + 2
Solve each equation for x. 1. (a) e7-4x = 6 (b) ln (3x - 10) = 2 2. (a) 2x-5 = 3 (b) ln x + ln(x - 1) = 1
Solve each inequality for x. (a) ln x < 0 (b) ex > 5
(a) Find the domain of f(x) = ln (ex - 3). (b) Find f-1 and its domain.
Graph the function f(x) √x3 + x2 + x + 1 and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for f-1 (x). (Your CAS will produce three possible
If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after hours is n = f(t) = 100 ∙ 2t/3. (a) Find the inverse of this function and explain
Find the exact value of each expression.1. (a) sin-1(√3 / 2)(b) cos-1 (-1)2. (a) arctan 1(b) sin-1 (1/√2)
Simplify the expression. Sin (tan-1 x)
Graph the given functions on the same screen. How are these graphs related? y = sin x, -π/2 ≤ x ≤ π/2: y = sin-1 x: y = x
Find the domain and range of the functiong(x) = sin-1 (3x + 1)
(a) If we shift a curve to the left, what happens to its reflection about the line y = x? In view of this geometric principle, find an expression for the inverse of g(x) = f(x + c), where f is a
Let f be the function whose graph is given.(a) Estimate the value of f(2).(b) Estimate the values of such that f(x) = 3.(c) State the domain of f.(d) State the range of f.(e) On what interval is f
Use transformations to sketch the graph of the function. y = - sin 2 x

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