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mathematics
calculus

Questions and Answers of Calculus

Find all solutions of the equation correct to two decimal places.
We saw in Example 9 that the equation cos x = x has exactly one solution.(a) Use a graph to show that the equation cos x = 0.3x has three solutions and find their values correct to two decimal
Use graphs to determine which of the functions f(x) = 10x2 and g(x) = x3/10 is eventually larger (that is, larger when is very large).
Use graphs to determine which of the functions f(x) = x4 – 100 x3 and g (x) = x3 is eventually larger.
For what values of x is it true that | sin x – x| < 0.1?
Graph the polynomials P(x) = 3x5 – 5x3 + 2x and Q(x) = 3x5 on the same screen, first using the viewing rectangle [– 2, 2] by [– 2, 2] and then changing to [– 10, 10] by [– 10,000, 10,000].
In this exerciser we consider the family of root functions f (x) = n√x, where n is a positive integer. (a) Graph the functions y = √x, y = 4√x, and y = 6√x on the same screen
In this exercise we consider the family of functions f(x) = 1x, where is a positive integer.(a) Graph the functions y = 1/x and y =1/x3 on the same screen using the viewing rectangle [-3, 3] by [-3,
Graph the function f(x) = x4 + cx2 + x for several values of c. How does the graph change when changes?
Graph the function f (x) = √1 + cx2 for various values of c. Describe how changing the value of affects the graph.
Graph the function y = xn2-x >> 0, for, n = 1, 2,3,4,5, and 6. How does the graph change as increases?
The curves with equations are called bullet-nose curves.Graph some of these curves to see why. What happens as increases?
What happens to the graph of the equation y = cx + x as c varies?
This exercise explores the effect of the inner function g on a composite function y = f (g(x)). (a) Graph the function y = sin (√x)).using the viewing rectangle [0, 400] by [– 1.5, 1.5]. How
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.
The first graph in the figure is that of y = sin45x as displayed by a TI-83 graphing calculator. It is inaccurate and so, to help explain its appearance, we replot the curve in dot mode in the second
(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
(a) How is the number defined?(b) What is an approximate value for e?(c) What is the natural exponential function?
Graph the given functions on a common screen. How are these graphs related?
Make a rough sketch of the graph of the function. Do not use a calculator, just use the graphs given in Figures 3 and 14 and, if necessary, the transformations of Section 1.3.
Starting with the graph of y = e2, 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 about
Starting with the graph of y = e2, find the equation of the graph that results from(a) Reflecting about the line y = 4(b) Reflecting about the line x = 2
Find the domain of each function.
Find the exponential function f(x) = Cax whose graph is given.
If f(x) = 5x show that.
Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer?I. One million dollars at the end of the monthII. One cent on the first day of the month,
Suppose the graphs of f(x) = x2 and g(x) = 2x 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) = x5 and g(x) = 5x by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function
Compare the functions f (x) = x10 and g(x) = ex by graphing both f and g in several viewing rectangles. When does the graph of g finally surpass the graph of f?
Use a graph to estimate the values of x such that ex > 1,000,000,000.
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) What
An isotope of sodium 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.(a) Find the amount remaining after 60 hours.(b) Find the amount remaining after hours.(c) Estimate the
Use a graphing calculator with exponential regression capability to model the population of the world with the data from 1950 to 2000 in Table 1 on page 58. Use the model to estimate the population
The table gives the population of the United States, in millions, for the years 1900€“2000. Use a graphing calculator with exponential regression capability to model the U.S. population
(a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one?
(a) Suppose f is a one-to-one function with domain and range B. How is the inverse function defined? What is the domain of f-1? What is the range of f-1? (b) If you are given a formula for f, how do
A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
Use a graph to decide whether f is one-to-one.
If f is a one-to-one function such that f(2) = 9, what is f-1(9)?
Let f(x) = 3 + x2 + tan (πx/2), where – 1 < x < 1. (a) Find f -1(3). (b) Find f (f-1(5)).
If g (x) = 3 + x + ex, find g-1 (4).
The graph of f is given.(a) Why is f one-to-one?(b) State the domain and range of f-1.(c) Estimate the value of f-1(1).
The formula C = 5/9 (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
In the theory of relativity, the mass of a particle with speed isWhere mo is the rest mass of the particle and is the speed of light in a vacuum. Find the inverse function of f and explain its
Find a formula for the inverse of the function.
Find an explicit formula for f €“1and 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
Use the given graph of f to sketch the graph of f-1.
Use the given graph of f to sketch the graphs of f-1 and 1/f.
(a) How is the logarithmic function y = log ax defined?(b) What is the domain of this function? (c) What is the range of this function?(d) Sketch the general shape of the graph of the function y =
(a) What is the natural logarithm?(b) What is the common logarithm?(c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.
Find the exact value of each expression.
Express the given quantity as s single logarithm.
Use Formula 10 to evaluate each logarithm correct to six decimal places.(a) log12 10 (b) log2 8.4
Use Formula 10 to graph the given functions on a common screen. How are these graphs related?
Suppose that the graph y = log 2x of 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
Compare the functions f(x) = x0.1 and g(x) = in x by graphing both f and in several viewing rectangles. When does the graph of f finally surpass the graph of g?
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.
Solve each equation for x.
Solve each inequality for x.
Find (a) the domain of f and (b) 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. (Your CAS will produce three possible
(a) If g(x) = x6 + x4, x > 0, use a computer algebra system to find an expression for g-1(x).(b) Use the expression in part (a) to graph y = g(x), y = x, and y = g-1(x) on the same screen.
(a) If g(x) = x6 + x4, x > 0, use a computer algebra system to find an expression for g-1(x). (b) Use the expression in part (a) to graph y = g(x), y = x, and y = g-1(x) on the same screen.
When a camera flash goes off, the batteries immediately begin to recharge the flash’s capacitor, which stores electric charge given by Q(t) = Qo(1 – e –t/a)(The maximum charge capacity is Qo
Starting with the graph of y = in x, find the equation of the graph that results from(a) Shifting 3 units upward(b) Shifting 3 units to the left(c) Reflecting about the x-axis(d) Reflecting about the
(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
Find the exact value of each expression.
Prove that cos (sin – 1x) = √1 – x2.
Simplify the expression.
Graph the given functions on the same screen. How are these graphs related?
Find the domain and range of the function. g(x) = sin-1(3x + 1)
(a) Graph the function f(x) = sin (sin-1x) and explain the appearance of the graph.(b) Graph the function f(x) = sin (sin-1x). How do you explain the appearance of this graph?
(a) What is a function? What are its domain and range?(b) What is the graph of a function?(c) How can you tell whether a given curve is the graph of a function?
Discuss four ways of representing a function. Illustrate your discussion with examples.
(a) What is an even function? How can you tell if a function is even by looking at its graph?(b) What is an odd function? How can you tell if a function is odd by looking at its graph?
What is an increasing function?
What is a mathematical model?
Give an example of each type of function.(a) Linear function (b) Power function(c) Exponential function (d) Quadratic function(e) Polynomial of degree 5 (f) Rational function
Sketch by hand, on the same axes, the graphs of the following functions.(a) f(x) = x (b) g(x) = x2 (c) h(x) = x3 (d) j(x) = x4
Draw, by hand, a rough sketch of the graph of each function. (a) y = sin x (b) y = tan x (c) y = ex (d) y = In x (e) y = 1/x (f) y = | x | (g) y = √ x (h) y = tan–1 x
Suppose that f has domain A and g has domain B.(a) What is the domain of f + g?(b) What is the domain of f g?(c) What is the domain of f/g?
How is the composite function f o g defined? What is its domain?
Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows.(a) Shift 2 units upward.(b) Shift 2 units downward.(c) Shift 2 units to the
(a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph?(b) If f is a one-to-one function, how is its inverse function f–1 defined? How do you obtain
(a) How is the inverse sine function f(x) = sin-1x defined? What are its domain and range?(b) How is the inverse cosine function f(x) = cos-1x defined? What are its domain and range?(c) How is the
If f is a function, then f(s + t) = f(s) + f (t).
If f(s) = f (t), then s = t.
If f is a function, then f (3x) = 2 f(x).
If x1 < x2 and f is a decreasing function, then f(x1) > f(x2)
A vertical line intersects the graph of a function at most once.
If f and are functions, then f o g = g o f.
If f is one-to-one, then f-1(x) 1 / f(x)
You can always divide by ex.
If 0 < a < b, then In a < In b
If x > 0, then (In x)6 = 6 In x.
If x > 0 and a > 1, then In x / In a = In x/a.
Let f be the function whose graph is given.(a) Estimate the value of f (2).(b) Estimate the values of x such that f(x) = 3.(c) State the domain of f.(d) State the range of f.(e) On what interval is f
The graph of is given.(a) State the value of g (2).(b) Why is one-to-one?(c) Estimate the value of g-1(2).(d) Estimate the domain of g-1.(e) Sketch the graph of g-1.
The distance traveled by a car is given by the values in the table.(a) Use the data to sketch the graph of d as a function of t.(b) Use the graph to estimate the distance traveled after 4.5 seconds.
Find the domain and range of the function.

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