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calculus

Questions and Answers of Calculus

Let (0, 0) represent a water fountain located in a city park. Each day you run through the park along a path given by x2 + y2 − 200x − 52,500 = 0 Where x and y are measured in meters. (a) What
Determine whether the statement is true or false. Justify your answer. 1. The domain of a rational function can never be the set of all real numbers. 2. The graph of the equation Ax2 +Bxy + Cy2 +Dx +
1. A ________ ________ is a quotient of polynomial functions. 2. When f (x) → ± ∞ as x → a from the left or the right, x = a is a ________ ________ of the graph of f. 3. When f (x) → b as
In Exercises 1-4, find all vertical and horizontal asymptotes of the graph of the function. 1. f (x) = 4 / x2 2. f (x) = 1 / (x - 2)3 3. f (x) = 5 + x / 5 - x 4. f (x) = 3 - 7x / 3 + 2x
In Exercises 1-4, match the rational function with its graph. [The graphs are labeled (a)-(h).]a.b. c. d. e. f. g. h. 1. f (x) = 4 / x + 2 2. f (x) = 5 / x - 2 3. f (x) = - 2x - 1 / x - 2 4. f (x)
In Exercises 1-2,(a) determine the domains of f and g,(b) simplify f and find any vertical asymptotes of the graph of f,(c) complete the table, and(d) explain how the two functions differ.1. f (x) =
The cost C (in dollars) of supplying recycling bins to p% of the population of a rural township is given byC = 25,000p / 100 - p, 0 ‰¤ p (a) Use a graphing utility to graph the cost
The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given by C = 225p / 100 - p, 0 ≤ p < 100. (a) Use a graphing utility to
The game commission introduces 100 deer into newly acquired state game lands. The population N of the herd is modeled by N = 20(5 + 3t) / 1 + 0.04t, t ≥ 0 Where t is the time in years. (a) Use a
A science class performs an experiment comparing the quantity of food consumed by a species of moth with the quantity of food supplied. The model for the experimental data is y = 1.568x - 0.001 /
Psychologists have developed mathematical models to predict memory performance as a function of the number of trials n of a certain task. Consider the learning curveP = 0.5 + 0.9(n - 1) / 1 + 0.9(n -
Consider a physics laboratory experiment designed to determine an unknown mass. A flexible metal meter stick is clamped to a table with 50 centimeters overhanging the edge (see figure). Known masses
In Exercises 1-3, determine whether the statement is true or false. Justify your answer. 1. The graph of a polynomial function can have infinitely many vertical asymptotes. 2. f (x) = x3 − 2x2 −
In Exercises 1-2, find the domain of the function and discuss the behavior of f near any excluded x-values. 1. f (x) = 1 / x - 1 2. f (x) = 4 / x + 3
In Exercises 51-54, a. determine the value that the function f approaches as the magnitude of x increases. Is f (x) greater than or less than this value when b. x is positive and large in
In Exercises 1 and 2, write a rational function f whose graph has the specified characteristics. (There are many correct answers.) 1. Vertical asymptote: None Horizontal asymptote: y = 2 2. Vertical
Describe the error. A real zero of the numerator of a rational function f is x = c. So, x = c must also be a zero of f.
Give an example of a rational function whose domain is the set of all real numbers. Give an example of a rational function whose domain is the set of all real numbers except x = 15.
1. Matching Match the graph of the rational functionf (x) = (ax + b) / (cx + d) with the given conditions.(a)(b) (c) (d) (i) a > 0 (ii) a > 0 (iii) a 0 b 0 b > 0 b c
Let (x1, y1) be the coordinates of a point on the parabola x2 = 4py. The equation of the line that just touches the parabola at the point (x1, y1), called a tangent line, is y − y1 = x1 / 2p (x −
Prove that the graph of the equation Ax2 + Cy2 + Dx + Ey + F = 0 is one of the following (except in degenerate cases). Conic .................................................... Condition (a) Circle
Consider the function f (x) = (ax) / (x − b)2. (a) Determine the effect on the graph of f when b ≠ 0 and a is varied. Consider cases in which a is positive and a is negative. (b) Determine the
The endpoints of the interval over which distinct vision is possible are called the near point and far point of the eye (see figure). With increasing age, these points normally change. The table
Statuary Hall is an elliptical room in the United States Capitol in Washington, D.C. The room is also called the Whispering Gallery because aperson standing at one focus of the room can hear even a
Use the figure to show that|d2 - d1| = 2a
Find an equation of a hyperbola such that for any point on the hyperbola, the difference between its distances from the points (2, 2) and (10, 2) is 6.
A tour boat travels between two islands that are 12 miles apart (see figure). For each trip between the islands, there is enough fuel for a 20-mile trip.(a) Explain why the region in which the boat
The filament of a light bulb is a thin wire that glows when electricity passes through it. The filament of a car headlight is at the focus of a parabolic reflector, which sends light out in a
Consider the parabola x2 = 4py. (a) Use a graphing utility to graph the parabola for p = 1, p = 2, p = 3, and p = 4. Describe the effect on the graph when p increases. (b) Locate the focus for each
1. For the rational function f (x) = N(x) / D(x), if the degree of N(x) is exactly one more than the degree of D(x), then the graph of f has a ________ (or oblique) ________. 2. The graph of g(x) = 3
In Exercises 1-4, use the graph of f (x) = 4 / x3 to sketch the graph of g.1. g (x) = 4 / (x + 2)32. g (x) = 4 / x3 - 23. g (x) = - 4 / x34. g (x) = 2 / x3
In Exercises 1-4, (a) State the domain of the function, (b) Identify all intercepts, (c) Find any vertical or horizontal asymptotes, and (d) Plot additional solution points as needed to
In Exercises 1-4, use the graph of f (x) = 2 / x to sketch the graph of g.1. g (x) = 2 / x + 42. f (x) = 2 / x - 43. g (x) = - 2 / x4. g (x) = 1 / x + 2
In Exercises 1 and 2, (a) State the domains of f and g, (b) Use a graphing utility to graph f and g in the same viewing window, and (c) Explain why the graphing utility may not show the difference
In Exercises 47-62, (a) State the domain of the function, (b) Identify all intercepts, (c) Find any vertical or slant asymptotes, and (d) Plot additional solution points as needed to sketch the
In Exercises 1-4, use a graphing utility to graph the rational function. Give the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a Line.
In Exercises 1-4,(a) Use the graph to determine any x-intercepts of the graph of the rational function and(b) Set y = 0 and solve the resulting equation to confirm your result in part (a).1. y = x +
In Exercises 1-4, use the graph of f (x) = 3 / x2 to sketch the graph of g.1. g (x) = 3 / x2 - 12. g (x) = - 3 / x23. g (x) = 3 / (x - 1)24. g (x) = 1 / x2
In Exercises 1-4, (a) Use a graphing utility to graph the function and determine any x-intercepts of the graph and (b) Set y = 0 and solve the resulting equation to confirm your result in
A rectangular region of length x and width y has an area of 600 square meters. (a) Write the width y as a function of x. (b) Determine the domain of the function based on the physical constraints of
A rectangular page contains 64square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are 1 1/2 inches wide. What should the dimensions
A page that is x inches wide and y inches high contains 30 square inches of print. The top and bottom margins are each 1 inch deep and the margins on each side are 2 inches wide (see figure).a. Show
A 1000-liter tank contains 50 liters of a 25% brine solution. You add x liters of a 75% brine solution to the tank. a. Show that the concentration C, the proportion of brine to total solution, in
In Exercises 1-4, use a graphing utility to graph the function and locate any relative maximum or minimum points on the graph 1. f (x) = 3(x + 1) / x2 + x + 1 2. g (x) = 6x / x2 + x + 1 3. C (x) = x
The ordering and transportation cost C (in thousands of dollars) for the components used in manufacturing a product is given by C = 100 (200 / x2 + x / x + 30), x ≥ 1 Where x is the order size (in
The cost C (in dollars) of producing x units of a product is given by C = 0.2x2 + 10x + 5 and the average cost per unit C̅ is given by C̅= C / x = 0.2x2 + 10x + 5 / x, x > 0. Sketch the graph of
A driver's average speed is 50 miles per hour on a round trip between two cities 100 miles apart. The average speeds for going and returning were x and y miles per hour, respectively.(a) Show that y
The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is Given by C = 3t2 + t / t3 + 50, t > 0.(a) Determine the horizontal asymptote of the graph of the
In Exercises 1-4, determine whether the statement is true or false. Justify your answer. 1. When the graph of a rational function f has a vertical asymptote at x = 5, it is possible to sketch the
Describe the error. The graph of h (x) = 6 - 2x / 3 - x has the line x = 3 as a vertical asymptote Because 3 − x = 0 when x = 3.
(a) Given a rational function f, how can you determine whether the graph of f has a slant asymptote? (b) You determine that the graph of a rational function f has a slant asymptote. How do you find
Write a rational function satisfying the criteria below. Then sketch the graph of your function. Vertical asymptote: x = 2 Slant asymptote: y = x + 1 Zero of the function: x = −2
1. Polynomial and rational functions are examples of ________ functions. 2. Exponential and logarithmic functions are examples of non-algebraic functions, also called ________ functions. 3. The
Match the exponential function with its graph. [The graphs are labeled (a), (b), (c), and (d).](a)(b) (c) (d) 1. f (x) = 2x 2. f (x) = 2x + 1 3. f (x) = 2ˆ’x 4. f (x) = 2xˆ’2
Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 1. f (x) = 7x 2. f (x) = 7−x 3. f (x) = (1/4)−x
Use the One-to-One Property to solve the equation for x. 1. 3x+1 = 27 2. 2x−2 = 64 3. (1/2)x = 32 4. 5x−2 = 1/125
Describe the transformation(s) of the graph of f that yield(s) the graph of g. 1. f (x) = 3x, g(x) = 3x + 1 2. f (x) = (7/2)x, g(x) = −(7/2)−x 3. f (x) = 10x, g(x) = 10−x+3 4. f (x)
Evaluate the function at the given value of x. Round your result to three decimal places. Function ......................................................................... Value 1. f (x) = ex
Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. 1. f (x) = 3ex+4 2. f (x) = 2e−1.5x 3. f (x) = 2ex−2 + 4 4. f (x) = 2 + ex−5
Use a graphing utility to graph the exponential function. 1. s(t) = 2e0.5t 2. s(t) = 3e−0.2t 3. g(x) = 1 + e−x 4. h(x) = ex−2
Use the One-to-One Property to solve the equation for x. 1. e3x+2 = e3 2. e2x−1 = e4 3. ex2−3 = e2x 4. ex2+6 = e5x
Complete the table by finding the balance A when P dollars is invested at rate r for t years and compounded n times per year.1. P = $1500, r = 2%, t = 10 years 2. P = $2500, r = 3.5%, t = 10 years
Complete the table by finding the balance A when $12,000 is invested at rate r for t years, compounded continuously.1. r = 4% 2. r = 6% 3. r = 6.5% 4. r = 3.5%
The projected population of the United States for the years 2025 through 2055 can be modeled by P = 307.58e0.0052t, where P is the population (in millions) and t is the time (in years), with t = 25
The population P (in millions) of Italy from 2003 through 2015 can be approximated by the model P = 57.59e0.0051t, where t represents the year,with t = 3 corresponding to 2003. (a) According to the
Use properties of exponents to determine which functions (if any) are the same. 1. f (x) = 3x−2 g(x) = 3x − 9 h(x) = 1/9 (3x) 2. f (x) = 4x + 12 g(x) = 22x+6 h(x) = 64(4x)
Evaluate the function at the given value of x. Round your result to three decimal places. Function ................................................................................. Value 1. f (x) =
Use a graphing utility to graph each function. Use the graph to find where the function is increasing and decreasing, and approximate any relative maximum or minimum values. (a) f (x) = x2e−x (b)
Use a graphing utility to graph y1 = [1 + (1/x)]x and y2 = e in the same viewing window. Using the trace feature, explain what happens to the graph of y1 as x increases.
Use a graphing utility to graph f (x) = [1 + 0.5/x]x and g(x) = e0.5 in the same viewing window. What is the relationship between f and g as x increases and decreases without bound?
Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs. (a) y1 = 2x, y2 = x2 (b) y1 = 3x, y2 = x3
The figure shows the graphs of y = 2x, y = ex, y = 10x, y = 2ˆ’x, y = eˆ’x, and y = 10ˆ’x. Match each function with its graph. [The graphs are labeled (a) through (f).]
Which functions are exponential? (a) f (x) = 3x (b) g(x) = 3x2 (c) h(x) = 3x (d) k(x) = 2−x
Use the formula A = P (1 +r/n)nt to calculate the balance A of an investment when P = $3000, r = 6%, and t = 10 years, and compounding is done (a) by the day, (b) by the hour, (c) by the minute,
The exponential function y = ax for a = 0.5, 1.2, and 2.0. Which of these curves intersects the line y = x? Determine all positive numbers a for which the curve y = ax intersects the line y = x.
Find a pattern for f-1(x) when f (x) = ax + 1/ax - 1 where a > 0, a ≠ 1.
Determine whether the graph represents equation (a), (b), or (c). Explain your reasoning.(a) y = 6e-x2/2 (b) y = 6/1 + e-x/2 (c) y = 6(1 - e-x2/2)
You have two options for investing $500. The first earns 7% interest compounded annually, and the second earns 7% simple interest. The figure shows the growth of each investment over a 30-year
Two different samples of radioactive isotopes are decaying. The isotopes have initial amounts of c1 and c2 and half-lives of k1 and k2, respectively. Find an expression for the time t required for
A lab culture initially contains 500 bacteria. Two hours later, the number of bacteria decreases to 200. Find the exponential decay model of the form B = B0akt that approximates the number of
The table shows the colonial population estimates of the American colonies for each decade from 1700 through 1780. (Source: U.S. Census Bureau)Let y represent the population in the year t, with t = 0
Slow that loga x/loga/b x = 1 + loga 1/b.
1. Solve (In x)2 = In x2. 2. Use a graphing utility to compare the graph of each function with the graph of y = ln x. (a) y1 = x - 1 (b) y2 = (x - 1) - ½(x - 1)2 (c) y3 = (x - 1) - ½(x - 1)2 +
Identify the pattern of successive polynomials given in Exercise 18. Extend the pattern one more term and compare the graph of the resulting polynomial function with the graph of y = ln x. What do
Use a graphing utility to graph each of the functions y1 = ex, y2 = x2, y3 = x3, y4 = √x, and y5 = |x|. Which function increases at the greatest rate as x approaches ∞?
Take the natural log of each side of each equation below. y = abx, y = axb (a) What are the slope and y-intercept of the line relating x and ln y for y = abx? (b) What are the slope and y-intercept
Use the model y = 80.4 -11 ln x, 100 ≤ x ≤ 1500 which approximates the minimum required ventilation rate in terms of the air space per child in a public school classroom. In the model, x is
(a) use a graphing utility to create a scatter plot of the data, (b) decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model, (c) explain
1. Use the result of Exercise 2 to make a conjecture about the rate of growth of y1 = ex and y = xn, where n is a natural number and x approaches ∞. 2. Use the results of Exercises 2 and 3 to
1. Given that f (x) = ex + e-x/2 and g (x) = ex - e-x/2 show that [f (x)]2 - [g (x)]2 = 1. 2. Use a graphing utility to compare the graph of the function y = ex with the graph of each function.[n!
Identify the pattern of successive polynomials given in Exercise 7. Extend the pattern one more term and compare the graph of the resulting polynomial function with the graph of y = ex. What do you
Graph the function f (x) = ex − e−x. From the graph, the function appears to be one-to-one. Assume that f has an inverse function and find f-1(x).
Write the exponential equation in logarithmic form. For example, the logarithmic form of 23 8 is log2 8 3. 1. 53 = 125 2. 93/2 = 27 3. 4-3 = 1/64 4. 240 = 1
Evaluate the logarithm at the given value of x without using a calculator. Function Value 1. f (x) = log2 x x = 64. 2. f (x) = log25 x x = 5 3. f (x) = log8 x x = 1 4. f (x) = log x x =
Use a calculator to evaluate f (x) = log x at the given value of x. Round your result to three decimal places. 1. x = 7/8 2. x = 1/500 3. x = 12.5 4. x = 96.75
Use the properties of logarithms to simplify the expression. 1. log8 8 2. logπ π2 3. log7.5 1 4. 5log5 3
Use the One-to-One Property to solve the equation for x. 1. log5(x + 1) = log5 6 2. log2(x − 3) = log2 9 3. log 11 = log(x2 + 7)
Sketch the graphs of f and g in the same coordinate plane. 1. f (x) = 7x, g(x) = log7 x 2. f (x) = 5x, g(x) = log5 x 3. f (x) = 6x, g(x) = log6 x 4. f (x) = 10x, g(x) = log x
Use the graph of g (x) = log3 x to match the given function with its graph. Then describe the relationship between the graphs of f and g. [The graphs are labeled (a), (b), (c), and (d).]a)b) c) d)
Find the domain, x-intercept, and vertical asymptote of the logarithmic function and sketch its graph. 1. f (x) = log4 x 2. g(x) = log6 x 3. y = log3 x + 1

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