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mathematics
calculus

Questions and Answers of Calculus

Solve the system graphically or algebraically. Explain your choice of method.1.2.
Determine whether each ordered pair is a solution of the system.a. (0, -4) b. (3, ˆ’1) c. (3/2, 1) d. (1/2, 5)
Use the equations for the total cost C and total revenue R to find the number x of units a company must sell to break even. (Round to the nearest whole unit.) 1. C = 8650x + 250,000, R = 9502x 2. C =
A small software development company invests $16,000 to produce a software package that will sell for $55.95. Each unit costs $9.45 to produce. (a) How many units must the company sell to break
You receive two sales job offers. One company offers a straight commission of 6% of sales. The other company offers a salary of $500 per week plus 3% of sales. How much would you have to sell per
1. Two new DVDs, a horror film and a comedy film, are released in the same week. The weekly number N of rentals decreases for the horror film and increases for the comedy film according to the
Describe the error in solving the system of equations.
The table shows the consumption C (in trillions of Btus) of geothermal energy and wind energy in the United States from 2004 through 2014. (Source: U.S. Energy Information Administration)(a) Use a
Use a system of equations to find the dimensions of the rectangle meeting the specified conditions. The perimeter is 56 meters and the length is 4 meters greater than the width.
What are the dimensions of a rectangular tract of land when its perimeter is 44 kilometers and its area is 120 square kilometers?
What are the dimensions of a right triangle with a two-inch hypotenuse and an area of 1square inch?
Determine whether the statement is true or false. Justify your answer. 1. In order to solve a system of equations by substitution, you must always solve for y in one of the two equations and then
When solving a system of equations by substitution, how do you recognize that the system has no solution?
Solve the system by the method of substitution. Check your solution(s) graphically.1.2.
The cost C of producing x units and the revenue R obtained by selling x units are shown in the figure.(a) Estimate the point of intersection. What does this point represent? (b) Use the figure to
A theorem from geometry states that if a triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle, then the triangle is a right triangle. Show that this
A hotel 35 miles from an airport runs a shuttle service to and from the airport. The 9:00 A.M. bus leaves for the airport traveling at 30 miles per hour. The 9:15 A.M. bus leaves for the airport
Solve each system of equations by letting X = 1 /x, Y = 1/y, and Z = 1/z.(a)(b)
For what values of a, b, and c does the linear system have (ˆ’1, 2, ˆ’3) as its only solution?
The following system has one solution: x = 1, y = ˆ’1, and z = 2.Solve each system of two equations that consists of (a) Equation 1 and Equation 2 (b) Equation 1 and Equation 3 (c) Equation
Solve the system of linear equations algebraically.
Each day, an average adult moose can process about 32 kilograms of terrestrial vegetation (twigs and leaves) and aquatic vegetation. From this food, it needs to obtain about 1.9 grams of sodium and
For a healthy person who is 4 feet 10 inches tall, the recommended minimum weight is about 91 pounds and increases by about 3.6 pounds for each additional inch of height. The recommended maximum
Cholesterol in human blood is necessary, but too much can lead to health problems. There are three main types of cholesterol: HDL (high-density lipoproteins), LDL (low-density lipoproteins), and VLDL
Find values of k1 and k2 such that the system of equations has an infinite number of solutions.
Under what condition(s) will the system of equations in x and y have exactly one solution?
Find values of a, b, and c (if possible) such that the system of linear equations has (a)a unique solution, (b)no solution, and (c)an infinite number of solutions.
Graph the lines determined by each system of linear equations. Then use Gaussian elimination to solve each system. At each step of the elimination process, graph the corresponding lines. How do the
A system of two equations in two variables has a finite number of solutions. Determine the maximum number of solutions of the system satisfying each condition. (a) Both equations are linear. (b) One
The Vietnam Veterans Memorial (or "The Wall") in Washington, D.C., was designed by Maya Ying Lin when she was a student at Yale University. This monument has two vertical, triangular sections of
Weights of atoms and molecules are measured in atomic mass units (u). A molecule of C2H6 (ethane) is made up of two carbon atoms and six hydrogen atoms and weighs 30.069 u. A molecule of C3H8
Connecting a DVD player to a television set requires a cable with special connectors at both ends. You buy a six-foot cable for $15.50 and a three-foot cable for $10.25. Assuming that the cost of a
1. The first step in solving a system of equations by the method of ________ is to obtain coefficients for x (or y) that differ only in sign. 2. Two systems of equations that have the same solution
Solve the system by the method of elimination and check any solutions algebraically.1.2. 3.
Match the system of linear equations with its graph. Describe the number of solutions and state whether the system is consistent or inconsistent. [The graphs are labeled (a), (b), (c) and
Use any method to solve the system. Explain your choice of method.1.2. 3.
An airplane flying into a headwind travels the 1800-mile flying distance between Indianapolis, Indiana, and Phoenix, Arizona, in 3hours. On the return flight, the airplane travels this distance in
Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts 12 hour after the first plane, but its speed is 80 kilometers per hour faster. Find the
Two cheeseburgers and one small order of fries contain a total of 1420 calories. Three cheeseburgers and two small orders of fries contain a total of 2290 calories. Find the caloric content of each
One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 179.2 milligrams of vitamin C. Two eight-ounce glasses of apple juice and three eight-ounce glasses
Find the equilibrium point of the demand and supply equations. 1. p = 500 − 0.4x p = 380 + 0.1x 2. p = 100 − 0.05x p = 25 + 0.1x
Thirty liters of a 40% acid solution is obtained by mixing a 25% solution with a 50% solution. (a) Write a system of equations in which one equation represents the total amount of final mixture
Solve the system by the method of elimination. Label each line with its equation. To print an enlarged copy of the graph, go to MathGraphs.com.1. 2x + y = 7x - y = -42. x + 3y = 1
Five hundred gallons of 89-octane gasoline is obtained by mixing 87-octane gasoline with 92 octane gasoline. (a) Write a system of equations in which one equation represents the total amount of final
A total of $24,000 is invested in two corporate bonds that pay 3.5% and 5% simple interest. The investor wants an annual interest income of $930 from the investments. What amount should be invested
A total of $32,000 is invested in two municipal bonds that pay 5.75% and 6.25% simple interest. The investor wants an annual interest income of $1900 from the investments. What amount should be
The numbers of prescriptions P (in thousands) filled at two pharmacies from 2012 through 2016 are shown in the table.(a) Use a graphing utility to create a scatter plot of the data for pharmacy A and
A store manager wants to know the demand for a product as a function of the price. The table shows the daily sales y for different prices x of the product.(a) Find the least squares regression line y
The sums have been evaluated. Solve the simplified system for a and b to find the least squares regression line for the points. Use a graphing utility to confirm the result. (Note: The symbol
An agricultural scientist used four test plots to determine the relationship between wheat yield y (in bushels per acre) and the amount of fertilizer x (in hundreds of pounds per acre). The table
The table shows the total gross domestic products y (in billions of dollars) of the United States for the years 2009 through 2015.(a) Find the least squares regression line y = at + b for the data,
Determine whether the statement is true or false. Justify your answer. 1. If two lines do not have exactly one point of intersection, then they must be parallel. 2. Solving a system of equations
Find the value of k such that the system of linear equations is inconsistent.1.2.
Briefly explain whether it is possible for a consistent system of linear equations to have exactly two solutions.
Give examples of systems of linear equations that have(a) No solution and(b) Infinitely many solutions.
Use the method of substitution to solve the system in Example 1. Do you prefer the method of substitution or the method of elimination? Explain.
Use the graphs of the two equations shown below.(a) Describe the graphs of the two equations.(b) Can you conclude that the system of equations whose graphs are shown is inconsistent? Explain.
The graphs of the two equations appear to be parallel. Yet, when you solve the system algebraically, you find that the system does have a solution. Find the solution and explain why it does not
Solve the system of equations for u and v. While solving for these variables, consider the trigonometric functions as constants. (Systems of this type appear in a course in differential
Use back-substitution to solve the system of linear equations.1.2.
Perform the row operation and write the equivalent system.1. Add Equation 1 to Equation 2.What did this operation accomplish? 2. Add ˆ’2 times Equation 1 to Equation 3. What did this
Solve the system of linear equations and check any solutions algebraically.1.2. 3. 4.
Solve the system of linear equations and check any solutions algebraically.1.2. 3.
Solve the system of linear equations and check any solutions algebraically.1.2.
An object moving vertically is at the given heights at the specified times. Find the position equation s = 1/2 at2 + v0t + s0 for the object. 1. At t = 1 second, s = 128 feet At t = 2 seconds, s = 80
Find the equation y = ax2 + bx + c of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. 1. (0, 0), (2, −2), (4,
Find the equation x2 + y2 + Dx + Ey + F = 0 of the circle that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. (0, 0), (5, 5), (10,
To expand its clothing line, a small corporation borrowed $775,000 from three different lenders. The money was borrowed at 8%, 9%, and 10% simple interest. How much was borrowed at each rate when the
A health insurance company advertises on television, on radio, and in the local newspaper. The marketing department has an advertising budget of $42,000 per month. A television ad costs $1000, a
Find the values of x, y, and z in the figure.1.2.
The perimeter of a triangle is 180 feet. The longest side of the triangle is 9 feet shorter than twice the shortest side. The sum of the lengths of the two shorter sides is 30 feet more than the
A chemist needs 10 liters of a 25% acid solution. The solution is to be mixed from three solutions whose concentrations are 10%, 20%, and 50%. How many liters of each solution will satisfy each
Applying Kirchhoff's Laws to the electrical network in the figure, the currents I1, I2, and I3, are the solution of the systemFind the currents.
A system of pulleys is loaded with 128-pound and 32-pound weights (see figure). The tensions t1 and t2 in the ropes and the acceleration a of the 32-pound weight are found by solving the systemwhere
The sums have been evaluated. Solve the simplified system for a, b, and c to find the least squares regression parabola for the points. Use a graphing utility to confirm the result. (Note: The symbol
In testing a new automobile braking system, engineers recorded the speed x (in miles per hour) and the stopping distance y (in feet). The table shows the results.(a) Find the least squares regression
Determine whether each ordered triple is a solution of the system of equations.(a) (0, 3, 1) (b) (ˆ’3, 0, 5) (c) (0, ˆ’1, 4) (d) (ˆ’1, 0, 5)
A wildlife management team studied the reproductive rates of deer in four tracts of a wildlife preserve. In each tract, the number of females x and the percent of females y that had offspring the
Find values of x, y and Î» that satisfy the system These systems arise in certain optimization problems in calculus, and Î» is called a Lagrange multiplier.1.2.
Every non square system of linear equations has a unique solution.
1. If a system of three linear equations is inconsistent, then there are no points common to the graphs of all three equations of the system.2. Are the following two systems of equations equivalent?
The number of sides x and the combined number of sides and diagonals y for each of three regular polygons are shown below. Write a system of linear equations to find an equation of the form y = ax2 +
Find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers.) 1. (2, 0, −1) 2. (−5, 3, −2) 3. (1/2, -3, 0) 4. (4, 2/5, 1/2)
1. The result of writing a rational expression as the sum of two or more simpler rational expressions is called the ________ ________ ________. 2. If the degree of the numerator of a rational
Write the partial fraction decomposition of the rational expression. Check your result algebraically. 1. 1 / (x2 + x) 2. 3 / (x2 - 3x) 3. 3 / (x2 + x -2) 4. (x + 1) / (x2 -x -6)
Write the partial fraction decomposition of the improper rational expression. 1. (x2 - x) / (x2 + x + 1) 2. (x2 - 4x) / (x2 + x + 6) 3. (2x3 - x2 + x + 5) / (x2 + 3x + 2)
Match the rational expression with the form of its decomposition. [The decompositions are labeled (a), (b), (c), and (d).] (a) A/x + B / (x + 2) + C / (x - 2) (b) A/x + B / (x - 4) (c) A/x + B/x2 + C
Write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. 1. (5 - x) / (3x2 + x - 1) 2. (4x2 - 1) / 2x (x + 1)2
The predicted cost C (in thousands of dollars) for a company to remove p% of a chemical from its waste water is given by the model C = 120p / (10,000 − p2), 0 ≤ p < 100. Write the partial
The magnitude of the range R of exhaust temperatures (in degrees Fahrenheit) in an experimental diesel engine is approximated by the modelR = 5000(4 ˆ’ 3x) / (11 ˆ’ 7x) (7
Determine whether the statement is true or false. Justify your answer. 1. For the rational expression x / {(x - 10) (x - 10)2}, the partial fraction decomposition is of the form A / (x + 10) + B (x
Identify the graph of the rational function and the graph representing each partial fraction of its partial fraction decomposition. Then state any relationship between the vertical asymptotes of the
Describe the error in writing the basic equation for the partial fraction decomposition of the rational expression.(x2 + 1) / x (x - 1) = A/x + B/(x - 1)x2 + 1 = A(x-1) + Bx
Describe two ways of solving for the constants in partial fraction decomposition.
Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 1. 3 / (x2 - 2x) 2. (x - 2) / (x2 + 4x + 3) 3. (6x + 5) / (x + 2)4 4. (5x2 + 3) / x2
1. An ordered pair (a, b) is a ________ of an inequality in x and y when the inequality is true after a and b are substituted for x and y, respectively. 2. The ________ of an inequality is the
Use a graphing utility to graph the inequality. 1. y ≥ −ln(x − 1) 2. y < ln(x + 3) − 1 3. y < 2x 4. y ≥ 3−x - 2 5. y ≤ 2 - 1/5 x 6. y > −2.4x + 3.3 7. 2/3 y + 2x2 - 5 ≥ 0 8. -1/6 x2
Write an inequality for the shaded region shown in the figure.1.2. 3. 4.
Sketch the graph of the solution set of the system of inequalities. Label the vertices of the region.1.2.
Sketch the graph of the solution set of the system.1.2.

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