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mathematics
calculus

Questions and Answers of Calculus

Simplify each expression so that only positive exponents remain. 1. y1/4 ( y1/2 2. x2/3 x1/5 3. z3/4 ( z4 4. x -2/3 ( x2
Simplify each expression. 1. (64x4 2. 3(-64x6y3 3. (128x4y5 4. 3(54x5x8
Perform the indicated operations and simplify. 1. (12x3y ( (3x2y 2. 3(16x2y ( 3(3x2y 3. (63x5y3 ( (28x2y 4. (10xz10 ( (30x17z
Determine a value for x that makes each statement true. 1. (A9)x = A 2. (B20)x = B 3. (7(R)x = R 4. ((T3)x = T
Rationalize each denominator and then simplify. 1. (2/3 2. (5/8 3. (m2x / (mx2 4. 5x3w / (4xw2
In calculus it is frequently important to write an expression in the form cxn, where c is a constant and n is a rational number. Write each expression in this form. 1. -2 / 3 3(x2 2. -2 (3 4(3x 3. 3x
In calculus problems, the answers are frequently expected to be in a form with a radical instead of a fractional exponent. Write each expression with radicals. 1. 3/2x1/2 2. 4/3 x 1/3 3. 1/2x -1/2 4.
The Richter scale reading for an earthquake measures its intensity (as a multiple of some minimum intensity used for comparison). The intensity I corresponding to a Richter scale reading R is given
The intensity of sound I (as a multiple of the average minimum threshold of hearing intensity) is related to the decibel level D (or loudness of sound) according toI = 10D/10(a) Express 10D/10 using
If $1000 is invested at r% compounded annually, the future value S of the account after two and a half years is given by(a) Express this equation with radical notation. (b) Find the value of this
The population P of India (in billions) for 2000-2050 can be approximated by the equationP = 0.924t0.13where t > 0 is the number of years past 2000.(a) Express this equation with radical
The percent p of paved roads and streets in the United States can be approximated with the equationp = 14.32t0.38where t is the number of years past 1950.(a) Express this equation with radical
Suppose it has been determined that the sales at Ewing Gallery decline after the end of an advertising campaign, with daily sales given by S = 2000(2-0.1x) where S is in dollars and x is the number
The growth of a company can be described by the equation N = 500(0.02)0.7t where t is the number of years the company has been in existence and N is the number of employees. (a) What is the number of
Simplify by combining like terms. 1. (16pq - 7p2) + (5pq + 5p2) 2. (3x3 + 4x2y2) + (3x2y2 - 7x3) 3. (4m2 - 3n2 + 5) - (3m2 + 4n2 + 8) 4. (4rs - 2r2s - 11rs2) - (11rs2 - 2rs + 4r2s)
Perform the indicated operations and simplify. 1. (5x3) (7x2) 2. (-3x2y) (2xy3) (4x2y2) 3. (39r3s2) ( (13r2s) 4. (-15m3n) ( (5mn4)
The expressions in Problems 1 and 2 are polynomials with the form anxn + an-1xn-1 + ..... + a1x + a0, where n is a positive integer. Complete the following.1. For 2x5 - 3x2 - 5,(a) 2 = a?(b) a3 =
Simplify each expression.1. (a) (3x - 2)2 - 3x - 2(3x - 2) + 5(b) (3x - 2)2 - (3x - 2) (3x - 2) + 52. (a) (2x - 3) (3x + 2) - (5x - 2) (x - 3)(b) 2x - 3(3x + 2) - 5x - 2(x - 3)
Perform the indicated operations with expressions involving fractional exponents and radicals and then simplify. 1. x1/2 (x1/2 + 2x3/2) 2. x-2/3(x5/3 - x-1/3) 3. (x1/2 + 1) (x1/2 - 2) 4. (x1/3 -
Evaluate each algebraic expression at the indicated values of the variables.1. 4x - x2 at x = - 22. 10 - 6(4 - x)2 at x = -13. 10xy - 4(x - y)2 at x = 5 and y = -24. 3x2 - 4y2 - 2xy at x = 3 and y =
Suppose a company's revenue R (in dollars) from the sale of x units of its product is given byR = 215xSuppose further that the total costs C (in dollars) of producing those x units is given byC = 65x
Suppose that a nurse needs 10 cc (cubic centimeters) of a 15.5% solution (that is, a solution that is 15.5% ingredient) of a certain medication, which must be obtained by mixing x cc of a 20%
Factor out the common monomial factor. 1. 9ab - 12a2b + 18b2 2. 8a2b - 160x + 4bx2 3. 4x2 + 8xy2 + 2xy3 4. 12y3z + 4yz2 - 8y2z3
Factor completely for the following questions1. 4x2 - x2. 2x5 + 18x33. x3 + 4x2 - 5x - 204. x3 - 2x2 - 3x + 6
Determine the missing factor.1. x3/2 + x1/2 = x1/2 (?)2. 2x1/4 + 4x3/4 = 2x1/4 (?)3. x-3 + x-2 x-3 (?)4. x-1 - x = x-1(?)
Factor by grouping for the following questions1. 7x3 - 14x2 + 2x - 42. 5y - 20 - x2y + 4x23. 6x - 6m + xy - my4. x3 - x2 - 5x + 5
Use the following factorization formulas involving cubes to factor each expression in Problems 51-58.Factorizations with CubesPerfect cubea3 + 3a2b + 3ab2 + b3 = (a + b)3Perfect cubea3 - 3a2b + 3ab2
1. The future value of a simple-interest investment of P dollars at an annual interest rate r for t years is given by the expression P + Prt. Factor this expression?2. When medicine is administered,
1. Factor the following expression for the maximum power in a certain electrical circuit?(R + r)2 - 2r(R + r)2. The consumer expenditure for a commodity is the product of its market price, p, and the
1. Revenue R from the sale of x units of a product is found by multiplying the price by the number of items sold. (a) Factor the right side of R = 300x - x2. (b) What is the expression for the price
Factor each expression in Problems 1-2 as a product of binomials. 1. x2 + 8x + 12 2. x2 - 2x - 8 3. x2 - 15x - 16 4. x2 - 21x + 20
Simplify each fraction. 1. 18x3y3 / 9x3z 2. 15α4b5 / 30α3b 3. x - 3y / 3x - 9y 4. x2 - 6x + 8 / x2 - 16
Simplify each complex fraction.1.2. 3. 4.
Rewrite each of the following so that only positive exponents remain and simplify.1. (a) (2-2 - 3-1)-1(b) (2-1 + 3-1)22. (a) (32 + 42)-1/2(b) (22 + 32)-13.
Rationalize the denominator of each fraction and simplify.1.2.
Rationalize the numerator of each fraction and simplify.1.2.
1. The formula for the future value of an annuity due involves the expression
Suppose a company's average costs are given by(a) Express the average-cost formula as a single fraction. (b) Write the expression that gives the company's total costs. Use this equation, A company's
Suppose a company's average costs are given by(a) Express the average-cost formula as a single fraction. (b) Write the expression that gives the company's total costs. Use this equation, A company's
1 Workers A, B, and C can complete a job in a, b, and c hours, respectively. Working together, they can completeof the job in 1 hour. Add these fractions over a common denominator to obtain an
Perform the indicated operations and simplify.1.2.
1. Is A ( B if A = {1, 2, 5, 7} and B = {x: x is a positive integer, x ( 8}? 2. Is it true that 3 ( {x: x 3}? 3. Are A = {1, 2, 3, 4} and B = {x: x ( 1} disjoint?
Evaluate each expression. 1. |5 - 11| 2. 44 ( 2 ( 11 - 102 3. (-3)2 - (-1)3 4. (3) (2) (15) - (5) (8) / (4) (10)
Write each inequality in interval notation, name the type of interval, and graph it on a real number line. (a) 0 ( x ( 5 (b) x ( - 3 and x < 7 (c) (-4, () ( (-(, 0)
1. Write an inequality that represents each of the following.(a) (-1, 16)(b) [-12, 8}(c).2. Find the following roots.(a) -3(-64(b) (4/49(c) 7 1.9487171
Evaluate the following without a calculator.(a)(b) 23 ( 2-5 (c) 49 / 43 (d)
Rewrite each of the following with positive exponents. Assume all variables are nonzero. (a) x5 ( x-7 (b) x8 / x-2 (c) (x3)3 (d) (y4)-2 (e) (-y-3)-2
Simplify each expression so that only positive exponents remain. Assume all variables are nonzero.1.2. 3. 4.
1. Write each of the following with an exponent and with the variable in the numerator. (a) (x (b) 3(x2 (c) 1/ 4(x 2. Write each of the following in radical form. (a) x3/7 (b) x-1/2 (c) -x3/2
Rationalize each of the following denominators and simplify. (a) 5xy / (2x (b) y / x 3(xy2
Simplify so that only positive exponents remain. Assume all variables are positive. 1. x1/2 ( x1/3 2. y-3/4 / y-7/4 3. x4 ( x1/4 4. 1/(x-4/3 ( x-7/3)
Simplify each expression. Assume all variables are positive. 1. (12x3y5 2. (1250x6y9 3. 3(24x4y4 ( 3(45x4y10 4. (16a2b3 ( (8a3b5
Use sets U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 9}, and B = {1, 3, 5, 6, 7, 8, 10} to find the elements of the sets described. 1. A ( B 2. A ( B 3. (A' ( B') 4. Does (A' ( B)' = A ( B?
Perform the indicated operations and simplify. 1. (3x + 5) - (4x + 7) 2. x(1 - x) + x[x - (2 + x)] 3. (3x3 - 4xy - 3) + (5xy + x3 + 4y - 1) 4. (4xy3) (6x4y2)
Factor each expression completely.1. 2x4 - x32. 4(x2 + 1)2 - 2(x2 + 1)33. 4x2 - 4x + 14. 16 - 9x2
Simplify each of the following (a) 2x / 2x + 4 (b) 4x2y3 - 6x3y4 / 2x2y2 - 3xy3
Perform the indicated operations and simplify.1.2. 3. 4.
State the property of the real numbers that is illustrated in each case.(a)(b) 2(3 ( 4) = (2 ( 3)4(c) 1/3 (6 + 9) = 2 + 3
Simplify each complex fraction.1.2. 3. Rationalize the denominator of And simplify. 4. Rationalize the numerator of And simplify?
In an attempt to determine some off-the-job factors that might be indicators of on-the-job effectiveness, a company made a study of 200 of its employees. It was interested in whether the employees
1. The percent of the U.S. population covered by an employment-based health insurance plan can be approximated by the expression - 0.75x + 63.8 where x is the number of years past 2000. Use this
1. Indicate whether each given expression is one or more of the following: rational, irrational, integer, natural, or undefined.(a) π(b) 0/6(c) 6/02. Insert the proper sign (<, =, or >) to
If an individual makes monthly deposits of $100 in an account that earns 9% compounded monthly, then the future value S of the account after n months is given by the formula
1. The per capita net cost C (in dollars) of private health insurance coverage in the United States from 2009 and projected to 2021 can be approximated by the formulaC = 31.9t + 310where t is the
Suppose you borrow $10,000 for n months to buy a car at an interest rate of 7.8% compounded monthly. The size of each month's payment R is given by the formula(a) Rewrite the expression on the
Suppose that in a study of water birds, the relationship between the number of acres of wetlands A and the number of species of birds S found in the wetlands area was given byS = kA1/3where k is a
1. Suppose that the total cost of producing and selling x units of a product is 300 + 4x and the total revenue from the sale of x units is 30x - 0.001 x2. Find the difference of these expressions
Suppose that the cost C (in dollars) of removing p% of the pollution from the waste water of a manufacturing process is given by(a) Express the right-hand side of this formula as a single
The average cost of producing a product is given by the expression 1200 + 56x + 8000 / x Write this expression with all terms over a common denominator?
In following question solve each equation.1. 4x - 7 = 8x + 22. 3x + 22 = 7x + 23. x + 8 = 8(x + 1)4. x + x + x = x
In following questions solve each fractional equation. Check your answers in the original equations?1.2.3.4.
Use a calculator to solve each equation. Round your answer to three decimal places.1. 3.259x - 8.638 = - 3.8(8.625x + 4.917)2. 3.319(14.1x - 5) = 9.95 - 4.6x3. 0.000316x + 9.18 = 2.1(3.1 - 0.0029x) -
Solve for y in terms of x.1. 3x - 4y = 152. 3x - 5y = 253.4.
Solve each inequality.1. 3(x - 1) < 2x - 12(x + 1) > x - 13. 1 - 2x > 94. 17 - x < - 4
Solve each inequality and graph the solution.1. 2(x - 1) - 3 > 4x + 12. 7x + 4 ( 2(x - 1)3.4.
1. A $648,000 property is depreciated for tax purposes by its owner with the straight-line depreciation method. The value of the building, y, after x months of use is given by y = 648,000 - 1800x
Credit card debt High interest rates make it difficult for people to pay off credit card debt in a reasonable period of time. The interest I (in dollars) paid on a $10,000 debt over 3 years when the
In seawater, the pressure p is related to the depth d according to 33p - 18d = 495 where d is in feet and p is in pounds per square inch. (a) Solve this equation for p in terms of d. (b) The Titanic
1. Burnem, Inc. manufactures blank CDs and sells them to a distributor in packs of 500 CDs. Burnem's total cost and total revenue (in dollars) for x packs of 500 CDs are given byTotal cost = 2x +
1. In its second year of operation, a local Internet provider's profits were $170,500. If this amount was 576% of the company's first-year profits, find the first year profits (to the nearest hundred
The percent P of the U.S. population with Internet access is given by25P - 34t = 1378where t is the number of years past 1990.(a) What percent of the U.S. population had Internet access in 2010?(b)
1. Data and projections from the Centers for Medicare and Medicaid Services from 2000 to 2018 indicate that the annual per capita out-of-pocket expenses for health care can be approximated byy =
Suppose a professor counts the final exam as being equal to each of the other tests in her course, and she will also change the lowest test score to match the final exam score if the final exam score
A retired woman has $120,000 to invest. She has chosen one relatively safe investment fund that has an annual yield of 9% and another, riskier fund that has a 13% annual yield. How much should she
One safe investment pays 10% per year, and a riskier investment pays 18% per year. A woman who has $145,600 to invest wants to have an income of $20,000 per year from her investments. How much should
A woman making $2000 per month has her salary reduced by 10% because of sluggish sales. One year later, after a dramatic improvement in sales, she is given a 20% raise over her reduced salary. Find
In wildlife management, the capture-mark-recapture technique is used to estimate the populations of fish or birds in an area or to measure the infestation of insects such as Japanese beetles. Suppose
1. For a certain product, the revenue is given by R = 40x and the cost is given by C = 20x + 1600. To obtain a profit, the revenue must be greater than the cost. For what values of x will there be a
1. Sean can spend at most $900 for a camera and some memory sticks. He plans to buy the camera for $695 and sticks for $5.75 each. Write an inequality that could be used to find the number of sticks
Severely obese persons (BMI > 40) are at serious health risk, and their number is increasing. Using data from 1990 and projected to 2030 the percent of severely obese U.S. adults can be
Using data from 2010 and projected to 2018, the federal income tax per capita can be approximated by the equationy = 486.48t + 3486.84where t is the number of years past 2010 and y is in dollars
1. During a particular summer, Dallas, Texas, endured 29 consecutive days on which the temperature was at least 110 oF. On many of those days, the combination of heat and humidity made it feel even
Use the values in the following table.1. (a) Explain why the table defines y as a function of x.(b) State the domain and range of this function.(c) If the table expresses y = f (x), find f(0) and
1. If R(x) = 8x - 10, find the following. (a) R(0) (b) R(2) (c) R(-3) (d) R(1.6) 2. If f (x) = 17 - 6x, find the following. (a) f(-3) (b) f(1) (c) f(10) (d) f(2/3)
1. If C(x) = 4x2 - 3, find the following.(a) C(0)(b) C(-1)(c) C(-2)(d) C(-3/2)2. If h(x) = 3x2 - 2x, find the following.(a) h(3)(b) h(-3)(c) h(2)(d) h(1/6)
If h(x) = x -2(4 - x)3, find the following. (a) h(-1) (b) h(0) (c) h(6) (d) h(2.5)
If R(x) = 100x - x3, find the following.(a) R(1)(b) R(10)(c) R(2)(d) R(-10)
If f(x) = x3 - 4/x, find the following. (a) f(-1/2) (b) f(2) (c) f(-2)
If C(x) = (x2 - 1) x, find the following. (a) C(1) (b) C(1/2) (c) C(-2)
Let f(x) = 1 + x + x2 and h ¹ 0.(a) Is f(2 + 1) = f(2) + f(1)?(b) Find f(x + h).(c) Does f(x + h) = f(x) + f(h)?(d) Does f(x + h) = f(x) + h?(e) FindAnd simplify it.

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