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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
It is often difficult to evaluate the quality of products that undergo a ripening or maturation process. Researchers have successfully used ultrasonic velocity to determine the maturation time of Mahon cheese. The age can be determined byM(v) = 0.0312443v2 - 101.39v + 82,264, v ≥ 1620,where M(v)
In Exercises, calculate the limit in the specified exercise, using a table such as in Exercises. Verify your answer by using a graphing calculator to zoom in on the point on the graph.Data from exercise 39In Exercises, use the properties of limits to help decide whether each limit exists. If a
Does a value of k exist such that the following limit exists?If so, find the value of k and the corresponding limit. If not,explain why not. 3x² + kx - 2 x²-3x + 2 lim x 2 x²
Repeat the instructions of Exercise 67 for the following limit.Data from Exercise 67Does a value of k exist such that the following limit exists?If so, find the value of k and the corresponding limit. If not,explain why not. 2x² + kx - 9 lim x 3 x² 2 4x + 3
In Exercises, calculate the limit in the specified exercise, using a table such as in Exercises. Verify your answer by using a graphing calculator to zoom in on the point on the graph.Data from exercise 40In Exercises, use the properties of limits to help decide whether each limit exists. If a
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.(a) Find(b) Find 5 Let g(x)=x²-2 7 if x < 0 if 0≤x≤ 3. if x > 3
In Exercises, calculate the limit in the specified exercise, using a table such as in Exercises. Verify your answer by using a graphing calculator to zoom in on the point on the graph.Data from exercise 41In Exercises, use the properties of limits to help decide whether each limit exists. If a
Use a graph of ƒ(x) = ln x to answer the following questions.(a) Find(b) Where does the function ln x have a vertical asymptote? lim In x. x->0*
In Exercises, calculate the limit in the specified exercise, using a table such as in Exercises. Verify your answer by using a graphing calculator to zoom in on the point on the graph.Data from exercise 42In Exercises, use the properties of limits to help decide whether each limit exists. If a
Use a graphing calculator to answer the following questions.(a) From a graph of y = x ln x, what do you think is the value ofSupport this by evaluating the function for several small values of x.(b) Repeat part (a), this time using the graph of y = x(ln x)2.(c) Based on your results from parts (a)
Use a graph of ƒ(x) = ex to answer the following questions.(a) Find(b) Where does the function ex have a horizontal asymptote? lim G(x). x-4
Use a graphing calculator to answer the following questions.(a) From a graph of y = xe-x, what do you think is the value ofSupport this by evaluating the function for several large values of x.(b) Repeat part (a), this time using the graph of y = x2e-x.(c) Based on your results from parts (a) and
Find each of the following limits (a) By investigating values of the function near the x-value where the limit is taken, and(b) Using a graphing calculator to view the function near that value of x. lim x² + 4x³9x² + 7x - 3 x - 1
Find each of the following limits (a) By investigating values of the function near the x-value where the limit is taken, and(b) Using a graphing calculator to view the function near that value of x. x1/3 + 1 X lim x1 x + 1 +1
Find each of the following limits (a) By investigating values of the function near the x-value where the limit is taken, and(b) Using a graphing calculator to view the function near that value of x. x² - 4 X 81 = x + x - X x-2 lim
Find each of the following limits (a) By investigating values of the function near the x-value where the limit is taken, and(b) Using a graphing calculator to view the function near that value of x. x3/2 - 8 - 6 lim x-4x + x¹/2 X
A friend who is confused about limits wonders why you investigate the value of a function closer and closer to a point, instead of just finding the value of a function at the point. How would you respond?
Explain in your own words why the rules for limits at infinity should be true.
Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f (x). lim √9x² + 5 2x
Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f (x). lim X10 2 √9x² + 5 2.x
Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f (x). lim X118 √36x² + 2x + 7 3x
Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f (x). lim X 00 V36x2 + 2x + 7 3x
Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f (x). lim (1 + 5x¹/3 + 2x5/3)³ 45
500 g of iodine-131 is decaying exponentially. After 3 days 386 g of iodine-131 is left.(a) Write a function in the form y = y0ekt giving the number of grams of iodine-131 after t days.(b) Write the function from part (a) in the form y = y0(386/500)ƒ(t).(c) Use your answer from part (a) to find
Using the graph of f (x) in Figure 10, show the graph of a f (x) where a satisfies the given condition.-1 < a < 0 Figure 10 y=f(x) X
Find the temperature of an object when t = 9 if T0 = 18, C = 5, and k = 0.6.Newton’s law of cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is introduced. This leads to an equation where the
Graph the following by hand.y = 1 + log3 x
Solve each equation in Exercises. Round decimal answers to four decimal places.logy 8 = 3/4
Isabella puts $10,500 into an account to save money to buy a car in 12 years. She expects the car of her dreams to cost $35,000 by then. Find the interest rate that is necessary if the interest is computed using the following methods.(a) Compounded quarterly (b) Compounded monthly
Using the graph of f (x) in Figure 10, show the graph of a f (x) where a satisfies the given condition.a < -1 Figure 10 y=f(x) X
If C = 100, k = 0.1, and t is time in minutes, how long will it take a hot cup of coffee to cool to a temperature of 25°C in a room at 20°C?Newton’s law of cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into
Graph the following by hand.y = -1n(x + 3)
Solve each equation in Exercises. Round decimal answers to four decimal places.logr 5 = 1/2
If C = -14.6 and k = 0.6 and t is time in hours, how long will it take a frozen pizza to thaw to 10°C in a room at 18°C?Newton’s law of cooling says that the rate at which a body cools is proportional to the difference in temperature between the body and an environment into which it is
Suppose the fixed cost to make coffee mugs is $200, and the marginal cost to make one mug is $0.85.(a) Write a formula for the average cost function.(b) As the number of mugs produced gets increasingly large, what does the average cost per mug approach?(c) Sketch a graph of the average cost
Graph the following by hand.y = 2 - 1n x2
Solve each equation in Exercises. Round decimal answers to four decimal places.log4 (5x + 1) = 2
On January 1, 2013, Jack deposited $1000 into Bank X to earn interest at the rate of j per annum compounded semiannually. On January 1, 2018, he transferred his account to Bank Y to earn interest at the rate of k per annum compounded quarterly. On January 1, 2021, the balance at Bank Y was
If r is an x-intercept of the graph of y = ƒ(x), what is an x-intercept of the graph of each of the following?(a) y = -ƒ(x) (b) y = ƒ(-x) (c) y = -ƒ(-x)
Solve equation.2x+2 = 1/8
Suppose the average cost per unit C(x), in dollars, to produce x units of yogurt is given by(a) Find C(10), C(20), C(50), C(75), and C(100).(b) Which of the intervals (0, ∞), and [0, ∞) would be a more reasonable domain for C? Why?(c) Give the equations of any asymptotes on (-∞ , ∞). Find
If b is the y-intercept of the graph of y = ƒ(x), what is the y-intercept of the graph of each of the following?(a) y = -ƒ(x) (b) y = ƒ(-x) (c) y = -ƒ(-x)
Solve equation.(9/16)x = 3/4
Salmonella bacteria are a common cause of food poisoning. A recent study found that the bacteria grow exponentially in rats and double in size every 5 hours.(a) Suppose a rat initially had 10,000 Salmonella bacteria. Write an equation of the form y(t) = y02t/k so that y(0) = 10,000 and y(5) =
Solve equation.1/2 = (b/4)1/4
Write each equation using logarithms.35 = 243
Write each equation using logarithms.51/2 = √5
Write each equation using logarithms.e0.8 ≈ 2.22554
Write each equation using logarithms.101.07918 ≈ 12
Write each equation using exponents.log2 32 = 5
Write each equation using exponents.log9 3 = 1/2
A technique for measuring cardiac output depends on the concentration of a dye after a known amount is injected into a vein near the heart. In a normal heart, the concentration of the dye at time t (in seconds) is given by the function g(t) = -0.006t4 + 0.140t3 - 0.053t2 + 1.79t.(a) Graph g(t) on
Write each equation using exponents.1n 82.9 ≈ 4.41763
Write each equation using exponents.log 3.21 ≈ 0.50651
Evaluate each expression without using a calculator. Then support your work using a calculator and the change-of-base theorem for logarithms.log3 81
Evaluate each expression without using a calculator. Then support your work using a calculator and the change-of-base theorem for logarithms.log32 16
Evaluate each expression without using a calculator. Then support your work using a calculator and the change-of-base theorem for logarithms.log4 8
Evaluate each expression without using a calculator. Then support your work using a calculator and the change-of-base theorem for logarithms.log100 1000
Simplify each expression using the properties of logarithms.log5 3k + log5 7k3
Simplify each expression using the properties of logarithms.log3 2y3 - log3 8y2
Simplify each expression using the properties of logarithms.4 log3 y - 2 log3 x
Simplify each expression using the properties of logarithms.3 log4 r2 - 2 log4 r
Solve each equation. If necessary, round each answer to the nearest thousandth.6 p = 17
Solve each equation. If necessary, round each answer to the nearest thousandth.3z-2 = 11
Solve each equation. If necessary, round each answer to the nearest thousandth.21-m = 7
Solve each equation. If necessary, round each answer to the nearest thousandth.12-k = 9
Solve each equation. If necessary, round each answer to the nearest thousandth.e-5-2x = 5
Draw a sketch of the arch or culvert on coordinate axes, with the horizontal and vertical axes through the vertex of the parabola. Use the given information to label points on the parabola. Then give the equation of the parabola and answer the question.A culvert is shaped like a parabola, 18 ft
Solve each equation. If necessary, round each answer to the nearest thousandth.e3x-1 = 14
Inflation is generally described as the increase over time of the cost of a particular product or service. The rate of inflation depends on many factors and does not remain constant. Inflation causes the value of a dollar to decrease over time. Over the past ten years, the average rate of inflation
Solve equation.92y+3 = 27y
Find all errors in the following calculation.(log(x + 2))2 = 2 log(x + 2) = 2(log x + log 2) = 2(log x + 100)
Solve each equation. If necessary, round each answer to the nearest thousandth. 1 + m 3, 5 = 15
Prove: loga (x/y) = loga x - loga y.
Solve each equation. If necessary, round each answer to the nearest thousandth. 1 + 2p 5 2 = 3
Prove: loga xr = r loga x.
Solve each equation. If necessary, round each answer to the nearest thousandth.logk 64 = 6
Solve each equation. If necessary, round each answer to the nearest thousandth.log3 (2x + 5) = 5
Solve each equation. If necessary, round each answer to the nearest thousandth.log(4p + 1) + log p = log 3
Solve each equation. If necessary, round each answer to the nearest thousandth.log2(5m - 2) - log2 (m + 3) = 2
Give the following properties of the exponential function ƒ(x) = ax; a > 0, a ≠ 1.(a) Domain (b) Range (c) y-intercept(d) Asymptote(s)(e) Increasing if a is ______(f) Decreasing if a is _____
Give the following properties of the logarithmic function ƒ(x) = loga x; a > 0, a ≠ 1.(a) Domain (b) Range (c) x-intercept(d) Asymptote(s) (e) Increasing if a is _____(f) Decreasing if a is _____
Compare your answers for Exercises 83 and 84. What similarities do you notice? What differences?Data from Exercises 83Give the following properties of the exponential function ƒ(x) = ax; a > 0, a ≠ 1.(a) Domain (b) Range (c) y-intercept(d) Asymptote(s)(e) Increasing if a
To rent a mid-size car from one agency costs $60 per day or fraction of a day. If you pick up the car in Boston and drop it off in Utica, there is a fixed $40 charge. Let C(x) represent the cost of renting the car for x days and taking it from Boston to Utica. Find the following.(a) C(3/4)(b)
The cost to remove x percent of a pollutant isin thousands of dollars. Find the cost of removing the following percents of the pollutant.(a) 80% (b) 50% (c) 90%(d) Graph the function.(e) Can all of the pollutant be removed? y = 7x 100 - x
Find the amount of interest earned by each deposit.$6902 if interest is 6% compounded semiannually for 8 years
Find the amount of interest earned by each deposit.$2781.36 if interest is 4.8% compounded quarterly for 6 years
Find the compound amount if $12,104 is invested at 6.2% compounded continuously for each period.2 years
Find the compound amount if $12,104 is invested at 6.2% compounded continuously for each period.4 years
Find the compound amounts for the following deposits if interest is compounded continuously.$1500 at 6% for 9 years
Find the compound amounts for the following deposits if interest is compounded continuously.$12,000 at 5% for 8 years
Find the compound amounts for the following deposits if interest is compounded continuously.How long will it take for $1000 deposited at 6% compounded semiannually to double? To triple?
Find the compound amounts for the following deposits if interest is compounded continuously.How long will it take for $2100 deposited at 4% compounded quarterly to double? To triple?
Find the effective rate to the nearest hundredth for each nominal interest rate.7% compounded quarterly
Find the effective rate to the nearest hundredth for each nominal interest rate.6% compounded monthly
Suppose the cost in dollars to produce x posters is given by(a) Sketch a graph of C(x).(b) Find a formula for C(x + 1) - C(x), the cost to produce an additional poster when x posters are already produced.(c) Find a formula for A(x), the average cost per poster.(d) Find a formula for A(x + 1) -
Find the effective rate to the nearest hundredth for each nominal interest rate.5% compounded continuously
Find the present value of each amount.$2000 if interest is 6% compounded annually for 5 years
Find the present value of each amount.$10,000 if interest is 8% compounded semiannually for 6 years
To help pay for college expenses, Julie borrowed $10,000 at 7% interest compounded semiannually for 8 years. How much will she owe at the end of the 8-year period?
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