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mathematics
applied calculus

Questions and Answers of Applied Calculus

Use Figure 3.22 to evaluate the derivative. 80 0 # f(x) 80 $6 0 Figure 3.22 80
Use the tangent line approximation.Given f(x) = x4 − x2 + 3 approximate f(1.04).
A new DVD is available for sale in a store one week after its release. The cumulative revenue, $R, from sales of the DVD in this store in week t after its release isR = f(t) = 350 ln t with t >
A patient’s total cholesterol level, T (t), and good cholesterol level, G(t), at t weeks after January 1, 2016, are measured in milligrams per deciliter of blood (mg/dl). The cholesterol ratio,
Let ℎ(x) = f(g(x)) and k(x) = g(f(x)). Use Figure 3.21 to estimate the derivatives.k¨(2) -3 3 -3 A 3 برا f(x) X g(x) -3 نیا 3 -3 3 X
Find the relative rate of change f¨(t)∕f(t) at the given value of t. Assume t is in years and give your answer as a percent.f(t) = 2t3 + 10; t = 4
The value of an automobile purchased in 2014 can be approximated by the function V (t) = 30(0.85)t, where t is the time, in years, from the date of purchase, and V (t) is the value, in thousands of
Find the relative rate of change f'(t)∕f(t) at the given value of t. Assume t is in years and give your answer as a percent.f(t) = 3t + 2; t = 5
The demand curve for a product is given byq = f(p) = 10,000e−0.25p,where q is the quantity sold and p is the price of the product, in dollars. Find f(2) and f¨(2). Explain in economic terms what
For Problems let ℎ(x) = f(g(x)) and k(x) = g(f(x)). Use Figure 3.21 to estimate the derivatives.ℎ'(2) -3 3 -3 A 3 برا f(x) X g(x) -3 نیا 3 -3 3 X
The quantity demanded of a certain product, q, is given in terms of p, the price, byq = 1000e−0.02p(a) Write revenue, R, as a function of price.(b) Find the rate of change of revenue with respect
With t in years since 2016, the height of a sand dune (in centimeters) is f(t) = 700 − 3t2. Find f(5) and f¨(5). Using units, explain what each means in terms of the sand dune.
For Problems let ℎ(x) = f(g(x)) and k(x) = g(f(x)). Use Figure 3.21 to estimate the derivatives.k¨(1) -3 3 -3 A 3 برا f(x) X g(x) -3 نیا 3 -3 3 X
The world’s population is about f(t) = 7.17e0.011t billion, where t is time in years since July 2014. Find f(0), f¨(0), f(10), and f¨(10). Using units, interpret your answers in terms of
Find the rate of change of a population of size P(t) = t3 + 4t + 1 at time t = 2.
The amount, A mg, of caffeine in the body t hours after drinking a cup of coffee can be approximated byA = f(t) = 120e−0.17t.(a) Find f¨(t).(b) Find f(2) and f¨(2). Give units.
If p is price in dollars and q is quantity, demand for a product is given byq = 5000e−0.08p.(a) What quantity is sold at a price of $10?(b) Find the derivative of demand with respect to price when
If $1000 is deposited in a bank account that pays 3% interest compounded continuously, the balance B after t years isB = f(t) = 1000e0.03t.(a) Find f¨(t).(b) Find f(10) and f¨(10). Give units.
For Problems let ℎ(x) = f(g(x)) and k(x) = g(f(x)). Use Figure 3.21 to estimate the derivatives.ℎ'(1) -3 3 -3 A 3 برا f(x) X g(x) -3 نیا 3 -3 3 X
Let f(t) = t2 − 4t + 5.(a) Find f'(t).(b) Find f'(1) and f'(2).(c) Use a graph of f(t) to check that your answers to part (b) are reasonable. Explain.
The concentration of carbon dioxide in the air, C(t), in parts per million at time, t, in months, since December 1, 2005 is given by(a) Is C(t) periodic? How about C'(t)?(b) Which of expressions
A drug concentration curve is given by C = f(t) = 20te−0.04t, with C in mg/ml and t in minutes.(a) Graph C against t. Is f'(15) positive or negative? Is f¨(45) positive or negative? Explain.(b)
A fish population is approximated by P(t) = 10e0.6t, where t is in months. Calculate and use units to explain what each of the following tells us about the population:(a) P(12) (b) P'(12)
Let f(x) = x3 − 4x2 + 7x − 11. Find f'(0), f'(2), f'(−1).
The quantity of a drug, Q mg, present in the body t hours after an injection of the drug is given isQ = f(t) = 100te−0.5t.Find f(1), f'(1), f(5), and f'(5). Give units and interpret the answers.
Find the equation of the tangent line to f(x) = 10e−0.2x at x = 4.
(a) Use a graph of P(q) = 6q−q2 to determine whether each of the following derivatives is positive, negative, or zero: P'(1), P'(3), P'(4). Explain.(b) Find P'(q) and the three derivatives in part
The concentration, C in ng/ml, of nicotine in the body t minutes after starting to smoke a cigarette can be approximated byC = f(t) = 4te−0.08t.(a) Find f¨(t).(b) Find f(15) and f¨(15). Give
Find the equation of the tangent line to y = e−2t at t = 0. Check by sketching the graphs of y = e−2t and the tangent line on the same axes.
For t in years since 2010, daily oil consumption in China, in thousands of barrels, was approximated byB = 8938e0.05t.(a) Is daily oil consumption increasing or decreasing with time?(b) How fast is
Find the derivative. Assume a, b, c, k are constants.y = 4.2q2 − 0.5q + 11.27
Find the derivative. Assume that a, b, c, and k are constants.y = te−t2
Differentiate the functions in Problems. Assume that A and B are constants.f(x) = 2x sin(3x)
Differentiate the functions in Problems. Assume that A, B, and C are constants.y = e0.7t
Find the derivative of the functions in Problems.R = (q2 + 1)4
Use the definition of the derivative to obtain the following results.If f(x) = 2x2 + 3, then f¨(x) = 4x.
For Problems find the derivative. Assume a, b, c, k are constants.y = x12
For Problems find the derivative. Assume that a, b, c, and k are constants.f(t) = te−2t
Differentiate the functions in Problems. Assume that A and B are constants.y = B + Asin t
Differentiate the functions in Problems. Assume that A, B, and C are constants.f(x) = x3 + 3x
Find the derivative of the functions in Problems.w = (t2 + 1)100
Use the definition of the derivative to obtain the following results.If f(x) = 5x2, then f¨(x) = 10x.
If f(x) = (2x + 1)(3x − 2), find f'(x) two ways: by using the product rule and by multiplying out. Do you get the same result?
Differentiate the functions in Problem. Assume that A and B are constants.y = 5 sin x
Differentiate the functions in Problem. Assume that A and B are constants.P = 3 + cos t
A developer has purchased a laundromat and an adjacent factory. To keep smoke, which ruins the clothes, out of the dryers the developer can protect the laundromat or install filters on the
Cost and revenue functions are given in Figure 4.56. Approximately what quantity maximizes profits? 60,000 40,000 20,000 0 Figure 4.56 CR 9 7000
A curve representing the total number of people, P, infected with a virus often has the shape of a logistic curve of the formwith time t in weeks. Suppose that 10 people originally have the virus and
True or false? Give an explanation for your answer. The global maximum of f(x) = x2 on every closed interval is at one of the endpoints of the interval.
The demand for yams is given by q = 5000 − 10p2, where q is in pounds of yams and p is the price of a pound of yams.(a) If the current price of yams is $2 per pound, how many pounds will be
In Problems graph the function and describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or
Find the point where the following curve is steepest: y = 50 1+6e-2 for t≥ 0.
Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or
Cost and revenue functions are given in Figure 4.56.(a) At a production level of q = 3000, is marginal cost or marginal revenue greater? Explain what this tells you about whether production should be
Plot the graph of f(x) = x3 − ex using a graphing calculator or computer to find all local and global maxima and minima for:(a) −1 ≤ x ≤ 4(b) −3 ≤ x ≤ 2
The demand for yams is given in Problem 14.(a) At a price of $2 per pound, what is the total revenue for the yam farmer?(b) Write revenue as a function of price, and then find the price that
Show analytically that if marginal cost is less than average cost, then the derivative of average cost with respect to quantity satisfies a¨(q) < 0.
Graph the function and describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing).
When production is 2000, marginal revenue is $4 per unit and marginal cost is $3.25 per unit. Do you expect maximum profit to occur at a production level above or below 2000? Explain.
Sketch the graph of a function on the interval 0 ≤ x ≤ 10 with the given properties.Has local minimum at x = 3, local maximum at x = 8, but global maximum and global minimum at the endpoints of
In Kazakhstan, the demand curve 12 of dairy products is q = kp−0.6 for some positive constant k. What is the elasticity of dairy products in Kazakhstan?
Show analytically that if marginal cost is greater than average cost, then the derivative of average cost with respect to quantity satisfies a'(q) > 0.
Graph the function and describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing).
Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or
Revenue and cost functions for a company are given in Figure 4.57.(a) Estimate the marginal cost at q = 400.(b) Should the company produce the 500th item? Why?(c) Estimate the quantity which
Sketch the graph of a function on the interval 0 ≤ x ≤ 10 with the given properties.Has local and global maximum at x = 3, local and global minimum at x = 10.
A reasonably realistic model of a firm’s costs is given by the short-run Cobb-Douglas cost curveC(q) = Kq1∕a + F,where a is a positive constant, F is the fixed cost, and K measures the technology
Graph the function and describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing).
If R is percent of maximum response and x is dose in mg, the dose-response curve for a drug is given by(a) Graph this function.(b) What dose corresponds to a response of 50% of the maximum? This is
Dose-response curves for three different products are given in Figure 4.82.(a) For the desired response, which drug requires the largest dose? The smallest dose?(b) Which drug has the largest maximum
Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or
A company has 100 units to spend for equipment and labor combined. The company spends x on equipment and 100 − x on labor, enabling it to produce Q items whereQ = 5x0.3(100 − x)0.8.How much
If E = 2 for all prices p, how can you maximize revenue?
Sketch the graph of a function on the interval 0 ≤ x ≤ 10 with the given properties.Has global maximum at x = 0, global minimum at x = 10, and no other local maxima or minima.
Graph the function and describe in words the interesting features of the graph, including the location of the critical points and where the function is monotonic (that is, increasing or decreasing).
Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or
A manufacturing process with a $12 million budget uses x kilograms of one raw material and y kilograms of a second raw material to make Q = 3 ln(x + 1) + 2 ln(y+1) units of product. The first raw
If E = 0.5 for all prices p, how can you maximize revenue?
The function y = t(x) is positive and continuous with a global maximum at the point (3, 3). Graph t(x) if t'(x) and t''(x) have the same sign for x < 3, but opposite signs for x > 3.
Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or
For the functions in Problems do the following:(a) Find f¨ and f¨¨ .(b) Find the critical points of f.(c) Find any inflection points of f.(d) Evaluate f at its critical points and at the endpoints
A person’s blood pressure, p, in millimeters of mercury (mm Hg) is given, for t in seconds, byp = 100 + 20 sin(2.5πt).(a) What are the maximum and minimum values of blood pressure?(b) What is the
Table 5.9 shows the upward vertical velocity v(t), in ft/min, of a small plane at time t seconds during a short flight.(a) When is the plane going up? Going down?(b) If the airport is located 110 ft
After a foreign substance is introduced into the blood, the rate at which antibodies are made is given bywhere time, t, is in minutes. Assuming there are no antibodies present at time t = 0, find the
Using Figure 5.88, list the following numbers from least to greatest:(a) f'(1)(b) The average value of f on 0 ≤ x ≤ 4(c) ∫10 f(x)dx 8 f(x), 1 + + x 2 3 4 Figure 5.88
Use Figure 5.51 to find limits a and b in the interval [0, 5] with a < b satisfying the given condition.∫b0 f(x) dx is largest 2 f(x) 3 4 5 Figure 5.51 X
Use Figure 5.51 to find limits a and b in the interval [0, 5] with a < b satisfying the given condition.∫4a f(x) dx is smallest OOIKJ - 2 f(x) 3 4 5 Figure 5.51 X
The graph of f(t) is in Figure 5.33. Which of the following four numbers could be an estimate of ∫10 f(t)dt accurate to two decimal places? Explain your choice.I. −98.35 II. 71.84 III. 100.12 IV.
Find the integrals in Problems. 3 Tx x³ lnx dx
Find the integrals in Problems. Check your answers by differentiation. s x -dx √x² + 4
Use the Fundamental Theorem to evaluate the definite integral exactly. 5 [²² 3x² dx
Supply and demand curves for a product are in Figure 6.41.(a) Estimate the equilibrium price and quantity.(b) Estimate the consumer and producer surplus. Shade them.(c) What are the total gains from
Let G'(t) = g(t) and G(0) = 4. Use Figure 6.6 to find the values of G(t) at t = 5, 10, 20, 25. 2 -2 -8(r)_ 10 15 20 Figure 6.6
Decide whether the expression is a number or a family of functions. (Assume f(x) is a function.) 5+ [250 2f(x) dx
Find the integrals in Problems. Check your answers by differentiation. [x√x² +1dx
Find the integrals in Problems. Check your answers by differentiation. 29e⁹² +1dq
Use the Fundamental Theorem to evaluate the definite integral exactly. [. 2e* dx
Supply and demand curves are in Figure 6.41. A price of $40 is artificially imposed.(a) At the $40 price, estimate the consumer surplus, the producer surplus, and the total gains from trade.(b)
A person deposits money into an account, which pays 5% interest compounded continuously, at a rate of $1000 per year for 15 years. Calculate:(a) The balance in the account at the end of the 15
Figure 6.7 shows the derivative g'. If g(0) = 0, graph g. Give (x, y)-coordinates of all local maxima and minima. - 2 3 نیا -g'(x)- + Figure 6.7 5 6 x

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