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mathematics
applied calculus

Questions and Answers of Applied Calculus

Table 8.7 gives the number of calories burned per minute, B = f(s,w), for someone roller-blading, as a function of the person’s weight, w, and speed, s.(a) Is fw positive or negative? Is fs
The function has a critical point at (0, 0). What sort of critical point is it?f(x, y) = x2 − cos y
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.f(x, y) = 3x + 3y
Values of f(x, y) are in Table 8.8. Assuming they exist, decide whether you expect the following partial derivatives to be positive or negative.(a) fx(−2, −1) (b) fy(2, 1)(c) fx(2,
The heat index is a temperature which tells you how hot it feels as a result of the combination of temperature and humidity. See Table 8.4. Heat exhaustion is likely to occur when the heat index
The quantity, Q, of a certain product manufactured depends on the quantity of labor, L, and of capital, K, used according to the functionQ = 900L1∕2K2∕3.Labor costs $100 per unit and capital
If f(x, y) = x3 + 3y2, find f(1, 2), fx(1, 2), fy(1, 2).
Using Table 8.4, graph heat index as a function of humidity with temperature fixed at 70◦F and at 100◦F. Explain the features of each graph and the difference between them in common-sense terms.
The function has a critical point at (0, 0). What sort of critical point is it?f(x, y) = x sin y
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.f(x, y) = x + y + 1
If f(u, v) = 5uv2, find f(3, 1), fu(3, 1), and fv(3, 1).
The Cobb-Douglas production function for a product isP = 5L0.8K0.2,where P is the quantity produced, L is the size of the labor force, and K is the amount of total equipment. Each unit of labor costs
Estimate zx(1, 0) and zx(0, 1) and zy(0, 1) from the contour diagram for z(x, y) in Figure 8.46. 1.2 1.0 0.8 0.6 0.4 0.2 y 0.5 z = 1 z = 2 1.0 Figure 8.46 z=3 1.5 2.0 X
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = x2 + 4x + y2
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.f(x, y) = 2x − y
Find all points where the partial derivatives of f(x, y) are both 0.f(x, y) = x2 + y2
A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities, q1 and q2, supplied by each factory, and is expressed by the joint cost
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = x2 + xy + 3y
Use the diagram from Problem 20 in Section 8.1 to estimate HT (T ,w) for T = 10, 20, 30 and w = 0.1, 0.2, 0.3.What is the practical meaning of these partial derivatives?Data from in problem 20 Figure
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.f(x, y) = −x − y
Match the graphs in Figure 9.3 with the following descriptions.(a) The temperature of a glass of ice water left on the kitchen table.(b) The amount of money in an interest-bearing bank account into
A person’s basal metabolic rate (BMR) is the minimal number of daily calories needed to keep their body functioning at rest. The BMR (in kcal/day) of a man of mass m (in kg), height ℎ (in cm) and
Find all points where the partial derivatives of f(x, y) are both 0.f(x, y) = xey
The quantity, Q, of a product manufactured by a company is given byQ = aK0.6L0.4,where a is a positive constant, K is the quantity of capital and L is the quantity of labor used. Capital costs are
The monthly mortgage payment in dollars, P, for a house is a function of three variables:P = f(A, r,N),where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years
Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these.f(x, y) = x2 + y2 + 6x − 10y + 8
The graphs in Figure 9.4 represent the temperature, H(◦C), of four eggs as a function of time, t, in minutes. Match three of the graphs with the descriptions (a)–(c). Write a similar description
Show that if S, I, and R satisfy the differential equations in Problem 1, the total population, S + I + R, is constant.Data from in problem 1 Let I be the number of infected people and S be
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.f(x, y) = y − x2
Repeat Problem 4 for a school with 350 students.Data from in problem 4 (a) In a school of 150 students, one of the students has the flu initially. What is In? What is So? (b) Use these values of Io
Sketch the slope field for dy∕dx = x∕y at the points marked in Figure 9.19. ● -1 1 -1 y Figure 9.19 X
The monthly cost, in dollars, of a cell phone bill isP = f(t, m, d) = 0.25t + 0.2m + 0.01dwhere t is the number of minutes talked, m is the number of messages sent and d is the number of kilobytes of
In Example 1, the maximum production is obtained when 2y = x no matter what the budget is. Assume the current values for x and y are those that maximize production.(a) If you increase x by 3000, by
Create a system of differential equations to model the situations in Problems. You may assume that all constants of proportionality are 1.The concentrations of two chemicals are denoted by x and y,
Let S and I satisfy the differential equations in Problem 1. Assume I ≠ 0.Data from in problem 1(a) If dI∕dt = 0, find S.(b) Show that I increases if S is greater than the value you found in part
Radioactive substances decay at a rate proportional to the quantity present. Write a differential equation for the quantity, Q, of a radioactive substance present at time t. Is the constant of
Create a system of differential equations to model the situations in Problems. You may assume that all constants of proportionality are 1.A population of fleas is represented by x, and a population
Problems give a graph of f'(x). Graph f(x). Mark the points x1, . . . , x4 on your graph and label local maxima, local minima and points of inflection. /X1 X₂ X3) X XA f'(x)
Consider a solution curve for each of the slope fields in Problem 7. Write one or two sentences describing qualitatively the long-run behavior of y. For example, as x increases, does y → ∞, or
Problems concern the graph of f' in Figure 6.21.Which is greater, f(0) or f(1)? y 2 3 X y = f'(x) Figure 6.21: Note: Graph of f', not f
Consider a solution curve for each of the slope fields in Problem 7. Write one or two sentences describing qualitatively the long-run behavior of y. For example, as x increases, does y → ∞, or
Find the integrals in Problems. Check your answers by differentiation. (In z)² N dz
Find the integrals in Problems. Check your answers by differentiation. et - ex - dx et + e-x
Consider a solution curve for each of the slope fields in Problem 7. Write one or two sentences describing qualitatively the long-run behavior of y. For example, as x increases, does y → ∞, or
Consider a solution curve for each of the slope fields in Problem 7. Write one or two sentences describing qualitatively the long-run behavior of y. For example, as x increases, does y → ∞, or
Consider a solution curve for each of the slope fields in Problem 7. Write one or two sentences describing qualitatively the long-run behavior of y. For example, as x increases, does y → ∞, or
Find an antiderivative.p(z) = (√z)3
Find the integrals in Problems. Check your answers by differentiation. y y² + 4 / dy
Consider the improper integral(a) Use a calculator or computer to find ∫b1 1∕(√x) dx for b = 100, 1000, 10,000. What do you notice?(b) Find ∫b1 1∕(√x) dx using the Fundamental Theorem of
The rate, r, at which people get sick during an epidemic of the flu can be approximated byr = 1000te−0.5t,where r is measured in people/day and t is measured in days since the start of the
Problems concern the graph of f' in Figure 6.21.List the following in increasing order: y 2 3 X y = f'(x) Figure 6.21: Note: Graph of f', not f
For Problems show the following quantities on Figure 6.22.A length representing f(b) − f(a). a f(x) b Figure 6.22 X
Find an antiderivative.g(t) = e−3t
An island has a carrying capacity of 1 million rabbits. (That is, no more than 1 million rabbits can be supported by the island.) The rabbit population is two at time t = 1 day and grows at a rate of
For Problems show the following quantities on Figure 6.22.A slope representing f(b) − f(a) /b − a. a f(x) b Figure 6.22 X
Find an antiderivative.ℎ(t) = cos t
Find the integrals in Problems. Check your answers by differentiation. 9 5q2 +8 -dq
Decide which function is an antiderivative of the other. f(x) = 1 Vx =;g(x) = 2√√x 8
Find the integrals in Problems. Check your answers by differentiation. et +1 et + t dt
Find an antiderivative.g(t) = 5 + cos t
For Problems show the following quantities on Figure 6.22.A length roughly approximating a f(x) b Figure 6.22 X
For Problems show the following quantities on Figure 6.22.An area representing F(b) − F(a), where F' = f. a f(x) b Figure 6.22 X
Find the integrals in Problems. Check your answers by differentiation. cos √√x - dx V x
Find the integrals in Problems. Check your answers by differentiation. evi dy
Find an antiderivative.g(θ) = sin θ − 2 cosθ
Decide which function is an antiderivative of the other. ƒ(x) = 1− —; g(x)=
Find the integrals in Problems. Check your answers by differentiation. e² et + 1 -dt
Decide which function is an antiderivative of the other. f(x) = ³ 3x. 2 3 *; g(x) = 2e³x
Find the integrals in Problems. Check your answers by differentiation. x + 1 x² + 2x + 19 -dx
Decide which function is an antiderivative of the other.f(x) = −sin x − cos x; g(x) = cos x − sin x
Find the integrals in Problems. Check your answers by differentiation. I sin sinº (50) cos(50) de
Find the integrals in Problems. Check your answers by differentiation. x cos(x²) sin(x²) dx
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 0. Is there only one possible solution?f(x) = 3
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 0. Is there only one possible solution?f(x) = 2 + 4x + 5x2
Suppose ∫20 g(t) dt = 5. Calculate the following:a.b. 4 g(t/2) dt
Use substitution to express each of the following integrals as a multiple of ∫ba (1∕w) dw for some a and b. Then evaluate the integrals.a.b. x 1+x² dx
(a) Using the density function in Example 2 on page 324, fill in values for the cumulative distribution function P(t) for the length of time people wait in the doctor’s office.(b) Graph P(t). t
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 0. Is there only one possible solution?f(x) = 1/4x
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 0. Is there only one possible solution?f(x) = √x
A point A is shown on a contour diagram of a function f(x, y).(a) Evaluate f(A).(b) Is fx(A) positive, negative, or zero?(c) Is fy(A) positive, negative, or zero? x A 7- 10 13- - 16- y
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 5.f(x) = x2 + 1
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 5.f(x) = 6x − 5
Use the contour diagram for the function z = f(x, y) in Figure 8.23.Approximate the coordinates of a point (x, y) with f(x, y) = 0.5. 3 2 A 1 y 0 0 0.1 0.25 0.5 1 B 1 2 Figure 8.23 3 برا X
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 5.f(x) = 8 sin(2x)
A point A is shown on a contour diagram of a function f(x, y).(a) Evaluate f(A).(b) Is fx(A) positive, negative, or zero?(c) Is fy(A) positive, negative, or zero? y A 5- 10 -15- 20- X
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 5.f(x) = 6e3x
In drilling an oil well, the total cost, C, consists of fixed costs (independent of the depth of the well) and marginal costs, which depend on depth; drilling becomes more expensive, per meter,
Figure 8.56 shows contours of f(x, y). List the x- and y-coordinates and the value of the function at any local maximum and local minimum points, and identify which is which. Are any of these local
Let p(x) be the density function for annual family income, where x is in thousands of dollars. What is the meaning of the statement p(70) = 0.05?
Use the contour diagram for the function z = f(x, y) in Figure 8.23.Find the value of f(A). 3 2 A 1 y 0 0 0.1 0.25 0.5 1 B 1 2 Figure 8.23 3 برا X
Concern the cost, C, of renting a car from a company which charges $40 a day and 15 cents a mile, so C = f(d, m) = 40d + 0.15m, where d is the number of days, and m is the number of miles.Make a
Use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint.f(x, y) = x + y, x2 + y2 = 1
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined.fx and fy if f(x, y) = x2 + 2xy + y3
Figure 8.57 shows contours of f(x, y). List x- and y- coordinates and the value of the function at any local maximum and local minimum points, and identify which is which. Are any of these local
A point A is shown on a contour diagram of a function f(x, y).(a) Evaluate f(A).(b) Is fx(A) positive, negative, or zero?(c) Is fy(A) positive, negative, or zero? y 54 58
Estimate the position and approximate value of the global maxima and minima on the region shown, including its boundary. 6 5 ال - بیا تا 4 3 2 3 1 2 3 4 5 6 7 *
Use the contour diagram for the function z = f(x, y) in Figure 8.23.Is f(B) greater than, equal to, or less than f(A)? 3 2 A 1 y 0 0 0.1 0.25 0.5 1 B 1 2 Figure 8.23 3 برا X
Concern the cost, C, of renting a car from a company which charges $40 a day and 15 cents a mile, so C = f(d, m) = 40d + 0.15m, where d is the number of days, and m is the number of miles.(a) Find
Use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint.f(x, y) = x2 + 4xy, x + y = 100
Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined.fx and fy if f(x, y) = 2x2 + 3y2

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