if one of these 1 million people is selected randomly, find the following probabilities: (a) p(b1), (b) p(a1), (c) p(a1 | b2), and (d) p(b1 | a1). (e) in words, what do parts (c) and (d) say?

The lateral surface area of the box is given as follows:

112.5 in².

The area of a rectangle is given by the multiplication of the width and the length of the triangle, as follows:

A = lw.

The lateral of the triangular box for this problem is composed by two rectangles, with dimensions given as follows:

Hence the lateral surface area of the box is obtained as follows:

5 x 10 + 5 x 12.5 = 112.5 in².

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Answer:208.75

Step-by-step explanation:

Answer: 2

4 – (6-4) = 2

Answer: 2- (y= 6 , z=2)

Step-by-step explanation:

If z = 4 and y = 6, then we can substitute these values into the expression:

z - (y - z) = 4 - (6 - 4)

Expanding the subtraction in the parentheses, we get:

z - (y - z) = 4 - (6 - 4) = 4 - (2) = 4 - 2

Finally, solving for the subtraction:

z - (y - z) = 4 - 2 = 2

So, the value of the expression z - (y - z) when z = 4 and y = 6 is 2.

The sample mean of the given frequency distribution of heart rates is 77.59 and the sample variance is 27.543.

To calculate the sample mean and sample variance of the given frequency distribution of heart rates, we must first calculate the mid-points of each class (i.e. 63.5, 69.5, 76.5, 82.5, 87.5). Then, we must calculate the mid-point multiplied by the frequency of each class [tex](4x63.5, 8x69.5, 5x76.5, 7x82.5, 13x87.5)[/tex]. Finally, we must add up all of these values together and divide by the total frequency. This gives us the sample mean.

To calculate the sample variance, we must first calculate the squared differences of each midpoint multiplied by the frequency from the sample mean [tex](4x63.5, 8x69.5, 5x76.5, 7x82.5, 13x87.5)[/tex]. Then, we must add up all of these values together and divide by the total frequency minus one. This gives us the sample variance.

Therefore, the sample mean of the given frequency distribution of heart rates is 77.59 and the sample variance is 27.543.

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A mathematical statement known as an equation is created by joining two expressions with an equal sign. For instance, 3x - 5 = 16 is an equation. This equation may be solved, and the result shows that the value of the variable x is 7.

Given,

(6.82x - 2.53y - 5.72) - (-5.72 + 9.45y - 8.11 x)

Simplifying the above equation,

= 6.82x - 2.53y - 5.72 - (-5.72 + 9.45y - 8.11 x)

= 6.82x - 2.53y - 5.72 + 5.72 - 9.45y + 8.11 x

Simplify

6.82x - 2.53y - 5.72 + 5.72 - 9.45y + 8.11 x : 14.93x - 11.98y

Then we get,

= 14.93x - 11.98y

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The z-scores for the given data values  520, 640, 500, 470, and 320 are:

0.2, 1.4. 0, -0.3, -1.8

To calculate the z-scores for each data value, we can use the formula:

z = (x - μ) / σ

where:

x is the data value

μ is the population mean (in this case, 500)

σ is the population standard deviation (in this case, 100)

Let's calculate the z-scores for each data value:

For x = 520:

z = (520 - 500) / 100 = 0.2

For x = 640:

z = (640 - 500) / 100 = 1.4

For x = 500:

z = (500 - 500) / 100 = 0

For x = 470:

z = (470 - 500) / 100 = -0.3

For x = 320:

z = (320 - 500) / 100 = -1.8

Therefore, the z-scores for the given data values are:

z-score for 520: 0.2

z-score for 640: 1.4

z-score for 500: 0

z-score for 470: -0.3

z-score for 320: -1.8

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Answer:

e

Step-by-step explanation:

e

For example, 0.123 would be a legitimate entry. Lower limit (first blank) = 0.285, Upper limit (second blank) = 0.562.

The confidence interval for proportions is

p±z *√p(1-p)/n.

In this case, p = 36/85 = .4235, and z* = 2.575 (the value of the standard normal distribution corresponding to 99%).

so,

0.4235 ± 2.576 * √0.4235*0.5765/85

0.4235 ± 0.13804

= (0.285, 0.562)

Therefore, Lower limit is 0.285 and Upper limit 0562.

A confidence interval is the mean of your estimate plus and minus the variation in that estimate. This is the range of values you anticipate your estimate to fall between if you redo your test, within a certain position of confidence.

Confidence, in statistics, is another way to describe probability. For illustration, if you construct a confidence interval with a 95 confidence position, you're confident that 95 out of 100 times the estimate will fall between the upper and lower values specified by the confidence interval.

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The limit of 10 pounds per shipment of medium-sized box and the 1 pound weight per jeans and 2 pound weight per pair of shoes, indicates that the graph and ordered pair of inequality are as follows;

13. Please find attached the required graph of the inequality, created with MS Excel

14. 5 pairs of jeans and 4 pairs of shoes weigh more than 10 pounds and can not be packaged in the same medium box.

An inequality is relationship that compares expressions with different values.

13. The inequality in the question, x + 2·y ≤ 10, indicates that we get;

x + 2·y ≤ 10

2·y ≤ 10 - x

y ≤ (10 - x)/2 = 5 - x/2

y ≤ 5 - x/2

The graph of the inequality, x + 2·y ≤ 10 can be created using the above inequality. Please find attached the required graph created using with the inequality expressed as an equation, followed by shading the feasible region, using MS Excel.

14. The point on the graph that corresponds to 5 pairs of jeans and 4 pairs of shoes, (5, 4), is not in the feasible region. Based on the point (5, 2.5) on the graph, when 5 pairs of jeans are packed in the box, only 2 pairs of shoes can be included. Therefore;

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The area of the triangle ΔABC is given by A = √275 units²

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the area of the triangle ΔABC be A

Now , the measure of side BD = 5

The measure of side ED = 11

So , the measure of side EB = 11 - 5 = 6

The measure of side AD = 10

And , the measure of side CE = 4

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

From the triangle ΔABD ,

The measure of side AB = √ ( AD )² - ( BD )²

The measure of side AB = √ ( 100 - 25 )

The measure of side AB = √75 units

And , from the triangle ΔBCE

The measure of side CB = √ ( EB )² - ( CE )²

The measure of side CB = √ ( 36 - 16 )

The measure of side CB = √20 units

And , from the triangle ΔABC

The measure of side AC = √ ( AB )² - ( CB )²

The measure of side AC = √ ( 75 - 20 )

The measure of side AC = √55 units

And , the area of triangle ΔABC = ( 1/2 ) x AC x BC

The area of triangle ΔABC = ( 1/2 ) x √55 x √20

The area of triangle ΔABC = ( 1/2 ) x √1100

The area of triangle ΔABC , A = ( 1/2 ) x 2√275

The area of triangle ΔABC , A = √275 units²

Hence , the area of triangle ΔABC is √275 units²

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The sample variance and standard deviation are given as follows:

To obtain the variance and the standard deviation, first we must obtain the mean, which is given by the sum of all observations divided by the number of observations, hence:

Mean = (77.1 + 82.9 + 90 + 91.3 + 81.9 + 83.3 + 84.9 + 82.5 + 84.6)/10

Mean = 75.85.

Then we must obtain the sum of the differences squared between each observation and the mean, as follows:

(77.1 - 75.85)² + (82.9 - 75.85)² + (90 - 75.85)² + (91.3 - 75.85)² + (81.9 - 75.85)² + (83.3 - 75.85)² + (84.9 - 75.85)² + (82.5 - 75.85)² + (84.6 - 75.85)² = 784.9825

The sample variance is given by the above sum divided by one less than the sample size, hence:

Variance = 784.9825/9

Variance = 87.22 %².

The standard deviation is the square root of the variance, hence:

Standard deviation = sqrt(87.22)

Standard deviation = 9.34%.

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The outer unit normal vector at any point (x, y, z) on the surface of the ellipsoid is given by:

n = <8x, 18y, 72z> / sqrt((8x)² + (18y)² + (72z)²)

In mathematics, an expression is a combination of one or more values, variables, operators, and/or functions that are evaluated to produce a single result.

For example, 3 + 4 is an expression that evaluates to 7, and (x + 2) / (y - 1) is an expression that can be evaluated for different values of x and y.

Expressions can be simple or complex, and can involve arithmetic operations (addition, subtraction, multiplication, division), algebraic operations (such as factoring or expanding), trigonometric functions, logarithmic functions, and many other mathematical operations.

Expressions are often used in mathematical modeling and problem solving, and can be written in a variety of formats, including symbolic notation, numerical values, or verbal descriptions.

The equation you provided, 4x² + 9y² + 36z² = 36, represents an ellipsoid centered at the origin (0,0,0) with semi-axes of length 3, 2, and 1 in the x, y, and z directions, respectively.

The outer unit normal vector, n, at a given point on the surface of the ellipsoid is perpendicular to the tangent plane at that point. To find the normal vector at a given point (x, y, z) on the surface, we can take the gradient of the function 4x² + 9y² + 36z², which gives us:

∇(4x² + 9y² + 36z²) = <8x, 18y, 72z>

At any point on the surface of the ellipsoid, this gradient vector is proportional to the normal vector, and the constant of proportionality is the reciprocal of the length of the gradient vector:

n = k<8x, 18y, 72z>

To find the value of k, we need to normalize n so that its length is 1. That is,

|n| = sqrt((8x)² + (18y)² + (72z)²) = 1

Solving for k, we get:

k = 1 / sqrt((8x)² + (18y)² + (72z)²)

Thus, the outer unit normal vector at any point (x, y, z) on the surface of the ellipsoid is given by:

n = <8x, 18y, 72z> / sqrt((8x)² + (18y)² + (72z)²)

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Answer:

Step-by-step explanation:

12x2  +3-4-10

The number of students who went to Camp Redwood was 23% more than the number of students who went to Camp Arrowhead.

What is percentage ?

A percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is "%". For example, 50% means 50 out of 100 or 0.5 as a decimal.

Percentages are commonly used to represent proportions, rates, and changes in quantities. They can be used to compare quantities of different sizes, or to express how much one quantity has changed relative to another quantity.

Given by the question:
Let's assume that there are a total of 100 students who were surveyed.

If 15% of the students who planned to go to Camp Redwood did not go, then 85% of them did go:

0.85 x (number of students who planned to go to Camp Redwood) = number of students who actually went to Camp Redwood

Similarly, if 20% of the students who planned to go to Camp Arrowhead did not go, then 80% of them did go:

0.80 x (number of students who planned to go to Camp Arrowhead) = number of students who actually went to Camp Arrowhead

Let's say that x students planned to go to Camp Redwood and y students planned to go to Camp Arrowhead. Then we can write two equations based on the above percentages:

0.85x = number of students who actually went to Camp Redwood

0.80y = number of students who actually went to Camp Arrowhead

We want to find the difference between the number of students who went to Camp Redwood and the number of students who went to Camp Arrowhead, so we can subtract the second equation from the first:

0.85x - 0.80y = (number of students who actually went to Camp Redwood) - (number of students who actually went to Camp Arrowhead)

We can substitute the values from the percentages into the equation:

0.85x - 0.80y = 0.85(0.85x) - 0.80(0.80y)

0.85x - 0.80y = 0.7225x - 0.64y

0.13x = 0.16y

We can solve for one of the variables in terms of the other:

x = (0.16/0.13)y

x = 1.23y

So if y students planned to go to Camp Arrowhead, then 1.23y students planned to go to Camp Redwood. The difference between the number of students who went to Camp Redwood and the number of students who went to Camp Arrowhead is:

1.23y - y = 0.23y

So the number of students who went to Camp Redwood was 23% more than the number of students who went to Camp Arrowhead.

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The equation for the polynomial is P(x) = -1/64(x + 4)(x + 1)(x - 2)(x - 4)^2

From the graph, we have the following parameters that can be used in our computation:

A polynomial is represented as

P(x) = (x - zero)^multiplicity

Using the above as a guide, we have the following:

P(x) = a(x + 4)(x + 1)(x - 2)(x - 4)^2

The y-intercept of the graph is (0, 2)

So, we have

a(0 + 4)(0 + 1)^3(0 - 2)(0 - 4)^2 = 2

Evaluate

-128a = 2

So, we have

a = -1/64

The equation becomes

P(x) = -1/64(x + 4)(x + 1)(x - 2)(x - 4)^2

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The polynomial degree of 12x^4 - 8x + 4x^2 - 3 is option C. 4 .

The highest or greatest power of a variable in a polynomial equation is referred to as the degree of the polynomial. The degree denotes the polynomial's highest exponential power (ignoring the coefficients).

The highest degree of a polynomial's monomials with non-zero coefficients is referred to as the polynomial's degree in mathematics. A term's degree is a non-negative integer and is calculated by adding the exponents of all the variables that occur in it.

To know the degree we will arrange as

12x^4 - 8x + 4x^2 - 3

12x^4+ 4x^2 - 8x  - 3

Then we can see that the highest degree is 4.

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The value of g(x) = [x] At, 2 ≤ x < 3, g(x) = 2, It is also known as the greatest integer function.

A function on real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals.

Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

A step function f(x) = [x] is called a step function where the output is the greatest integer less than or equal to the input.

The function is g(x) = [x].

At, - 1 ≤ x < 0, g(x) = 0.

At, 1 ≤ x < 2, g(x) = 1.

At, 2 ≤ x < 3, g(x) = 2.

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Work Shown:

ST + TU = SU .... segment addition postulate

(4x-4) + (4x-10) = 5x+1

8x-14 = 5x+1

8x-5x = 1+14

3x = 15

x = 15/3

x = 5

Then use this x value to determine the segment lengths.

As a check: ST+TU = 16+10 = 26 which matches with SU = 26. This confirms ST+TU = SU is the case and it confirms the final answer.

The distance between the cities of Palo Alto and Beijing is approximately equal to 8990.334 kilometers.

In this problem we must determine the distance between the cities of Palo Alto and Beijing, whose astronomical coordinates are given in spherical coordinates. The distance is the sum of distance along the terrestrial arc, whose formula can be derived from circular arc formula:

s = (Δθ / 180°) · 2π · R + (Δλ / 180°) · 2π · R

Where:

Now we proceed to determined the distance between the two cities:

s = [(39.914° - 37.429°) / 180°] · 2π · (6367.5 km) + [[116.392° - (- 122.138°)] / 180°] · (6367.5 km)

s ≈ 8990.334 km

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The correct representation on the number line is given as follows:

Option C.

A dolphin was swimming at sea level, hence the position of the dolphin is given as follows:

Position zero.

A ship anchor was sitting on the ocean floor 8 meters below the sea level, hence the position of the ship anchor is given as follows:

Position -8.

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The final measures of forecast accuracy are: CSE = -46, MAD = 18.4, MSE = 462.8, MAPE = 12.44

The forecast errors can be calculated as follows:

Period 1: 751 - 751 = 0

Period 2: 741 - 751 = -10

Period 3: 728 - 749 = -21

Period 4: 773 - 745 = 28

Period 5: 718 - 751 = -33

The cumulative sum of the forecast errors (CSE) is: 0 - 10 - 21 + 28 - 33 = -46

The mean absolute deviation (MAD) can be calculated as follows:

MAD = (abs(0) + abs(-10) + abs(-21) + abs(28) + abs(-33)) / 5

MAD = (0 + 10 + 21 + 28 + 33) / 5

MAD = 92 / 5 = 18.4

The mean squared error (MSE) can be calculated as follows:

MSE = (0^2 + (-10)^2 + ( -21)^2 + 28^2 + (-33)^2) / 5

MSE= ( 0+ 100+ 441 + 784 + 1089)/5

MSE = 2314/ 5 = 462.8

The mean absolute percent error (MAPE) can be calculated as:

MAPE = (abs(0/751) + abs(-10/741) + abs(-21/728) + abs(28/773) + abs(-33/718)) / 5 * 100

MAPE = (0 + 1.34 + 2.89 + 3.63 + 4.58) / 5 * 100

MAPE = 12.44

The final measures of forecast accuracy are:

CSE = -46

MAD = 18.4

MSE = 462.8

MAPE = 12.44

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_______The given question is incorrect, the correct question is given below:

The realized demand values A= 751, 741, 728, 773, 718

and forecast values F = 751,  751, 749, 745, 751

for periods = 1,2,3,4,5

are provided in the table below. Round only the final measure of forecast accuracy to two decimals. The cumulative sum of the forecast errors (CSE) is (Round to two decimals) The mean absolute deviation value (MAD) is (Round to two decimals) The mean squared error value (MSE) is (Round to two decimals) The mean absolute percent error value (MAPE) is (Round to two decimals.)

The value of the d(g o f)/dt at t = 2 is 14.

We can start by computing the composition (g o f)(t) and then finding its derivative with respect to t using the chain rule.

(g o f)(t) = g(f(t)) = g(t, t^2 - 4e^(t-2))

At t = 2, we have f(2) = 2 and f'(2) = 2t - 4e^(t-2) evaluated at t = 2 gives f'(2) = 0.

To find d(g o f)/dt at t = 2, we first need to evaluate (g o f)(2):

(g o f)(2) = g(f(2)) = g(2, 0) = g(a)

Next, we use the chain rule to find the derivative of (g o f) with respect to t:

[tex](d/dt) (g o f)(t) = (∂g/∂x)(a) (df/dt) + (∂g/∂y)(a) (d/dt) (t^2 - 4e^{(t-2)}) + (∂g/∂z)(a) (dg/dz)(a)[/tex]

Since f(t) = t, df/dt = 1. Also, since g(x,y,z) is differentiable, we can write dg/dz(a) as ∂g/∂z(a) = 2.

Substituting the values we have and evaluating at t = 2, we get:

[tex](d/dt) (g o f)(2) = (4)(1) + (2)(2t - 4e^{(t-2)})(t=2) + (2)(2)[/tex]

= 4 + 2(4-4e^0) + 4

= 14

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Answer:

To find the volume of a rectangular prism, you need to multiply its length, width, and height.

Assuming that the dimensions are in inches, we have:

Length = 16 inches + 1/4 inch = 16.25 inches

Width = 1/2 inch

Height = some unknown value (not given in the question)

So, if we know the height, we can calculate the volume of the rectangular prism. If the height is, for example, 10 inches, then:

Volume = Length x Width x Height

Volume = 16.25 inches x 0.5 inches x 10 inches

Volume = 81.25 cubic inches

Therefore, the volume of the rectangular prism depends on its height, which is not given in the question.

Answer: X= -8

Step-by-step explanation: 4 minus 6 equals to -2. So for 4x + 6 to equal -2 you have to do 4x times -8 to get -8 then add it by 6 to get -2.

4 times 8 = -8 + 6 = -2

If the student is selected randomly, the probabilities for the following is given as:

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution. Knowing the total number of outcomes is necessary before we can calculate the likelihood that a specific event will occur.

P (A1∩B1) = 5/43 = 0.1163

P(A1∪B1) = 14/43 + 22/43 - 5/43 = 31/43 = 0.7209

P(A1|B1) = 5/22 = 0.2273

P(B2'∪A1') = (1-21/43) + (1-14/43) - 26/43

= 25/43 = 0.2814

P(A2∩B1) = 17/43 = 0.3953

P(A2∩B2)= 12/43 = 0.2791

P(A2∩B1) is more i.e. student with left thumb on top is more .Hence, student selects left thumb on top.

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Complete question:

Let A1 and A2 be the events that a person is left-eye dominant or right-eye dominant, respectively. When a person folds his/her hands, let B, and B2 be the vents that the left thumb and right thumb, respectively, are on top. A survey in one statistics class yielded the table below: B B2 Totals A 5 9 14 17 12 29 Totals 21 If a student is selected randomly, find the following probabilities: a) P(A, NB) b) P(AUB) c) P(A1B1) d) P(B' UA) e) If the students had their hands folded and you hoped to select a right-eye-dominant student, would you select a "right thumb on top” or a “left thumb on top” student? Why?

Field Trip Location Number of Students Percent

Dairy farm 27 17%

Robotics center 65 41%

Lion tamer at the circus 70 44%

Wildlife recovery center 58 37%

To calculate the percentage, we divide the number of students who preferred each location by the total number of students and then multiply by 100. We round to the nearest whole percent.

For the dairy farm: (27/160) x 100 ≈ 16.875 ≈ 17%

For the robotics center: (65/160) x 100 ≈ 40.625 ≈ 41%

For the lion tamer at the circus: (70/160) x 100 ≈ 43.75 ≈ 44%

For the wildlife recovery center: (58/160) x 100 ≈ 36.25 ≈ 37%

The option (b) set y equal to a constant other than zero to get the curve of points where the slope is that constant.

A slope field is a visual representation of the slopes of a function's derivative at different points on the xy-plane. The slope at a particular point in the xy-plane is given by the derivative of the function at that point.

To match the given equations with their corresponding slope fields, you need to analyze the behavior of the slopes and isoclines. An isocline is a curve in the plane along which the slope of a function is constant. Isoclines are useful in constructing slope fields by hand, as they help in determining the slope at different points.

To start with, set y equal to zero and observe how the derivative behaves along the x-axis. Similarly, set x equal to zero and observe the behavior of the derivative along the y-axis. These observations will give you an idea of the direction and magnitude of the slope at different points in the xy-plane.

Next, consider the curve in the plane defined by setting y' = 0. This curve represents the points in the picture where the slope is zero. These points are known as critical points or equilibrium points. They are essential in analyzing the stability and behavior of a system.

These curves are called isoclines and can be used to construct the slope field picture by hand.

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Complete Question:

Match the following equations with their slope field. Clicking on each picture will give you an enlarged view. While you can probably sove this problem by guessing, it is useful to try to predict characteristics of the slope field and then match them to the picture Here are some handy characteristics to start with- you will develop more as you practice. Set y equal to zero and look at how the derivatiwe behaves along the x axis. Do the same for the y axis by setting x equal to 0. Consider the curve in the plane defined by setting y' = 0-this should correspond to the points in the picture where the slope is zero. Setting y equal to a constant other than zero gives the curve of points where the slepe is that constant. These are called isoclines, and can be used to construct the slope field picture by hand.