The directional derivative duf at the point (3, 4) for the function f(x, y) = x² + y², with u being the unit vector in the same direction as (1, -1), is -sqrt(2).
to evaluate the directional derivative, denoted as duf, of the function f(x, y) = x² + y² at the point (3, 4), where u is the unit vector in the same direction as (1, -1), we need to find the dot product between the gradient of f at the given point and the unit vector u.
let's calculate it step by step:
step 1: find the gradient of f(x, y).
the gradient of f(x, y) is given by the partial derivatives of f with respect to x and y. let's calculate them:
∂f/∂x = 2x
∂f/∂y = 2yso, the gradient of f(x, y) is ∇f(x, y) = (2x, 2y).
step 2: normalize the vector (1, -1) to obtain the unit vector u.
to normalize the vector (1, -1), we divide it by its magnitude:
u = sqrt(1² + (-1)²) = sqrt(1 + 1) = sqrt(2)
u = (1/sqrt(2), -1/sqrt(2)) = (sqrt(2)/2, -sqrt(2)/2)
step 3: evaluate duf at the point (3, 4).
to find the directional derivative, we take the dot product of the gradient ∇f(3, 4) = (6, 8) and the unit vector u = (sqrt(2)/2, -sqrt(2)/2):
duf = ∇f(3, 4) · u = (6, 8) · (sqrt(2)/2, -sqrt(2)/2)
= (6 * sqrt(2)/2) + (8 * -sqrt(2)/2)
= 3sqrt(2) - 4sqrt(2)
= -sqrt(2)
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please solve
1 3. If r(t)= (1.-1 ) find the curvature of 7(1) at * = .
To find the
curvature
of a curve at a given point, we can use the formula for curvature: K = |dT/ds| / |ds/dt|, where T is the unit
tangent vector
, s is the arc length parameter, and t is the parameter of the curve.
To find the curvature, we first need to compute the unit tangent vector T. The unit tangent vector T is given by T = dr/ds, where dr/ds is the derivative of the
vector function
r(t) with respect to the arc length parameter s. Since we are not given the arc
length
parameter, we need to find it first.
To find the arc length parameter s, we integrate the
magnitude
of the
derivative
of r(t) with respect to t. In this case, r(t) = (1, -1), so dr/dt = (0, 0), and the magnitude of dr/dt is 0. Therefore, the arc length parameter is simply s = t.
Now that we have the arc length parameter s, we can find the unit tangent vector T = dr/ds. Since dr/ds = dr/dt = (1, -1), the unit tangent vector T is (1, -1)/sqrt(2).
Next, we need to find ds/dt. Since s = t, ds/dt = 1.
Finally, we can calculate the curvature K using the formula K = |dT/ds| / |ds/dt|. In this case, dT/ds = 0, and |ds/dt| = 1. Therefore, the curvature at t = 1 is K = |dT/ds| / |ds/dt| = 0/1 = 0.
Hence, the curvature of the
curve
at t = 1 is 0.
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the last three blanks are
,
lim n goes to infinty A,n (equal
or not equal)
0 and n+1 ( <
,>,<=,>=,= , not = , n/a)
for all n the series ( converges
, divergers, inconclusive)
"The limit as n approaches infinity of A,n is equal to 0, and n+1 is greater than or equal to 0 for all n. The series converges."
As n approaches infinity, the value of A,n approaches 0. Additionally, the value of n+1 is always greater than or equal to 0 for all n. Therefore, the series formed by the terms A,n converges, indicating that its sum exists and is finite.
Sure! Let's break down the explanation into three parts:
1. Limit of A,n: The statement "lim n goes to infinity A,n = 0" means that as n gets larger and larger, the values of A,n approach 0. In other words, the terms in the sequence A,n gradually become closer to 0 as n increases indefinitely.
2. Relationship between n+1 and 0: The statement "n+1 >= 0" indicates that the expression n+1 is greater than or equal to 0 for all values of n. This means that every term in the sequence n+1 is either greater than or equal to 0.
3. Convergence of the series: Based on the previous two statements, we can conclude that the series formed by adding up all the terms of A,n converges. The series converges because the individual terms approach 0, and the terms themselves are always non-negative (greater than or equal to 0). This implies that the sum of all the terms in the series exists and is finite.
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Find the indefinite integral. (Remember to use absolute values where appropriate. Use for the constant of integration) | Cacax mtan(2x)+ c
The indefinite integral of |cosec(x) tan(2x)| dx is |cosec(x)| + C.
To find the indefinite integral of |cosec(x) tan(2x)| dx, we can split the absolute value into two cases based on the sign of cosec(x).Case 1: If cosec(x) > 0, then the integral becomes ∫(cosec(x) tan(2x)) dx. By using the substitution u = cos(x), du = -sin(x) dx, we can rewrite the integral as ∫(-du/tan(2x)). The integral of -du/tan(2x) can be evaluated using the substitution v = 2x, dv = 2dx. Substituting these values, we get -∫(du/tan(v)) = -ln|sec(v)| + C = -ln|sec(2x)| + C.Case 2: If cosec(x) < 0, then the integral becomes ∫(-cosec(x) tan(2x)) dx.
By using the substitution u = -cos(x), du = sin(x) dx, we can rewrite the integral as ∫(du/tan(2x)). Using the same substitution v = 2x, dv = 2dx, we get ∫(du/tan(v)) = ln|sec(v)| + C = ln|sec(2x)| + C.Combining the results from both cases, the indefinite integral of |cosec(x) tan(2x)| dx is |cosec(x)| + C, where C is the constant of integration.
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Let AB be the line segment beginning at point A(2, 1) and ending at point B(-11, -13). Find the point P on the line segment that is of the distance from A to B.
The point P on the line segment AB that is equidistant from A and B is approximately (-287/30, 571/210).
To find the point P on the line segment AB that is of the same distance from point A as it is from point B, we can use the concept of midpoint.
Point A(2, 1)
Point B(-11, -13)
To find the midpoint of the line segment AB, we can use the formula:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Let's substitute the coordinates of A and B into the formula to find the midpoint:
Midpoint = ((2 + (-11)) / 2, (1 + (-13)) / 2)
Midpoint = (-9/2, -12/2)
Midpoint = (-9/2, -6)
Now, we want to find the point P on the line segment AB that is of the same distance from point A as it is from point B.
Since P is equidistant from both A and B, it will lie on the perpendicular bisector of AB, passing through the midpoint.
To find the equation of the perpendicular bisector, we need the slope of AB.
The slope of AB can be calculated using the formula:
Slope = (y₂ - y₁) / (x₂ - x₁)
Slope of AB = (-13 - 1) / (-11 - 2)
Slope of AB = -14 / -13
Slope of AB = 14/13 (or approximately 1.08)
The slope of the perpendicular bisector will be the negative reciprocal of the slope of AB:
Slope of perpendicular bisector = -1 / (14/13)
Slope of perpendicular bisector = -13/14 (or approximately -0.93)
Now, we have the slope of the perpendicular bisector and a point it passes through (the midpoint).
We can use the point-slope form of a line to find the equation of the perpendicular bisector:
y - y₁ = m(x - x₁)
Using the midpoint (-9/2, -6) as (x₁, y₁) and the slope -13/14 as m, we can write the equation of the perpendicular bisector:
y - (-6) = (-13/14)(x - (-9/2))
y + 6 = (-13/14)(x + 9/2)
Simplifying the equation:
14(y + 6) = -13(x + 9/2)
14y + 84 = -13x - 117/2
14y = -13x - 117/2 - 84
14y = -13x - 117/2 - 168/2
14y = -13x - 285/2
Now, we have the equation of the perpendicular bisector.
To find the point P on the line segment AB that is equidistant from A and B, we need to find the intersection of the perpendicular bisector and the line segment AB.
Substituting the x-coordinate of P into the equation, we can solve for y:
-13x - 285/2 = 2x + 1
-15x = 1 + 285/2
-15x = 2/2 + 285/2
-15x = 287/2
x = (287/2)(-1/15)
x = -287/30
Substituting the y-coordinate of P into the equation, we can solve for x:
14y = -13(-287/30) - 285/2
14y = 287/30 + 285/2
14y = (287 + 855)/30
14y = 1142/30
y = (1142/30)(1/14)
y = 571/210
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x2 + 2x = 2x + x2 what property does this demonstrate
The equation x² + 2x = 2x + x² demonstrates the commutative property of addition.
The commutative property of addition states that the order of the terms does not affect the result when adding.
In this case, the terms x² and 2x on the left side of the equation are switched to 2x and x² on the right side of the equation, and the equation still holds true.
This shows that the terms can be rearranged without changing the sum.
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Evaluate the integral 12 2 fa? (2 (23 – 2)"?dat by making the substitution u = : 23 – 2. + C
Therefore, the integral ∫2^(3 – 2x) dx, with the substitution u = 2^(3 – 2x), evaluates to:
(-1 / (2(ln 2))) ln (8) + (1 / ln 2) x + C, where C is the constant of integration.
To evaluate the integral ∫2^(3 – 2x) dx using the substitution u = 2^(3 – 2x), let's proceed with the following steps:
Let u = 2^(3 – 2x)
Differentiate both sides with respect to x to find du/dx:
du/dx = d/dx [2^(3 – 2x)]
To simplify the derivative, we can use the chain rule. The derivative of 2^x is given by (ln 2) * 2^x. Applying the chain rule, we have:
du/dx = d/dx [2^(3 – 2x)] = (ln 2) * 2^(3 – 2x) * (-2) = -2(ln 2) * 2^(3 – 2x)
Now, we can solve for dx in terms of du:
du = -2(ln 2) * 2^(3 – 2x) dx
dx = -du / [2(ln 2) * 2^(3 – 2x)]
Substituting this value of dx and u = 2^(3 – 2x) into the integral, we have:
∫2^(3 – 2x) dx = ∫-du / [2(ln 2) * u]
= -1 / (2(ln 2)) ∫du / u
= (-1 / (2(ln 2))) ln |u| + C
Finally, substituting u = 2^(3 – 2x) back into the expression:
∫2^(3 – 2x) dx = (-1 / (2(ln 2))) ln |2^(3 – 2x)| + C
= (-1 / (2(ln 2))) ln |2^(3) / 2^(2x)| + C
= (-1 / (2(ln 2))) ln |8 / 2^(2x)| + C
= (-1 / (2(ln 2))) ln (8) - (-1 / (2(ln 2))) ln |2^(2x)| + C
= (-1 / (2(ln 2))) ln (8) - (-1 / (2(ln 2))) (2x ln 2) + C
= (-1 / (2(ln 2))) ln (8) + (1 / ln 2) x + C
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From 2000 through 2005, the rate of change in the number of cattle on farms C (in millions) in a certain country can be modeled by the equation shown below, where t is the year, with t = 0 corresponding to 2000. dC = = 0.69 – 0.132t2 + 0.044et dt In 2002, the number of cattle was 96.5 million. (a) Find a model for the number of cattle from 2000 through 2005. C(t) = = (b) Use the model to predict the number of cattle in 2007. (Round your answer to 1 decimal place.) million cattle
a. The model equation for the number of cattle from 2000 through 2005 is C(t) = 0.69t - (0.132/3)t^3 + 0.044e^t + 95.472 - 0.044e^2
b. The predicted number of cattle in 2007 (rounded to 1 decimal place) is 79.9 million cattle.
a. To find a model for the number of cattle from 2000 through 2005, we need to integrate the given rate of change equation.
dC = 0.69 - 0.132t^2 + 0.044e^t dt
Integrating both sides with respect to t:
∫dC = ∫(0.69 - 0.132t^2 + 0.044e^t) dt
C = 0.69t - (0.132/3)t^3 + 0.044e^t + C
Since the number of cattle in 2002 was 96.5 million, we can use this information to find the constant C. Plugging in t = 2 and C = 96.5 into the model equation:
96.5 = 0.692 - (0.132/3)(2^3) + 0.044e^2 + C
96.5 = 1.38 - 0.352 + 0.044e^2 + C
C = 96.5 - 1.38 + 0.352 - 0.044e^2
C = 95.472 - 0.044e^2
Now we have the model equation for the number of cattle from 2000 through 2005:
C(t) = 0.69t - (0.132/3)t^3 + 0.044e^t + 95.472 - 0.044e^2
b. To predict the number of cattle in 2007 (corresponding to t = 7), we can plug t = 7 into the model:
C(7) = 0.697 - (0.132/3)(7^3) + 0.044e^7 + 95.472 - 0.044e^2
C(7) = 4.83 - 20.412 + 0.044e^7 + 95.472 - 0.044e^2
C(7) = 79.89 + 0.044e^7 - 0.044e^2
Therefore, the predicted number of cattle in 2007 (rounded to 1 decimal place) is 79.9 million cattle.
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A certain drug is being administered intravenously to a hospitalpatient. fluid containing 5 mg/cm^3 of the drug enters thepatient's bloodstream at a rate of 100 cm^3/h. The drug isabsorbed by body tissues or otherwise leaves the bloodstream at arate proportional to the amount present, with a rate constant of0.4/hr.
A. assuming that the drug is always uniformly distributedthroughout the blood stream, write a differential equation for theamount of drug that is present in the blood stream at any giventime.
B. How much of the drug is present in the bloodstream after a longtime?
A. The differential equation for the amount of drug present in the bloodstream at any given time can be written as follows: dA/dt = 5 * 100 - 0.4 * A where A represents the amount of drug in the bloodstream at time t.
The first term, 5 * 100, represents the rate at which the drug enters the bloodstream, calculated by multiplying the concentration (5 mg/cm^3) with the rate of fluid entering (100 cm^3/h). The second term, 0.4 * A, represents the rate at which the drug is leaving the bloodstream, which is proportional to the amount of drug present in the bloodstream.
B. To determine the amount of drug present in the bloodstream after a long time, we can solve the differential equation by finding the steady-state solution. In the steady state, the rate of drug entering the bloodstream is equal to the rate of drug leaving the bloodstream.
Setting dA/dt = 0 and solving the equation 5 * 100 - 0.4 * A = 0, we find A = 500 mg. This means that after a long time, the amount of drug present in the bloodstream will reach 500 mg. This represents the equilibrium point where the rate of drug entering the bloodstream matches the rate at which it is leaving the bloodstream, resulting in a constant amount of drug in the bloodstream.
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show full work please
18. A company claims it can extract N gallons of contaminated water per day from a deep well at the rate modeled by N(t)=61¹-720r³ +21600r² where t is the number of days since the extraction begins
The company's extraction rate of contaminated water from a deep well is modeled by the function N(t) = 61¹ - 720r³ + 21600r², where t represents the number of days since the extraction began.
The given function N(t) = 61¹ - 720r³ + 21600r² represents the extraction rate of contaminated water, measured in gallons per day, from the deep well. The variable t represents the number of days since the extraction process started. The function is defined in terms of the variable r.
To understand the behavior of the extraction rate, we need to analyze the properties of the function. The function is a polynomial of degree 3, indicating a cubic function. The coefficient values of 61¹, -720r³, and 21600r² determine the shape of the function.
The first term, 61¹, is a constant representing a base extraction rate that is independent of time or any other variable. The second term, -720r³, is a cubic term that indicates the influence of the variable r on the extraction rate. The third term, 21600r², is a quadratic term that also affects the extraction rate.
The cubic and quadratic terms introduce variability and complexity into the extraction rate function. The values of r determine the specific rate of extraction at any given time. By manipulating the values of r, the company can adjust the extraction rate according to its requirements.
In summary, the company's extraction rate of contaminated water from the deep well is modeled by the function N(t) = 61¹ - 720r³ + 21600r², where t represents the number of days since the extraction began. The function incorporates a cubic term and a quadratic term, allowing the company to control the extraction rate by manipulating the variable r.
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Question 6. Find the area of the portion of the plane 3x + 4y + 2z = 24 that lies in the first octant.
Answer: The limits of integration for x and y in the first octant are:
0 ≤ x ≤ 8
0 ≤ y ≤ 6
Step-by-step explanation:
To find the area of the portion of the plane 3x + 4y + 2z = 24 that lies in the first octant, we need to determine the limits of integration for the coordinates x, y, and z.
The first octant is defined by positive values of x, y, and z. Therefore, we need to find the values of x, y, and z that satisfy the equation 3x + 4y + 2z = 24 in the first octant.
For x, we have:
x ≥ 0
For y, we have:
y ≥ 0
For z, we have:
z ≥ 0
Now, let's solve the equation 3x + 4y + 2z = 24 for z to find the upper limit for z in the first octant:
2z = 24 - 3x - 4y
z = (24 - 3x - 4y)/2
Therefore, the limits of integration for x, y, and z in the first octant are as follows:
0 ≤ x ≤ ?
0 ≤ y ≤ ?
0 ≤ z ≤ (24 - 3x - 4y)/2
To find the upper limits for x and y, we need to determine the points of intersection between the plane and the coordinate axes.
When x = 0, the equation becomes:
4y + 2z = 24
2y + z = 12
y = (12 - z)/2
When y = 0, the equation becomes:
3x + 2z = 24
x = (24 - 2z)/3
To find the upper limits for x and y, we substitute z = 0 into the equations:
For x, we have:
x = (24 - 2(0))/3
x = 8
For y, we have:
y = (12 - 0)/2
y = 6
Therefore, the limits of integration for x and y in the first octant are:
0 ≤ x ≤ 8
0 ≤ y ≤ 6
Now, we can calculate the area using a triple integral:
Area = ∫∫∫ (24 - 3x - 4y)/2 dy dx dz, over the region R in the first octant.
Area = ∫[0,8] ∫[0,6] ∫[0,(24 - 3x - 4y)/2] (24 - 3x - 4y)/2 dz dy dx
Evaluating the triple integral will give us the area of the portion of the plane in the first octant.
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a. Use the product rule to find the derivative of the given function b. Find the derivative by expanding the product first h(z)= (4 -z?) (22 -32+4) a. Use the product rule to find the derivative of th
a)Using the product rule to find the derivative of the function: Simplifying this expression, we get d/dz(h(z)) = -8z³ + 20z² + 24z - 88.
The product rule states that for two functions u(x) and v(x), the derivative of their product is given by d/dx(u(x) * v(x))
= u(x) * dv/dx + v(x) * du/dx.
Let's apply this to the given function: h(z) = (4 - z²)(22 - 32z + 4z²)
Now, let's denote the first function as u(z) = 4 - z² and the second function as v(z) = 22 - 32z + 4z².
So, we have h(z) = u(z) * v(z).
Now, let's apply the product rule, d/dz(u(z) * v(z)) = u(z) * dv/dz + v(z) * du/dz, where du/dz is the derivative of the first function and dv/dz is the derivative of the second function with respect to z.
The derivative of u(z) is given by du/dz = -2z and the derivative of v(z) is given by dv/dz = -32 + 8z.
Putting these values in the product rule formula, we get:
d/dz(h(z)) = (4 - z²) * (-32 + 8z) + (22 - 32z + 4z²) * (-2z).
Simplifying this expression, we get d/dz(h(z)) = -8z³ + 20z² + 24z - 88.
b)Finding the derivative by expanding the product first: We can also find the derivative by expanding the product first and then taking its derivative.
This is done as follows:
h(z) = (4 - z²)(22 - 32z + 4z²)= 88 - 128z + 16z² - 22z² + 32z³ - 4z⁴
Taking the derivative of this expression,
we get d/dz(h(z)) = -8z³ + 20z² + 24z - 88, which is the same result as obtained above using the product rule.
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05. Evaluate Q4. Evaluate For f(x, y, z) = xyʻz + 4x*y, defined for x,y,z20, compute fx. fry and fax: Find all second-order partial derivatives of f(x,y) = x+y – y + Inx
The partial derivatives for f(x, y, z) = xyʻz + 4xy with respect to x, y, and z are fx = yz, fy = xz + 4x, and fz = xy. The second-order partial derivatives of f(x, y) = x + y - y + ln(x) are fx = 0, fxy = 1, fyx = 1, fyy = -1, and fyx = 0.
To find partial derivatives, we take the derivative of the function with respect to each variable while keeping the other variables constant.
To find the partial derivatives of f(x, y, z) = xyʻz + 4xy:
fx = ∂f/∂x = yz
fy = ∂f/∂y = xz + 4x
fz = ∂f/∂z = xy
For f(x, y) = x + y - y + ln(x), the partial derivative with respect to x is f = 1 + 1/x, and the partial derivative with respect to y is f_y = 1.
To find the second-order partial derivatives of f(x, y) = x + y - y + ln(x):
fx = ∂²f/∂x² = 0
fxy = ∂²f/∂x∂y = 1
fyx = ∂²f/∂y∂x = 1
fyy = ∂²f/∂y² = -1
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A plane intersects one nappe of a double-napped cone such that the plane is not perpendicular to the axis and is not parallel to the generating line.
Which conic section is formed?
1. circle
2. hyperbola
3. ellipse
4. parabola
The conic section formed in this case is a hyperbola. So, option 2 is the right choice.
When a plane intersects one nappe of a double-napped cone and is neither perpendicular to the axis nor parallel to the generating line, the conic section formed is a hyperbola.
A hyperbola is characterized by its two separate branches that are symmetrically curved and open. The plane intersects the cone in such a way that the resulting curve is non-circular and has two distinct branches. The branches of the hyperbola curve away from each other and do not form a closed loop like a circle or an ellipse.
In contrast, a circle is formed when the plane intersects the cone perpendicular to the axis, an ellipse is formed when the plane intersects the cone at an angle and is parallel to the generating line, and a parabola is formed when the plane intersects the cone parallel to the axis.
Therefore, the conic section formed in this scenario is a hyperbola.
The right answer is 2. hyperbola
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Let f(x) = x - 8x? -4. a) Find the intervals on which f is increasing or decreasing b) Find the local maximum and minimum values of . c) Find the intervals of concavity and the inflection points. d) Use the information from a-c to make a rough sketch of the graph
There are no local minimum values, inflection points, or intervals of concavity. The graph of f(x) will resemble an inverted parabola opening downwards, with a maximum point at x = 1/16 and a y-value of -4.
To analyze the function f(x) = x - 8x^2 - 4, we will perform the following steps:
a) Find the intervals on which f is increasing or decreasing:
To determine the intervals of increasing and decreasing, we need to analyze the sign of the derivative of f(x).
First, let's find the derivative of f(x):
f'(x) = 1 - 16x
To find the intervals of increasing and decreasing, we set f'(x) = 0 and solve for x:
1 - 16x = 0
16x = 1
x = 1/16
The critical point is x = 1/16.
Now, we analyze the sign of f'(x) in different intervals:
For x < 1/16: Choose x = 0, f'(0) = 1 - 0 = 1 (positive)
For x > 1/16: Choose x = 1, f'(1) = 1 - 16 = -15 (negative)
Therefore, f(x) is increasing on the interval (-∞, 1/16) and decreasing on the interval (1/16, ∞).
b) Find the local maximum and minimum values of f(x):
To find the local maximum and minimum values, we need to analyze the critical points and the endpoints of the given interval.
At the critical point x = 1/16, we can evaluate the function:
f(1/16) = (1/16) - 8(1/16)^2 - 4 = 1/16 - 1/128 - 4 = -4 - 1/128
Since the function is decreasing on the interval (1/16, ∞), the value at x = 1/16 will be a local maximum.
As for the endpoints, we consider f(0) and f(∞):
f(0) = 0 - 8(0)^2 - 4 = -4
As x approaches ∞, f(x) approaches -∞.
Therefore, the local maximum value is -4 at x = 1/16, and there are no local minimum values.
c) Find the intervals of concavity and the inflection points:
To find the intervals of concavity and the inflection points, we need to analyze the second derivative of f(x).
The second derivative of f(x) can be found by differentiating f'(x):
f''(x) = -16
Since the second derivative is a constant (-16), it does not change sign. Thus, there are no inflection points and no intervals of concavity.
d) Sketch the graph:
Based on the information obtained, we can sketch a rough graph of the function f(x):
The function is increasing on the interval (-∞, 1/16) and decreasing on the interval (1/16, ∞).
There is a local maximum at x = 1/16 with a value of -4.
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For a Goodness of Fit Test for a fair dice, does the following
code produce?
(throws2a, p = c(1/6, 1/6, 1/6, 1/6, 1/6, 1/6))
a. the alternative hypothesis
b. the p-value
c. the test statist
The given code does not directly produce the alternative hypothesis, p-value, or test statistic for a Goodness of Fit Test for a fair dice. Additional steps and code are required to perform the test and obtain these values.
To conduct a Goodness of Fit Test for a fair dice, you need to compare the observed frequencies of each outcome (throws2a) with the expected probabilities (p) assuming a fair dice. The code provided only defines the expected probabilities for a fair dice, but it does not include the observed frequencies or perform the actual test.
To obtain the alternative hypothesis, p-value, and test statistic, you would need to use a statistical test specifically designed for Goodness of Fit, such as the chi-squared test. This test compares the observed frequencies with the expected frequencies and calculates a test statistic and p-value.
The code for conducting a chi-squared test would involve additional steps, such as calculating the observed frequencies, creating a contingency table, and using a statistical function or package to perform the test. The output of the test would include the alternative hypothesis, p-value, and test statistic, which can be interpreted to determine if the observed data significantly deviate from the expected probabilities for a fair dice.
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Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C 3y + 7e (x)^1/2 dx + 10x + 7 cos(y2) dy C is the boundary of the region enclosed by the parabolas y = x2 and x = y2
The line integral along the curve C can be evaluated using Green's Theorem, which relates it to a double integral over the region enclosed by the curve.
In this case, the curve C is the boundary of the region enclosed by the parabolas[tex]y = x^2 and x = y^2[/tex]. To evaluate the line integral, we can first find the partial derivatives of the given vector field:
[tex]F = (3y + 7e^(√x)/2) dx + (10x + 7cos(y^2)) dy[/tex]
Taking the partial derivative of the first component with respect to y and the partial derivative of the second component with respect to x, we obtain:
∂F/∂y = 3
[tex]∂F/∂x = 10 + 7cos(y^2)[/tex]
Now, we can calculate the double integral over the region R enclosed by the curve C using these partial derivatives. By applying Green's Theorem, the line integral along C is equal to the double integral over R of the difference of the partial derivatives:
∮C F · dr = ∬R (∂F/∂x - ∂F/∂y) dA
By evaluating this double integral, we can determine the value of the line integral along the given curve.
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Assume the probability of Lukas Podolski scores in a soccer match is 25%.
a) Assuming that Lukas performs independently in different matches, what is the probability that Lukas will score in world cup quarter final match and semifinal match? Use 4 decimal places _______
b) Assume again that Lukas performs independently in different games, what is the probability of Lukas scoring in quarter final OR semi final? Use 4 decimal places _______
(a) The probability that Lukas Podolski will score in both the World Cup quarter-final and semi-final matches is 0.0625 (or 6.25%).
(b) The probability of Lukas Podolski scoring in either the World Cup quarter-final OR the semi-final match is 0.5 (or 50%).
What is Probability?
Probability is a branch of mathematics in which the chances of experiments occurring are calculated.
a) To find the probability that Lukas Podolski will score in both the World Cup quarter-final and semi-final matches, we multiply the probabilities of him scoring in each match since the events are independent.
Probability of scoring in the quarter-final match = 0.25 (or 25%)
Probability of scoring in the semi-final match = 0.25 (or 25%)
Probability of scoring in both matches = 0.25 * 0.25 = 0.0625
Therefore, the probability that Lukas Podolski will score in both the World Cup quarter-final and semi-final matches is 0.0625 (or 6.25%).
b) To find the probability of Lukas Podolski scoring in either the quarter-final OR the semi-final match, we can use the principle of addition. Since the events are mutually exclusive (he can't score in both matches simultaneously), we can simply add the probabilities of scoring in each match.
Probability of scoring in the quarter-final match = 0.25 (or 25%)
Probability of scoring in the semi-final match = 0.25 (or 25%)
Probability of scoring in either match = 0.25 + 0.25 = 0.5
Therefore, the probability of Lukas Podolski scoring in either the World Cup quarter-final OR the semi-final match is 0.5 (or 50%).
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can
you please answer these questions and write all the steps legibly.
Thank you.
Series - Taylor and Maclaurin Series: Problem 10 (1 point) Find the Taylor series, centered at c= 3, for the function 1 f(x) = 1-22 f(α) - ΣΟ The interval of convergence is: Note: You can earn part
The Taylor series for the function f(x) = 1/(1-2x), centered at c = 3 the interval of convergence is (-1/2, 1/2).
Let's find the Taylor series centered at c = 3 for the function f(x) = 1/(1-2x).
To find the Taylor series, we need to compute the derivatives of the function and evaluate them at the center (c = 3).
The general formula for the nth derivative of f(x) is given by:[tex]f^{n}(x) = (n!/(1-2x)^{n+1})[/tex]
where n! denotes the factorial of n.
Step 1: Compute the derivatives of f(x):
f'(x) = ([tex]1!/(1-2x)^{1+1}[/tex])
f''(x) = ([tex]2!/(1-2x)^{2+1}[/tex])
f'''(x) = ([tex]3!/(1-2x)^{3+1}[/tex])
Step 2: Evaluate the derivatives at x = 3:
f'(3) = ([tex]1!/(1-2(3))^{1+1}[/tex])
f''(3) = ([tex]2!/(1-2(3))^{2+1}[/tex])
f'''(3) = ([tex]3!/(1-2(3))^{3+1}[/tex])
Step 3: Simplify the expressions obtained from step 2:
f'(3) = 1/(-11)
f''(3) = 2/(-11)²
f'''(3) = 6/(-11)³
Step 4: Write the Taylor series using the simplified expressions from step 3:
f(x) = f(3) + f'(3)(x-3) + f''(3)(x-3)² + f'''(3)(x-3)³ + ...
Substituting the simplified expressions:
f(x) = 1 + (1/(-11))(x-3) + (2/(-11)²)(x-3)² + (6/(-11)³)(x-3)³ + ...
Step 5: Determine the interval of convergence.
The interval of convergence for a Taylor series can be determined by analyzing the function's convergence properties. In this case, the function f(x) = 1/(1-2x) has a singularity at x = 1/2. Therefore, the interval of convergence for the Taylor series centered at c = 3 will be the interval (-1/2, 1/2), excluding the endpoints.
To summarize, the Taylor series for the function f(x) = 1/(1-2x), centered at c = 3, is given by:
f(x) = 1 + (1/(-11))(x-3) + (2/(-11)²)(x-3)² + (6/(-11)³)(x-3)³ + ...
The interval of convergence is (-1/2, 1/2).
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Graph the function f(t) = 5t(h(t-1) - h(t – 7)) for 0
The graph of the function f(t) = 5t(h(t-1) - h(t – 7)) for 0 < t < 10. Since the slope of the line for 1 ≤ t < 7 is 0.
The function f(t) = 5t(h(t-1) - h(t – 7)) for 0
Graph of the function f(t) = 5t(h(t-1) - h(t – 7)) for 0 < t < 10:
The graph of the function f(t) = 5t(h(t-1) - h(t – 7)) for 0 < t < 10 is given as follows:
First, let us determine the y-intercept of the function f(t).
Since t > 0, we have:h(t - 1) = 1, if t ≥ 1, and h(t - 7) = 0, if t ≥ 7.
This implies:f(t) = 5t (h(t - 1) - h(t - 7)) = 5t [1 - 0] = 5t for t ≥ 1.
This means the graph of f(t) is a straight line that passes through (1, 5).
Now, let us determine the point at which the graph of f(t) changes slope.
Since h(t - 1) changes from 1 to 0 when t = 7, and h(t - 7) changes from 0 to 1 when t = 7, we can split the function into two parts, as follows:
For 0 < t < 1:f(t) = 5t(1 - 0) = 5t.
For 1 ≤ t < 7:
f(t) = 5t(1 - 1) = 0.
For 7 ≤ t < 10:f
(t) = 5t(0 - 1) = -5t + 50.
Since the slope of the line for 1 ≤ t < 7 is 0, the graph of the function changes slope at t = 1 and t = 7.The final graph is shown below:Therefore, this is the graph of the function f(t) = 5t(h(t-1) - h(t – 7)) for 0 < t < 10.
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Evaluate •S 4 cos x sin x dx Select the better substitution: (A) uecos x, (B) u = 4 cos x, or (C) u = sin x. O(A) O(B) (C) With this substitution, the limits of integration are updated directly as f
The better substitution for evaluating the integral of 4 cos x sin x dx is u = 4 cos x (option B). This substitution simplifies the integral and makes the integration process easier.
To evaluate the integral of 4 cos x sin x dx, we can consider the given substitutions and determine which one leads to simpler integration.
Let's evaluate each of the given substitutions and see how they affect the integral.
(A) u = ecos x
Taking the derivative, we have du = -sin x dx. This substitution does not simplify the integral since we still have sin x in the integrand.
(B) u = 4 cos x
Taking the derivative, we have du = -4 sin x dx. This substitution simplifies the integral as it eliminates the sin x term.
(C) u = sin x
Taking the derivative, we have du = cos x dx. This substitution also simplifies the integral as it eliminates the cos x term.
Comparing the substitutions, both (B) and (C) simplify the integral by eliminating one of the trigonometric functions. However, (B) u = 4 cos x leads to a more direct simplification since it eliminates the sin x term directly.
Therefore, the better substitution for evaluating the integral of 4 cos x sin x dx is u = 4 cos x (option B). This substitution simplifies the integral and makes the integration process easier.
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Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. Suppose that a cup of coffee begins at 183 degrees and, after sitting in room temperature of 67 degrees for 17 minutes, the coffee reaches 175 degrees. How long will it take before the coffee reaches 163 degrees? Include at least 2 decimal places in your answer. minutes
The time it takes for the coffee to reach a temperature of 163 degrees from its initial temperature of 183 degrees, given the provided conditions.
To find the time it takes for the coffee to reach 163 degrees, we need to set up an equation using the exponential decay formula derived from Newton's Law of Cooling. The equation is given by T(t) = T_s + (T_0 - T_s) * e^(-kt), where T(t) is the temperature at time t, T_s is the surrounding temperature, T_0 is the initial temperature, k is the proportionality constant, and e is the base of the natural logarithm.
Using the given information, we can substitute the values into the equation. T(t) = 163 degrees, T_s = 67 degrees, T_0 = 183 degrees, and t is the unknown time we want to find. We can rearrange the equation to solve for t: t = -ln((T(t) - T_s)/(T_0 - T_s))/k.
Substituting the values into the equation, we have t = -ln((163 - 67)/(183 - 67))/k. To find k, we can use the information that the coffee reaches 175 degrees after 17 minutes: 175 = 67 + (183 - 67) * e^(-k * 17). Solving this equation will give us the value of k.
With the value of k, we can now substitute it into the equation for t: t = -ln((163 - 67)/(183 - 67))/k. Evaluating this equation will provide the time it takes for the coffee to reach a temperature of 163 degrees from its initial temperature of 183 degrees, given the provided conditions.
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Question
Allison rolls a standard number cube 30 times and records her results. The number of times she rolled a 4 is 6. What is the experimental probability of rolling a 4? What is the experimental probability of not rolling a 4?
P(4) =
p(not 4)=
Answer:
P(4) = 0.2 or 20%.
p(not 4)= 0.8 or 80%
Step-by-step explanation:
To calculate the experimental probability of rolling a 4, we divide the number of times a 4 was rolled (6) by the total number of rolls (30).
Experimental probability of rolling a 4:
P(4) = Number of favorable outcomes / Total number of outcomes
= 6 / 30
= 1 / 5
= 0.2
Therefore, the experimental probability of rolling a 4 is 0.2 or 20%.
To calculate the experimental probability of not rolling a 4, we subtract the probability of rolling a 4 from 1.
Experimental probability of not rolling a 4:
P(not 4) = 1 - P(4)
= 1 - 0.2
= 0.8
Therefore, the experimental probability of not rolling a 4 is 0.8 or 80%.
someone pls complete this. I will give brainliest
The values of the variables are:
1.
a = 14.69
b = 20.22
2.
p = 11.28
q = 4.08
3.
x = 18.25
y = 17
4.
a = 9
b = 16.67
We have,
1.
Sin 36 = a / 25
0.59 = a/25
a = 0.59 x 25
a = 14.69
Cos 36 = b / 25
0.81 = b / 25
b = 0.81 x 25
b = 20.22
2.
Sin 20 = q / 12
0.34 = q / 12
q = 0.34 x 12
q = 4.08
Cos 20 = p / 12
0.94 = p / 12
p = 0.94 x 12
p = 11.28
3.
Sin 43 = y/25
0.68 = y / 25
y = 0.68 x 25
y = 17
Cos 43 = x/25
0.73 = x / 25
x = 0.73 x 25
x = 18.25
4.
Sin 57 = 14 / b
0.84 = 14 / b
b = 14 / 0.84
b = 16.67
Cos 57 = a / b
0.54 = a / 16.67
a = 0.54 x 16.67
a = 9
Thus,
The values of the variables are:
1.
a = 14.69
b = 20.22
2.
p = 11.28
q = 4.08
3.
x = 18.25
y = 17
4.
a = 9
b = 16.67
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dy = 9e²+7, y(-7)= 0 dz Solve the initial value problem above. (Express your answer in the form y=f(x).)
The solution to the initial value problem is y = 9/2 * e^(2x) + 7x + 49 - 9/2 * e^(-14).
To solve the initial value problem, we need to find the function y(x) that satisfies the given differential equation and initial condition.
The given differential equation is dy/dx = 9e^(2x) + 7.
To solve this, we can integrate both sides of the equation with respect to x:
∫ dy = ∫ (9e^(2x) + 7) dx
Integrating, we get:
y = 9/2 * e^(2x) + 7x + C
where C is the constant of integration.
To find the specific value of C, we use the initial condition y(-7) = 0. Substituting x = -7 and y = 0 into the equation, we can solve for C:
0 = 9/2 * e^(2*(-7)) + 7*(-7) + C
0 = 9/2 * e^(-14) - 49 + C
C = 49 - 9/2 * e^(-14)
Now we have the complete solution:
y = 9/2 * e^(2x) + 7x + 49 - 9/2 * e^(-14)
Therefore, the solution to the initial value problem is y = 9/2 * e^(2x) + 7x + 49 - 9/2 * e^(-14).
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Find the derivative of the trigonometric function f(x) = 7x cos(-x). Answer 2 Points f'(x) = =
The derivative of the trigonometric function f(x) = 7x cos(-x) can be found using the product rule and the chain rule.
The product rule states that the derivative of the product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. In this case, let's consider the functions u(x) = 7x and v(x) = cos(-x). Taking the derivatives of these functions, we have u'(x) = 7 and v'(x) = -sin(-x) * (-1) = sin(x).
Applying the product rule, we can find the derivative of f(x):
f'(x) = u'(x) * v(x) + u(x) * v'(x)
= 7 * cos(-x) + 7x * sin(x)
Simplifying the expression, we have: f'(x) = 7cos(-x) + 7xsin(x)
Therefore, the derivative of the trigonometric function f(x) = 7x cos(-x) is f'(x) = 7cos(-x) + 7xsin(x).
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Compute the determinant using cofactor expansion along the first row and along the first column.
1 2 3
4 5 6
7 8 9
The determinant of the given matrix using cofactor expansion along the first row and first column is 0.
To compute the determinant of the matrix using cofactor expansion along the first row, we multiply each element of the first row by its cofactor and sum the results. The cofactor of each element is determined by the sign (-1)^(i+j) multiplied by the determinant of the submatrix obtained by removing the row and column containing that element. In this case, the first row elements are 1, 2, and 3. The cofactor of 1 is 5*(-1)^(2+2) = 5, the cofactor of 2 is 6*(-1)^(2+3) = -6, and the cofactor of 3 is 0*(-1)^(2+4) = 0. Therefore, the determinant using cofactor expansion along the first row is 1*5 + 2*(-6) + 3*0 = 0.
Similarly, to compute the determinant using cofactor expansion along the first column, we multiply each element of the first column by its cofactor and sum the results. The cofactor of each element is determined using the same method as above. The first column elements are 1, 4, and 7. The cofactor of 1 is 5*(-1)^(2+2) = 5, the cofactor of 4 is 9*(-1)^(3+2) = -9, and the cofactor of 7 is 0*(-1)^(3+3) = 0. Therefore, the determinant using cofactor expansion along the first column is 1*5 + 4*(-9) + 7*0 = 0.
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Determine the end behavior for function f(x) —x3(x 9)3(x + 5).
The end behavior of the function f(x) = -x^3(x + 9)^3(x + 5) indicates that as x approaches positive or negative infinity, the function approaches negative infinity.
To determine the end behavior of the function, we examine the behavior of the function as x becomes very large (approaching positive infinity) and as x becomes very small (approaching negative infinity).
As x approaches positive infinity, the dominant term in the function is -x^3. Since x is being raised to an odd power (3), the term -x^3 approaches negative infinity. The other factors, (x + 9)^3 and (x + 5), do not change the sign or the behavior of the function significantly. Therefore, as x approaches positive infinity, f(x) also approaches negative infinity.
Similarly, as x approaches negative infinity, the dominant term in the function is also -x^3. Again, since x is being raised to an odd power (3), the term -x^3 approaches negative infinity. The other factors, (x + 9)^3 and (x + 5), do not change the sign or the behavior of the function significantly. Therefore, as x approaches negative infinity, f(x) also approaches negative infinity.
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1. Find k such that f(x) = kx is a probability density function over the interval (0,2). Then find the probability density function.
To determine the value of P(x) based on the given expression, we need to equate the integrand to the given expression and solve for P(x). By comparing the coefficients of the terms on both sides of the equation, we find that P(x) = x + 3.
Let's rewrite the given expression as an integral:
∫(2x^2 - x + 3) / P(x) dx + 5(2x^2 - 2x + 10x).
To find P(x), we compare the terms on both sides of the equation.
On the left side, we have ∫(2x^2 - x + 3) / P(x) dx + 5(2x^2 - 2x + 10x).
On the right side, we have x + 3.
By comparing the coefficients of the corresponding terms, we can equate them and solve for P(x).
For the x^2 term, we have 2x^2 = 5(2x^2), which implies 2x^2 = 10x^2. This equation is true for all x, so it does not provide any information about P(x).
For the x term, we have -x = -2x + 10x, which implies -x = 8x. Solving this equation gives x = 0, but this is not sufficient to determine P(x).
Finally, for the constant term, we have 3 = 5(-2) + 5(10), which simplifies to 3 = 50. Since this equation is not true, there is no solution for the constant term, and it does not provide any information about P(x).
Combining the information we obtained, we can conclude that the only term that provides meaningful information is the x term. From this, we determine that P(x) = x + 3.
Therefore, the value of P(x) is x + 3, which corresponds to option A.
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Matrix A is factored in the form PDP Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 1「-40-113001001 2 0 -4 A2 3 8 0 0 3 0 1 2 0 3 02 1 8 Select the correct choice below and fill in the answer boxes to complete your choice.
The eigenvalues of matrix A are λ1 = -1, λ2 = 2, and λ3 = 3. The basis for each eigenspace can be determined by finding the corresponding eigenvectors.
To find the eigenvalues and eigenvectors of matrix A, we can use the Diagonalization Theorem. The first step is to find the eigenvalues by solving the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix.
After solving the characteristic equation, we find the eigenvalues of A. Let's denote them as λ1, λ2, and λ3.
Next, we can find the eigenvectors corresponding to each eigenvalue by solving the system of equations (A - λI)X = 0, where X is a vector. The solutions to these systems will give us the eigenvectors. Let's denote the eigenvectors corresponding to λ1, λ2, and λ3 as v1, v2, and v3, respectively.
Finally, the basis for each eigenspace can be formed by taking linear combinations of the corresponding eigenvectors. For example, if we have two linearly independent eigenvectors v1 and v2 corresponding to the eigenvalue λ1, then the basis for the eigenspace associated with λ1 is {v1, v2}.
In summary, the Diagonalization Theorem allows us to find the eigenvalues and eigenvectors of matrix A, which can be used to determine the basis for each eigenspace.
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A company manufactures and sets x cellphones per week. The weekly price demand and cost equations are given below p=600 -0.1x and Cox) - 20,000+ 140x (A) What price should the company charge for the p
a) The company should produce 49 phones with price of $300.1
Maximum weekly revenue: $14,707.9
b) The company should produce 38 phones with price of $368.2.
Maximum weekly profit: $3,231.6
(A) To maximize the weekly revenue, we need to find the value of x that maximizes the revenue function R(x), where R(x) is the product of the price and the quantity sold (x).
The revenue function is given by:
R(x) = x p(x)
where p(x) = 600 - 6.1x
Substitute p(x) into the revenue function:
R(x) = x (600 - 6.1x)
Now, we can find the value of x that maximizes the revenue by taking the derivative of R(x) with respect to x and setting it equal to zero:
dR/dx = 600 - 12.2x
Setting dR/dx = 0 and solving for x:
600 - 12.2x = 0
12.2x = 600
x = 600 / 12.2
x = 49.18
Since we cannot produce a fraction of a cellphone, we round down to 49 phones.
Now, to find the price, substitute the value of x back into the price-demand equation:
p = 600 - 6.1 x 49
= 600 - 299.9
= 300.1
So, the company should produce 49 phones each week and charge a price of $300.1 to maximize the weekly revenue.
Maximum weekly revenue:
R(49) = 49 x 300.1
= $14,707.9
(B) The profit function is given by:
P(x) = R(x) - C(x)
where C(x) = 20 + 140x
Substitute the expressions for R(x) and C(x) into the profit function:
P(x) = (x (600 - 6.1x)) - (20 + 140x)
Now, take the derivative of P(x) with respect to x and set it equal to zero
dP/dx = 600 - 12.2x - 140
Setting dP/dx = 0 and solving for x:
600 - 12.2x - 140 = 0
-12.2x = -460
x = -460 / -12.2
= 37.7
Since we cannot produce a fraction of a cellphone, we round up to 38 phones.
Now, to find the price, substitute the value of x back into the price-demand equation:
p = 600 - 6.1 x 38
= 600 - 231.8
= 368.2
So, the company should produce 38 phones each week and charge a price of $368.2 to maximize the weekly profit.
Now, Maximum weekly profit:
P(38) = (38 x (600 - 6.1 x 38)) - (20 + 140 * 38)
= $3,231.6
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The question attached here seems to be incomplete, the complete question is:
company manufactures and sells x cellphones per week. The weekly price-demand and cost equations are given below
p = 600 - 6.1x and C(x) = 20 + 140x
(A) What price should the company charge for the phones, and how many phones should be produced to maximize the weekly revenue? What is the maximum weekly revenue?
The company should produce phones each week at a price of (Round to the nearest cent as needed) Box
The maximum weekly revenue is $ (Round to the nearest cent as needed)
(B) What price should the company charge for the phones, and how many phones should be produced to maximize the weekly profit? What is the maximus weekly prof
Box s The company should produce phones each week at a price of (Round to the nearest cent as needed) root(, 5) Box
The maximum weekly profit is $ (Round to the nearest cent as needed