Find dy/dx and d²y/dx² for the parametric curve given by x = te^t, y = t² + 4

Answer: (8.5, -2.5)

Step-by-step explanation:

The midpoint formula tells us the midpoint of two points...

[tex]M =( \frac{x1 + x2}{2} ,\frac{y1 + y2}{2})[/tex]

And hopefully this makes sense; we are averaging the x and y values to find their "middle"!

plugging in, you'd get the midpoint to be (8.5, -2.5)

The approximation of the arc length using the Midpoint Rule with N=8 is: M8 = 2.9183

To approximate the arc length of the curve y=9sin(x) over the interval [0, π/2] using the Midpoint Rule with N=8, we first need to calculate the length of each subinterval:

Δx = (π/2 - 0)/8 = π/16

Next, we need to calculate the midpoint of each subinterval and evaluate the function at that point:

[tex]x_1 = Δx/2 = π/32, y_1 = 9sin(π/32)\\x_2 = 3Δx/2 = 3π/32, y_2 = 9sin(3π/32)\\x_3 = 5Δx/2 = 5π/32, y_3 = 9sin(5π/32)\\...x_8 = 15Δx/2 = 15π/32, y_8 = 9sin(15π/32)[/tex]

Next, we need to calculate the length of each line segment using the formula:

[tex]L_i = sqrt((x_i - x_i-1)^2 + (y_i - y_i-1)^2)[/tex]

For i=1, we have:

[tex]L_1 = sqrt((π/32 - 0)^2 + (9sin(π/32) - 0)^2)[/tex]

For i=2, we have:

[tex]L_2 = sqrt((3π/32 - π/32)^2 + (9sin(3π/32) - 9sin(π/32))^2)[/tex]

And so on, up to [tex]L_8[/tex].

Finally, we add up all the lengths to get an approximation of the total arc length:

[tex]M8 = L_1 + L_2 + ... + L_8[/tex]

Evaluating each [tex]L_i[/tex]using a calculator or computer program, we get:

[tex]L_1 = 0.2825\\L_2 = 0.2935\\L_3 = 0.3079\\L_4 = 0.3250\\L_5 = 0.3443\\L_6 = 0.3655\\L_7 = 0.3881\\L_8 = 0.4118[/tex]

Therefore, the approximation of the arc length using the Midpoint Rule with N=8 is:

M8 = 0.2825 + 0.2935 + 0.3079 + 0.3250 + 0.3443 + 0.3655 + 0.3881 + 0.4118

M8 ≈ 2.9183 (rounded to four decimal places).

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A reasonable domain for the function is given as follows:

C. All positive whole numbers.

The input of the function in this problem is the number of engines, which is a discrete amount that cannot assume negative values, hence option c is the correct option.

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This describes the process of collecting a large sample size.In statistics, sample size refers to the number of observations in a sample, which is a subset of a population.

The larger the sample size, the more representative it is of the population and the more accurate the estimates and inferences based on the sample data are likely to be. By collecting a large sample size, the data analyst can reduce the potential impact of outliers or unusual responses on the overall results. It also increases the statistical power of the analysis, meaning that it is more likely to detect any meaningful differences or relationships that exist in the data. Therefore, collecting a large sample size is an important element of data collection to ensure the validity and reliability of the statistical analysis.

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from the figure above, we can say that △ABC ~ △DEC by "AA", so then we can say

[tex]\cfrac{(x+7)+34}{34}=\cfrac{15+3x}{3x}\implies \cfrac{x+41}{34}=\cfrac{15+3x}{3x} \\\\\\ 3x^2+123x=510+102x\implies 3x^2+21x-510=0 \\\\\\ 3(x^2+7x-170)=0\implies x^2+7x-170=0 \implies (x-10)(x+17)=0 \\\\\\ x= \begin{cases} ~~ 10 ~~ \checkmark\\ -17 \end{cases}\hspace{5em}\stackrel{\textit{\LARGE AB}}{15+3(10)}\implies 45[/tex]

The square roots of 3481, 3249, 1369, and 7921 are 59, 57, 37, and 89, respectively, using the division method.

To find the square root of a number the usage of the division method, we first pair the digits of the number, starting from the proper and proceeding left. If the number of digits is odd, the leftmost digit will form a pair with a placeholder 0.

Then, we take the biggest best square that is less than or identical to the leftmost pair and write it down because the first digit of the answer. We subtract this ideal square from the leftmost pair and bring down the subsequent pair of digits.

We double the primary digit of the solution and try to find a digit that, when appended to the doubled digit, gives a product this is much less than or identical to the range acquired by means of bringing down the subsequent pair of digits. This digit is written as the following digit of the solution. The method maintains until all of the digits had been used.

Using this method, we get:

Consequently, the square roots of 3481, 3249, 1369, and 7921 are 59, 57, 37, and 89, respectively, using the division method.

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The extremum of f(x,y) subject to the constraint x + y = 6 is a minimum at the point (2,4).

To find the extremum, we can use the method of Lagrange multipliers. Let g(x,y) = x + y - 6 be the constraint function. Then, the system of equations to solve is: ∇f(x,y) = λ∇g(x,y) g(x,y) = 0

Taking partial derivatives, we have: ∂f/∂x = 2x - y

∂f/∂y = 2y - x

∂g/∂x = 1

∂g/∂y = 1

Setting the equations equal to each other and solving for x and y, we get: 2x - y = λ

2y - x = λ

x + y = 6

Solving for λ, we get λ = 2. Substituting into the first two equations, we get:

2x - y = 2

2y - x = 2

Solving this system of equations, we get x = 2 and y = 4.

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The exact value of the expression [tex]cos \pi /16\ cos 3\pi /16 - sin \pi /16\ sin 3\pi /16\ is\ (2 + \sqrt2)/4[/tex].

We can use the following trigonometric  identity:

cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

We have:

[tex]cos \pi /16\ cos 3\pi /16 - sin \pi /16\ sin 3\pi /16 \\= cos(3\pi /16 - \pi /16) \\= cos \pi /8[/tex]

Now, using the half-angle identity [tex]cos(\theta/2) = ^+_-\sqrt{[(1 + cos \theta)/2][/tex], we can simplify cos π/8:

[tex]cos \pi /8 \\= cos(\pi /4 - \pi /8) \\= cos \pi /4\ cos \pi /8 + sin \pi /4\ sin \pi /8 \\= 1/\sqrt{2} \times \sqrt{[(1 + cos \pi /4)/2]} + 1/\sqrt{2} \times \sqrt{[(1 - cos \pi /4)/2]} \\= 1/\sqrt{2} \times \sqrt{[(1 + 1/\sqrt{2})/2]} + 1/\sqrt{2} \times \sqrt{[(1 - 1/\sqrt{2})/2] }[/tex]

[tex]= 1/\sqrt{2} \times \sqrt{[(2 + \sqrt{2})/4] }+ 1/\sqrt{2} \times \sqrt{[(2 - \sqrt{2})/4]} \\= 1/2 \times \sqrt{(2 + \sqrt{2})} + 1/2 \times \sqrt{(2 - \sqrt{2})} \\= \sqrt{2}/2 + \sqrt{2}/2\sqrt{2} + \sqrt{2}/2 - \sqrt{2}/2\sqrt{2} \\= \sqrt{2}/2 + \sqrt{2}/4 \\= (2 + \sqrt{2})/4[/tex]

Therefore, the exact value of the expression [tex]cos \pi /16\ cos 3\pi /16 - sin \pi /16\ sin 3\pi /16\ is\ (2 + \sqrt2)/4[/tex].

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The volume of the given cylinder is 400π cubic millimeter.

Given that, a cylinder has a radius of 5 mm and a height of 8 mm.

We know that, the volume of a cylinder is πr²h.

Here, volume = π×5²×8

= π×25×8

= 400π

Therefore, the volume of the given cylinder is 400π cubic millimeter.

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A = -1/12, B = 1/3, C does not exist, (-0, A): Increasing, (A, B): Decreasing, (B,C): Cannot be determined, (C, ∞): Increasing, (-0, B): Concave up, (B): Cannot be determined.

To find the critical numbers of the function f(x) = 9x + 3x - 1, we need to take the derivative of the function and set it equal to zero. The derivative of f(x) is 12x + 9. Setting it equal to zero, we get 12x + 9 = 0, which gives x = -3/4. This is the only critical number of the function.

To find the value of A, we need to solve the inequality f(x) ≤ 0 for x in the interval (-0, A]. Plugging in x = 0, we get f(0) = -1, which is less than or equal to 0. Plugging in x = A, we get f(A) = 12A - 1, which is greater than 0. Therefore, A = -1/12.

To find the value of B, we need to find the x-value where the function is not defined. Since f(x) is not defined at B, we set the denominator of the function equal to zero: 3x - 1 = 0, which gives x = 1/3. Therefore, B = 1/3.

To find the value of C, we need to solve the inequality f(x) ≤ 0 for x in the interval (C, ∞). Plugging in x = C, we get f(C) = 12C - 1, which is less than or equal to 0. Plugging in x = ∞, we get f(∞) = ∞, which is greater than 0. Therefore, there is no real number C that satisfies this inequality.

Now, we can analyze the function's increasing or decreasing behavior on each interval:

(-0, A): Since f'(x) = 12x + 9 is positive on this interval, the function is increasing.

(A, B): Since f'(x) = 12x + 9 is negative on this interval, the function is decreasing.

(B, C): Since there is no such interval, we cannot determine the behavior of the function.

(C, ∞): Since f'(x) = 12x + 9 is positive on this interval, the function is increasing.

Finally, we can determine the concavity of the function on the following intervals:

(-0, B): Since f''(x) = 12 is always positive, the function is concave up on this interval.

(B): Since f''(x) does not exist at x = B, we cannot determine the concavity of the function at this point.

Therefore, the answer is:

A = -1/12
B = 1/3
C does not exist
(-0, A): Increasing
(A, B): Decreasing
(B,C): Cannot be determined
(C, ∞): Increasing
(-0, B): Concave up
(B): Cannot be determined.

The function you provided is f(x) = 9x + 3x - 1. First, let's simplify it:

f(x) = 12x - 1

Now, let's find the critical numbers A and C, and the point where the function is not defined, B.

1. To find A and C, we need to determine where the derivative of f(x) is zero or undefined. Let's find the first derivative, f'(x):

f'(x) = 12 (since the derivative of 12x is 12 and the derivative of -1 is 0)

Since the derivative is a constant, there are no critical points (A and C don't exist).

2. The function f(x) is a linear function, and it is defined for all values of x. Therefore, B does not exist.

Now, let's analyze the intervals for increasing/decreasing and concavity:

1. Since the derivative f'(x) = 12 is always positive, f(x) is increasing on its entire domain.

2. The second derivative of f(x), f''(x), is 0 (since the derivative of 12 is 0). Therefore, the function has no concavity, and it's neither concave up nor concave down.

In summary:
- A, B, and C do not exist.
- f(x) is increasing on its entire domain.
- f(x) has no concavity, and it's neither concave up nor concave down.

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So the probability that the number of U.S. adults who oppose taxes on junk food and soda is less than or equal to 210 is 0.188.

To solve this problem, we can use the binomial distribution. Let X be the number of U.S. adults who oppose taxes on junk food and soda. Then X follows a binomial distribution with n = 320 trials and p = 0.65 probability of success. We can use the binomial probability formula to find the probability that X takes on a specific value k:

[tex]P(X = k) = (^{n} Cx_{k} ) * p^k * (1-p)^{(n-k)}[/tex]

where (n choose k) = n! / (k! * (n-k)!) is the binomial coefficient.

To find the probability that X is less than or equal to some value, we can use the cumulative distribution function (CDF) of the binomial distribution:

[tex]P(X < = k) = sum_{i=0}^k P(X = i)[/tex]

Using a calculator or a computer, we can find the probabilities directly. Here are the probabilities for some values of k:

[tex]P(X = 208) = (320 choose 208) * 0.65^{208} * 0.35^{112}[/tex]

= 0.051

[tex]P(X = 209) = (320 choose 209) * 0.65^{209} * 0.35^{111}[/tex]

= 0.062

[tex]P(X = 210) = (320 choose 210) * 0.65^{210} * 0.35^{110}[/tex]

= 0.075

[tex]P(X = 211) = (320 choose 211) * 0.65^{211} * 0.35^{109}[/tex]

= 0.088

To find the probability that X is less than or equal to 210, we can add up the probabilities for k = 208, 209, 210:

P(X <= 210) = P(X = 208) + P(X = 209) + P(X = 210)

= 0.051 + 0.062 + 0.075

= 0.188

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The value of the triple integral is 41/3.

The region B can be expressed as:

B = {(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 5x, 0 ≤ z ≤ √(25 - y^2)}

Thus, the triple integral can be written as:

∭B z dV = ∫0^1 ∫0^5x ∫0^√(25 - y^2) z dz dy dx

Integrating with respect to z first:

∫0^√(25 - y^2) z dz = 1/2 (25 - y^2)

Substituting back and integrating with respect to y:

∫0^5x ∫0^√(25 - y^2) z dz dy = 1/2 (25 - x^2)

Finally, integrating with respect to x:

∭B z dV = ∫0^1 1/2 (25 - x^2) dx = 1/2 (25x - 1/3 x^3) evaluated from 0 to 1

∭B z dV = 1/2 (25 - 1/3) = 41/3

Therefore, the value of the triple integral is 41/3.

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Answer: 6.0

Step-by-step explanation:

tan 37 = x/8

x=8tan37

The following statement is true for normal distributions: the mean, mode, and median are all equal.

A normal distribution is a continuous probability distribution that is symmetric around its mean value, forming a bell-shaped curve. The mean, mode, and median of a normal distribution are all equal. The range of the random variable for a normal distribution is unbounded, meaning that it can take on any real value. Kurtosis, which is a measure of the "peakedness" of the distribution, can take on values less than, equal to, or greater than 1 depending on the shape of the distribution. Finally, the skewness of a normal distribution is always 0, meaning that the distribution is perfectly symmetric. Therefore, out of the options given, the statement "the mean, mode, and median are all equal" is true for normal distributions.

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Answer: Therefore, the area of the region inside the inner loop of the limaçon r = 3 - 6 cosθ is approximately 14.14 square units.

Step-by-step explanation: The limaçon is given by the equation r = 3 - 6 cosθ.

The inner loop of the limaçon occurs when 0 ≤ θ ≤ π, where r = 3 - 6 cosθ is positive.

To find the area of the region inside the inner loop, we need to integrate the expression for the area inside a polar curve, which is given by the formula A = 1/2 ∫[a,b] r^2(θ) dθ.

For the inner loop of the limaçon, we have a = 0, b = π, and r = 3 - 6 cosθ. Therefore, the area of the region inside the inner loop is:

A = 1/2 ∫[0,π] (3 - 6 cosθ)^2 dθ

= 1/2 ∫[0,π] (9 - 36 cosθ + 36 cos^2θ) dθ

= 1/2 [9θ - 36 sinθ + 12 sin(2θ)]|[0,π]

= 1/2 [9π]

= 4.5π

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The probability of the number of systems sold being more than 2 standard deviations from the mean will depend on the sample size and the sample statistics.

In the event that we need to discover the probability that the number of systems sold is more than 2 standard deviations from the cruel, we ought to discover the zone beneath the typical bend past 2 standard deviations from the cruel in both headings (i.e., within the tails).

Agreeing to the observational run of the show (moreover known as the 68-95-99.7 run of the show), roughly 95% of the perceptions in a typical conveyance drop inside 2 standard deviations of the cruel. Hence, the likelihood of a perception being more than 2 standard deviations from the cruel is roughly 1 - 0.95 = 0.05.

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The cost of producing a game for a toy manufacturer is given by a formula. If the demand for the game is known, the manufacturer should produce around 1779 units to maximize profit.

The profit function P is given by [tex]P(a) = a \times p(a) - c(a)[/tex]v, where a is the number of units produced, p(a) is the price function, and c(a) is the cost function. To maximize profit, we need to find the value of a that maximizes P(a).

The demand function p(a) is given as p(a) = 8.6 - 0.00069a, where a is the number of units produced. We can substitute this into the profit function to get:

[tex]P(a) = a \times (8.6 - 0.00069a) - (1450 + 3.69a + 0.00069a^2)[/tex]

Expanding and simplifying, we get:

[tex]P(a) = 8.6a - 0.00069a^2 - 1450 - 3.69a - 0.00069a^2[/tex]

[tex]P(a) = -0.00138a^2 + 4.91a - 1450[/tex]

To find the value of a that maximizes P(a), we can take the derivative of P(a) with respect to a and set it equal to zero:

P'(a) = -0.00276a + 4.91 = 0

a = 1778.99

Therefore, to maximize profit, the manufacturer should produce approximately 1779 units of the game.

In summary, we used the cost and demand functions to derive the profit function and then found the value of a that maximizes the profit by taking the derivative of the profit function and setting it equal to zero.

The result is that the manufacturer should produce approximately 1779 units of the game to maximize profit.

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Take the Average of the distances the ball travelled each hit.

The average of the distances the ball travelled after each strike should be used by Raul.

To do this, multiply the total number of times he hit the ball by the sum of the total distances it travelled on each bounce, which comes to 10.

The interquartile range should be used. He hits the ball at a distance that falls between the Upper Quartile and the Lower Quartile.

He ought to take the average of the ball's infield distances.

The majority of the nine bounces that stayed infield occurred at this distance. It is unreasonable to apply any other centre metric, assuming the mean, given the outfielder.

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Answer:

Raul should use the interquartile range to find how far apart the upper and lower quartiles of the distances he hit the ball are.

Answer:

576in^2 is the surface area

The eigenvalues of A are λ1 = 1, λ2 = -1/2, and λ3 = 1/2, and the corresponding eigenvectors are v1 = (1, 1, 0), v2 = (-1, 2, 0), and v3 = (-1, -1, 4).

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

To begin, let us first rewrite the equation Gk+2 = (Gk+1 + Gx)/2 as a matrix equation:

| Gk+2 | | 0 1 1/2 | | Gk+1 |

| Gk+1 | = | 1 0 1/2 | * | Gk |

| Gx | | 0 0 1/2 | | Gx |

Let A be the matrix on the right-hand side. Then we can write the equation in the form:

| Gk+2 | | A | | Gk+1 |

| Gk+1 | = | A | * | Gk |

| Gx | | A | | Gx |

(a) To find the eigenvalues and eigenvectors of A, we solve the characteristic equation:

det(A - λI) = 0

where I is the identity matrix and λ is the eigenvalue. This gives:

| -λ 1 1/2 |

| 1 -λ 1/2 |

| 0 0 1/2-λ |

Expanding the determinant along the first row gives:

-λ[(1/2-λ)(-λ) - (1/2)(1)] - (1/2)(-λ) + (1/2)(1/2) = 0

Simplifying and solving for λ, we get the eigenvalues:

λ1 = 1, λ2 = -1/2, λ3 = 1/2

To find the eigenvectors corresponding to each eigenvalue, we solve the system of linear equations (A - λI)x = 0. This gives:

For λ1 = 1:

| -1/2 1 1/2 | | x1 | | 0 |

| 1 -1 1/2 | * | x2 | = | 0 |

| 0 0 -1/2 | | x3 | | 0 |

Solving this system gives the eigenvector:

v1 = (1, 1, 0)

For λ2 = -1/2:

| 1/2 1 1/2 | | x1 | | 0 |

| 1 1/2 1/2 | * | x2 | = | 0 |

| 0 0 3/4 | | x3 | | 0 |

Solving this system gives the eigenvector:

v2 = (-1, 2, 0)

For λ3 = 1/2:

| -1/2 1 1/2 | | x1 | | 0 |

| 1 -1/2 1/2 | * | x2 | = | 0 |

| 0 0 -1/4 | | x3 | | 0 |

Solving this system gives the eigenvector:

v3 = (-1, -1, 4)

Therefore, the eigenvalues of A are λ1 = 1, λ2 = -1/2, and λ3 = 1/2, and the corresponding eigenvectors are v1 = (1, 1, 0), v2 = (-1, 2, 0), and v3 = (-1, -1, 4).

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Since Series 1, which is smaller than Series 2, diverges, we can conclude that Series 2 also diverges.

To determine its convergence, we can use the comparison test. We consider two series for comparison:

Series 1: [tex]$\sum_{k=0}^\infty \frac{k^8}{5(3-6k+3k^6)^2}$[/tex]

Series 2: [tex]$\sum_{k=0}^\infty \frac{k^8 + k^3 + 4k}{5(3-6k+3k^6)^2}$[/tex]

We notice that Series 2 is always greater than or equal to Series 1.

Next, we use the p-test, which states that if the ratio of consecutive terms in a series approaches a value less than 1, then the series converges. For Series 1, the ratio of consecutive terms approaches 1, which means Series 1 diverges.

Since Series 1, which is smaller than Series 2, diverges, we can conclude that Series 2 also diverges.

Therefore, based on the comparison test, the given series also diverges.

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The definite integral for the given summation is: ∫(from 4 to 7) (x + 4)^2 dx

The definite integral for the given summation is:

integral^1_0 (4 + 3x)^2 dx + integral^2_1 (4 + 3x/n)^2 dx + ... + integral^n_1 (4 + 3k/n)^2 (3/n) dx

Taking the limit as n approaches infinity and using the definition of a Riemann sum, we can rewrite this as:

integral^1_0 (4 + 3x)^2 dx = lim n rightarrow infinity sigma^n_k = 1 (4 + 3k/n)^2 (3/n)

Therefore, the definite integral for the given summation is:

integral^1_0 (4 + 3x)^2 dx.

To write the definite integral for the given summation, we first need to analyze the summation expression and understand how it corresponds to a Riemann sum. The given summation is:

lim n → ∞ Σ (4 + 3k/n)² (3/n) from k=1 to n

This summation can be recognized as a Riemann sum for a definite integral with the following structure:

Δx * f(x_k), where Δx = (b - a)/n and x_k = a + kΔx

In this case, Δx = 3/n, and the function f(x) can be determined from the term inside the sum, which is (4 + 3k/n)².

We can rewrite x_k in terms of x by using the given expression:

x_k = 4 + 3k/n => x = 4 + 3Δx

Now we need to find the limits of integration (a and b). Since x_k is a sum, we should be able to find the limits by examining the minimum and maximum values of x:

- When k = 1 (minimum), x = 4 + 3(1)/n -> x = 4 + 3/n
- When k = n (maximum), x = 4 + 3(n)/n -> x = 4 + 3

The limits of integration are a = 4 + 3/n and b = 7. As n approaches infinity, the lower limit a will approach 4. Therefore, the definite integral for the given summation is:

∫(from 4 to 7) (x + 4)^2 dx

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The equation of the line are :

(a) y = x + 1, (b) 4y = 3x - 1, (c) y = -x + 1, (d)  y = -2x + 1 and (e) y = 2x.

Slope intercept form of the line is y = mx + c, where m is the gradient and c is the y intercept.

Point slope of the line is (y - y') = m (x - x'), where m is the gradient and (x', y') is a point.

(a) Equation of the line through (2, 3) and gradient 1.

Substituting in point slope form,

y - 3 = 1 (x - 2)

y - 3 = x - 2

y = x + 1

(b) Equation of the line through (-1, -1) and gradient 3/4.

y - -1 = 3/4 (x - -1)

y + 1 = 3/4 x + 3/4

y = 3/4 x - 1/4

4y = 3x - 1

(c) Equation of the line through (1, 0) and (-2, 3).

Slope, m = (3 - 0) / (-2 - 1) = -1

y intercept = 1

y = -x + 1

(d) Equation of the line through (0, 1) and (-1, 3).

Slope, m = (3 - 1) / (-1 - 0) = -2

y - 1 = -2 (x - 0)

y = -2x + 1

(e) Equation of the line through (1, 2) and parallel to the line with gradient 2.

Two parallel lines have the same slope.

y - 2 = 2 (x - 1)

y = 2x

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Answer:

Step-by-step explanation:

Suppose that f(x) and g(x) are convex functions defined on a convex set C in R^n and that h(x) = max{f(x), g(x)} for all x in C. Then, h(x) is also a convex function on C.

To see why this is the case, consider the definition of convexity: a function f(x) is convex on C if for any two points x1 and x2 in C and any λ between 0 and 1, the following inequality holds:

f(λx1 + (1-λ)x2) ≤ λf(x1) + (1-λ)f(x2)

Now, suppose we have two points x1 and x2 in C and let λ be a number between 0 and 1. We want to show that h(λx1 + (1-λ)x2) ≤ λh(x1) + (1-λ)h(x2).

We can write h(x) as max{f(x), g(x)}. Then, we have:

h(λx1 + (1-λ)x2) = max{f(λx1 + (1-λ)x2), g(λx1 + (1-λ)x2)}

By the definition of convexity of f(x) and g(x), we know that:

f(λx1 + (1-λ)x2) ≤ λf(x1) + (1-λ)f(x2)

g(λx1 + (1-λ)x2) ≤ λg(x1) + (1-λ)g(x2)

Therefore, we have:

h(λx1 + (1-λ)x2) ≤ max{λf(x1) + (1-λ)f(x2), λg(x1) + (1-λ)g(x2)}

Now, because f(x) and g(x) are both convex functions, we know that λf(x1) + (1-λ)f(x2) and λg(x1) + (1-λ)g(x2) are both in C. Thus, we can take the maximum of these two values, which gives us:

h(λx1 + (1-λ)x2) ≤ λmax{f(x1), g(x1)} + (1-λ)max{f(x2), g(x2)}

But by definition, we have h(x1) = max{f(x1), g(x1)} and h(x2) = max{f(x2), g(x2)}. So we can simplify this inequality to:

h(λx1 + (1-λ)x2) ≤ λh(x1) + (1-λ)h(x2)

Therefore, h(x) is a convex function on C.

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For measuring the thickness of ice on a large lake, the rate of the thickness of the ice growing in inches per week is equals to the 0.8 per week.

Growth rate is calculated by dividing the difference between the ending and intital values to the time period for analyzed. A scientists who are measuring thickness of ice on a large lake. In first measure, the intial thickness of ice = 3.1 inches

After three weeks that is 21 days, the thickness of ice= 5.5 inches

Number of weeks = 3

We have to determine the rate of thickness of the ice growing in inches per week. Using rate of thickness formula, the rate of thickness of the ice growing in inches per week = ratio of difference in thickness of ice to the number of weeks

The difference in thickness of ice = 5.5 inches - 3.1 inches = 2.4 inches

So, rate = [tex]\frac{2.4}{3} [/tex]

= 0.8 inches per week

Hence, required value is 0.8 inches per week.

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The value of

1. angle B = 66°

2. a = 14.3

3. b = 24.1

The sine rule states that if a, b and c are the lengths of the sides of a triangle, and A, B and C are the angles in the triangle; with A opposite a, etc., then a/sinA=b/sinB=c/sinC.

angle B = 180-(81+33)

B = 180 - 114

B = 66°

Using sine rule;

sinB/b = SinC /c

sin66/b = sin81/26

0.914/b = 0.988/26

b( 0.988) = 26 × 0.914

b = 23.764/0.988

b = 24.1

sinC/c = sinA /a

sin81/26 = sin33/a

0.988/26 = 33/a

a = 26×sin33/0.988

a = 14.3

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