Find the derivative, dy/dx for the given Implicit relation. 2tan^3(5y) + x^5e^2y = l^4x

The solution to this equation 4x² – 15x = 4 include the following: D. x = 4, -1/4.

In Mathematics and Geometry, the general form of a quadratic function can be modeled and represented by using the following quadratic equation;

y = ax² + bx + c

Where:

Next, we would solve the quadratic function by using the factorization method as follows;

4x² – 15x = 4

Subtract 4 from both sides of the quadratic function:

0 = 4x² – 15x - 4

0 = 4x² - 16x + x - 4

(4x + 1)(x - 4)

x = 4 or x = -1/4.

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The function of the trigonometric graph plotted is

The equation is written by the general formula

y = A cos (Bx + C) + D

where:

A = amplitude.

B = 2π/T, where T = period

C = phase shift.

D = vertical shift.

amplitude

A = (maximum - minimum) / 2

from the graph,

maximum = 1

minimum = -1

A = |-1 - (-3)| / 2 = 2/2 = 1

B = 2π/T

where T = 2π

B = 2π/(2π) = 1

C = phase shift

= 0 - π/4

= - π/4

D = vertical shift

= 0 - 2 = -2

plugging in the results of the parameters to the equation

y = 1 cos (1x + (-π/4)) + (-2)

this is written as

y = cos (x - π/4) - 2

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A one-tailed t-test is used when the researcher has a specific directional hypothesis.

If you are comparing the difference between two separate populations, such as children who attend two different Elementary schools, you should use an independent-measures design. In an independent-measures design, two separate groups of participants are sampled, and each participant is only tested once. The purpose of this design is to compare the means of two independent populations to determine if there is a statistically significant difference between them. In contrast, a within-groups design would involve testing the same group of participants twice under different conditions, while a repeated-measures design would involve testing the same group of participants under all conditions. A one-tailed t-test is a specific type of statistical test that can be used in either an independent-measures or within-groups design to test a directional hypothesis.

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The degree measure of the larger angle is 50⁰.

Complementary angles are two angles whose sum is 90 degrees.

So we have two angles; let the smaller angle = x

then bigger angle = (x + 10)

The two angles will add up to 90 degrees;

x + (x + 10) = 90

2x + 10 = 90

2x = 90 - 10

2x = 80

x = 80/2

x = 40⁰

The measure of the larger angle = 40⁰ + 10⁰ = 50⁰

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Is not enough evidence to conclude that the mean price of the shares in the month is different from 165.

To test whether the mean price of the shares in the month is 165, we will perform a one-sample t-test.

The null hypothesis H0: μ = 165, where μ is the true population mean price of the shares in the month.

The alternative hypothesis Ha: μ ≠ 165.

We will use a significance level of α = 0.05.

First, we will calculate the sample mean and sample standard deviation:

sample mean (x) = (163 + 163 + 164 + 164 + 165 + 169 + 170 + 171) / 9 = 166.11

sample standard deviation (s) = √[((163-166.11)² + (163-166.11)² + (164-166.11)² + (164-166.11)² + (165-166.11)² + (169-166.11)² + (170-166.11)² + (171-166.11)²) / (9-1)] = 2.791

Next, we will calculate the t-statistic:

t = (x - μ) / (s / √n) = (166.11 - 165) / (2.791 / √9) = 1.441

Using a t-table or a calculator, we find that the p-value associated with a t-statistic of 1.441 and 8 degrees of freedom is approximately 0.186.

Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that the mean price of the shares in the month is different from 165.

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Let help you with that.

To find the distance between two points, we can use the distance formula:

```

d = √(x2 - x1)2 + (y2 - y1)2

```

Where:

* `d` is the distance between the two points

* `x1` and `y1` are the coordinates of the first point

* `x2` and `y2` are the coordinates of the second point

In this case, the points are (8, -4) and (-1, -2):

```

d = √((8 - (-1))^2 + ((-4) - (-2))^2)

```

```

d = √(9^2 + (-2)^2)

```

```

d = √(81 + 4)

```

```

d = √85

```

```

d = 9.2 (rounded to the nearest tenth)

```

Therefore, the distance between the two points is 9.2 units.

Answer:

Step-by-step explanation:

The proportion that can be used to find the length of the side y for the similar triangle DEF is y/16 = 38/32, which makes option C correct.

Similar triangles are two triangles that have the same shape, but not necessarily the same size. This means that corresponding angles of the two triangles are equal, and corresponding sides are in proportion.

The side DE corresponds to AB, also the side DF corresponds to AC, so;

DE/AB = DF/AC

y/38 = 16/32

by cross multiplication;

y/16 = 38/32

Therefore, the proportion that can be used to find the length of the side y for the similar triangle DEF is y/16 = 38/32.

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Answer:

Jonah used $17.50 (14% of $125) to buy music on iTunes.

Step-by-step explanation:

The image of point A after it is dilated with a scale factor 3 is (6, 12)

Dilation is a transformation, which is used to resize the object.

A scale factor is when you enlarge a shape and each side is multiplied by the same number. This number is called the scale factor.

Let the coordinates of A are (2, 4)

We have to find the coordinates of point A after dilation with scale factor 3

A'=(3×2, 3×4)

=(6, 12)

Hence,  the image of point A after it is dilated with a scale factor 3 is (6, 12)

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i) The equation in vertex form is f(x) = (x - 2)² - 1.

ii) The vertex of the parabola is (2, -1).

iii)The x-intercepts are (1, 0) and (3, 0) and the y-intercept is (0, 3).

b. The quadratic function is: f(x) = (1/8)(x + 3)² + 2.

A quadratic function is a function of form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a U-shaped curve called a parabola.

The vertex of a parabola is the point where the parabola changes direction. It lies on the axis of symmetry, which is a vertical line that divides the parabola into two equal halves.

Here we have

The parabola f(x) = x² - 4x +3

a. Consider the parabola f(x) = x^2 - 4x + 3

i) To write the equation in vertex form, we complete the square:

f(x) = x² - 4x + 3

= (x² - 4x + 4) - 1

= (x - 2)² - 1

Therefore, the equation in vertex form is f(x) = (x - 2)² - 1.

ii) The vertex of the parabola is (2, -1). The axis of symmetry is the vertical line passing through the vertex, which is x = 2.

iii) To find the x-intercepts, we set f(x) = 0:

(x - 2)² - 1 = 0

(x - 2)² = 1

x - 2 = ±1

x = 1, 3

Therefore, the x-intercepts are (1, 0) and (3, 0).

To find the y-intercept, we set x = 0:

f(0) = 0² - 4(0) + 3 = 3

Therefore, the y-intercept is (0, 3).

b. To write the quadratic function for the parabola that has vertex (-3, 2) and passes through (1, 4), we use the vertex form of the quadratic equation:

f(x) = a(x - h)² + k,

where (h, k) is the vertex.

Substituting the given values, we get:

f(x) = a(x + 3)² + 2

To find the value of a, we substitute the point (1, 4) into the equation:

4 = a(1 + 3)² + 2

2 = 16a

a = 1/8

Therefore,

i) The equation in vertex form is f(x) = (x - 2)² - 1.

ii) The vertex of the parabola is (2, -1).

iii)The x-intercepts are (1, 0) and (3, 0) and the y-intercept is (0, 3).

b. The quadratic function is: f(x) = (1/8)(x + 3)² + 2.

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A.i. The equation in vertex form is f(x) = (x - 2)² - 1.

ii. The axis of symmetry is the vertical line passing through the vertex, which is x = 2.

iii. The x-intercepts are x = 1 and x = 3. The y-intercept is y = 3.

B. The quadratic function for the parabola is:

f(x) = (1/8)(x + 3)^2 + 2.

a.

i) To write the equation in vertex form, we need to complete the square. The general vertex form of a parabola is given by f(x) = a(x - h)² + k, where (h, k) represents the vertex.

Let's complete the square for the given parabola f(x) = x² - 4x + 3:

f(x) = x² - 4x + 3

= (x² - 4x + 4) - 4 + 3 [Adding and subtracting (4/2)² = 4 to complete the square]

= (x - 2)² - 1

So, the equation in vertex form is f(x) = (x - 2)² - 1.

ii) Comparing the equation f(x) = (x - 2)² - 1 with the vertex form f(x) = a(x - h)² + k, we can see that the vertex is (h, k) = (2, -1). The axis of symmetry is the vertical line passing through the vertex, which is x = 2.

iii) To find the x-intercepts, set f(x) = 0 and solve for x:

(x - 2)² - 1 = 0

(x - 2)² = 1

x - 2 = ±√1

x - 2 = ±1

x = 2 ± 1

So, the x-intercepts are x = 1 and x = 3.

To find the y-intercept, set x = 0 in the equation:

f(0) = (0 - 2)² - 1

= (-2)² - 1

= 4 - 1

= 3

So, the y-intercept is y = 3.

b.

To write the quadratic function for the parabola with a vertex at (-3, 2) and passing through (1, 4), use the vertex form of a parabola.

The vertex form of a parabola is f(x) = a(x - h)² + k, where (h, k) represents the vertex.

Using the given vertex (-3, 2):

h = -3 and k = 2.

Substituting the values of h and k:

f(x) = a(x - (-3))² + 2

= a(x + 3)² + 2

Now, use the point (1, 4) to find the value of 'a'.

Substituting x = 1 and f(x) = 4 in the equation:

4 = a(1 + 3)² + 2

4 = a(4²) + 2

4 = 16a + 2

16a = 4 - 2

16a = 2

a = 2/16

a = 1/8

Therefore, the quadratic function for the parabola is: f(x) = (1/8)(x + 3)² + 2.

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The probability of A appearing before B is [tex]\frac{4}{7}[/tex].

To find the probability that TA < TB, we can use the fact that the probability of a certain pattern appearing in a sequence of coin flips is independent of the position in the sequence. In other words, the probability of A appearing at time n is the same as the probability of A appearing at time n+k for any k.

Using this fact, we can set up a system of equations to solve for the probability of TA < TB. Let p be the probability of A appearing before B, and q be the probability of B appearing before A. Then we have:

[tex]p = \frac{1}{2} + \frac{1}{2q}[/tex]  (since the first flip can be either 0 or 1 with equal probability)
[tex]q= \frac{1}{4p} + \frac{1}{2q} + \frac{1}{4}[/tex] (if the first two flips are 0, the sequence B has appeared; if the first flip is 1 and the second is 0, the sequence is neither A nor B and we start over; if the first flip is 1 and the second is 1, we have a new chance for A to appear before B)

Solving for p, we get:
[tex]p=\frac{4}{7}[/tex]

Therefore, the probability of A appearing before B is [tex]\frac{4}{7}[/tex].

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To prove F(c)(A + B) → (Vx) A (3.c)B, we need to show that if the union of sets A and B is finite, then there exists an element x in A such that for all elements y in B, (x, y) is in the relation C.

Assume F(c)(A + B) is true. Then for any (x, y) in C, x belongs to A + B, which means x belongs to either A or B. If x belongs to A, then we have found an element x in A such that for all elements y in B, (x, y) is in C, and we are done. If x belongs to B, then we need to find another element in A such that the condition holds.

Since A and B are finite, their union A + B is also finite. Let n be the size of A + B. Then there are n distinct elements in A + B, say a1, a2, ..., an. Since there are more elements in A than in B (or equal if they have the same size), there must be at least one element of A among a1, a2, ..., an. Call this element x.

Now, consider any element y in B. Since x belongs to A + B and y belongs to B, their sum x + y belongs to A + B as well. But we know that x + y cannot be equal to x, since y is not in A. Therefore, x + y must be equal to one of the remaining n-1 elements of A + B, say ai. But then ai - x = y, so (x, y) is in C.

Therefore, we have shown that F(c)(A + B) → (Vx) A (3.c)B is true.

For the second part of the question, we need to show that for any set A in N, there exists a set B in N such that A is a subset of B and B is infinite.

Let B be the set of all natural numbers greater than the maximum element in A. Then A is clearly a subset of B, and B is infinite since it contains all natural numbers greater than a certain number.

Therefore, we have shown that for any set A in N, there exists a set B in N such that A is a subset of B and B is infinite.

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Answer:

Step-by-step explanation:

Answer

Equation of a straight line

y = -0.33x + 4.995

Answer:

Step-by-step explanation:

Find the midpoint of the segment by using the formula [(x1 + x2)/2, (y1 + y2)/2]. The midpoint is [(2 + 6)/2, (1 + 3)/2] = (4, 2).

Find the slope of the segment by using the formula (y2 - y1)/(x2 - x1). The slope is (3 - 1)/(6 - 2) = 1/2.

Find the negative reciprocal of the slope by flipping the fraction and changing the sign. The negative reciprocal is -2/1 or -2.

Find the equation of the perpendicular bisector by using the point-slope form y - y1 = m(x - x1), where m is the negative reciprocal and (x1, y1) is the midpoint. The equation is y - 2 = -2(x - 4).

Simplify the equation by distributing and rearranging. The equation is y = -2x + 10. This is the equation of the perpendicular bisector in slope-intercept form.

The maximum modulus is the maximum value of |z| within the given domain.

To find the maximum modulus, we need to find the point(s) within the unit circle where the modulus is the highest.

(a) ƒ(z) = z² - 2z + 3

We can write ƒ(z) as ƒ(z) = (z - 1)² + 2, which is a parabola that opens upwards. The maximum modulus occurs at the vertex, which is located at z = 1, and the maximum modulus is ƒ(1) = 2.

(b) ƒ(z) = z² + z - 1

We can write ƒ(z) as ƒ(z) = (z + 1/2)² - 5/4, which is a parabola that opens upwards. The maximum modulus occurs at the vertex, which is located at z = -1/2, and the maximum modulus is ƒ(-1/2) = 1/4.

(c) ƒ(z) = (z + 1)/(2z - 2)

We can write ƒ(z) as ƒ(z) = 1/2 + 3/(2z - 2), which is a hyperbola that opens downwards. The maximum modulus occurs at the point where the real part of z is 1/2, and the imaginary part of z is 0, which is located at z = 1. The maximum modulus is ƒ(1) = 2.

(d) ƒ(z) = cos(z)

The maximum modulus of cos(z) occurs at z = 0 or z = π, where the modulus is 1.

Therefore, the maximum modulus for each function is:

(a) 2

(b) 1/4

(c) 2

(d) 1

Note: The modulus of a complex number z is defined as |z| = sqrt(x^2 + y^2), where x and y are the real and imaginary parts of z, respectively. The maximum modulus is the maximum value of |z| within the given domain.

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The GCF of 20xyz and 30x^2z is: 2*5*x*z = 10xz

To find the greatest common factor (GCF) of 20xyz and 30x^2z, we can factor out the common factors that they share.

Identifying the common factors in the two monomials:

20xyz = 2 * 2 * 5 * x * y * z

30x^2z = 2 * 3 * 5 * x * x * z

The GCF is the product of the common factors raised to the lowest power that they appear in both monomials.

Therefore, the GCF of 20xyz and 30x^2z is:

2 * 5 * x * z = 10xz

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Step-by-step explanation:

Simple interest = prt÷100

Here,

Principle = 12000

Rate = 6

Time = 7

So to find the simple interest,

You just apply the formula.

[tex]\frac{p \times r \times t}{100} = \frac{12000 \times 6 \times 7}{100} = 5040[/tex]

Interest = 5040

1. The volume of the box is given by:

Volume = length x width x height = 9 in. x 2 in. x 12 in. = 216 cubic inches

So, the answer is 216 cubic inches.

2. The surface area of the box is given by:

Surface Area = 2lw + 2lh + 2wh = 2(9)(12) + 2(9)(2) + 2(2)(12) = 216 + 36 + 48 = 300 square inches

So, the answer is 300 square inches.

3. The surface area of the cereal box is 300 sq. in.

If cardboard costs $0.01 per square inch, then the cost of manufacturing this cereal box would be:

300 sq. in. x $0.01/sq. in. = $3.00

Therefore, the correct answer is -3.00.

4. The volume of the rectangular prism is given by:

Volume = length x width x height = 9 in. x 2 in. x 12 in. = 216 cubic inches

To find the surface area of the sphere, use the formula:

A = 4πr²

Plugging in r = 3.7 in.:

A = 4π(3.7 in.)² ≈ 172.11 square inches

Rounding this to the nearest square inch gives us:

A ≈ 172 square inches

5. The rectangular prism with dimensions of 9 in. x 2 in. x 12 in. would cost less in packaging.

6. The volume of the cylinder is given by:

Volume = πr²h = 390 cubic inches

Solving for r:

r ≈ 4 inches

Therefore, the approximate radius of the cylinder should be 4 inches to hold the same volume of cereal as the rectangular prism.

7. The surface area of the travel-size cereal box is given by:

Surface Area = 2lw + 2lh + 2wh = 2(3)(1.5) + 2(3)(4.5) + 2(1.5)(4.5) = 9 + 27 + 13.5 = 49.5 square inches

To find the length of each side in the cube with the same surface area, we use the formula:

Surface Area of Cube = 6s²

49.5 = 6s²

s ≈ 2.2 inches

So, the length of each side in the cube would be approximately 2.2 inches.

8. The rectangular prism with dimensions of 3 in. x 1.5 in. x 4.5 in. holds more volume than the cube with a side length of approximately 2.2 inches.

9. Some factors that a cereal company may consider in creating their packaging include how well it stacks for shipping and storing on shelves, how easy it is to hold and pour, and how much advertising space there is on the front of the design.

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(a) According to Neyman-Pearson's lemma, the rejection region of the type I error rate (α) for testing H0: θ = θ0 against HA: θ = θA, where θ0 < θA, is given by:

{X: f(x; θA) / f(x; θ0) > k}

where f(x; θ) is the probability mass function (PMF) of the distribution of the data, given the parameter θ, and k is chosen such that the type I error rate is α.

For this problem, we have H0: p = 0 and HA: p > 0.4, with α = 0.05. Therefore, we need to find the value of k such that P(X ∈ R | H0) = α, where R is the rejection region.

Using the fact that X follows a binomial distribution with n trials and success probability p, we have:

f(x; p) = (n choose x) * p^x * (1-p)^(n-x)

Then, the likelihood ratio is:

L(x) = f(x; pA) / f(x; p0) = (pA / p0)^x * (1-pA / 1-p0)^(n-x)

We want to find k such that:

P(X ∈ R | p = p0) = P(L(X) > k | p = p0) = α

Using the distribution of L(X) under H0, we have:

P(L(X) > k | p = p0) = P(X > k') = 1 - Φ(k')

where Φ is the cumulative distribution function (CDF) of a standard normal distribution, and k' is the value of k that satisfies:

(1 - pA / 1 - p0)^(n-x) = k'

k' can be found using the fact that X ~ Bin(n, p0) and P(X > k') = α, which yields:

k' = qbinom(α, n, 1-pA/1-p0)

Therefore, the rejection region R is given by:

R = {X: X > qbinom(α, n, 1-pA/1-p0)}

(b) We have n = 20, p0 = 0.45, pA = 0.65, and X = 11. Using the rejection region R defined in part (a), we have:

R = {X: X > qbinom(0.05, 20, 1-0.65/1-0.45)} = {X: X > 12}

Since X = 11 is not in R, we fail to reject H0 at the 5% level of significance.

(c) The p-value is the probability of observing a test statistic as extreme as the one computed from the data, assuming H0 is true. For this problem, the test statistic is X = 11, and we want to find the probability of observing a value as extreme or more extreme than 11, under the null hypothesis H0: p = 0. Using the binomial distribution with p = 0.45, we have:

p-value = P(X >= 11 | p = 0) = 1 - P(X <= 10 | p = 0)

= 1 - pbinom(10, 20, 0.45)

= 0.151

Therefore, the p-value is 0.151, which is greater than the level of significance α = 0.05, so we fail to reject H0.

The probability that no more than 6 women dislike their mother-in-law is very low, at 0.00002

We can model the number of women who dislike their mother-in-law out of a sample of 9 married women using a binomial distribution with parameters n=9 and p=0.85.

(a)   [tex]P(all 9 women dislike their mother-in-law) = (0.85)^9 = 0.322[/tex]

(b) [tex]P(none of the 9 women dislike their mother-in-law) = (1-0.85)^9 = 0.0001[/tex]

(c) P(at least one woman dislikes her mother-in-law) = 1 - P(none of the 9 women dislike their mother-in-law) = 1 - 0.0001 = 0.9999

(d) P(no more than 6 women dislike their mother-in-law) = P(X <= 6) where X follows a binomial distribution with parameters n=9 and p=0.85. We can use a calculator or binomial distribution table to find:

P(X <= 6) = 0.00002

Therefore, the probability that no more than 6 women dislike their mother-in-law is very low, at 0.00002.

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Answer:

80%

Step-by-step explanation:

Solve for x:

x = ($0.80 / $1.00) * 100 x = 0.8 * 100 x = 80

So, the value of 8 dimes is 80% the value of a dollar.

Hope this helps :)

Pls brainliest...

Answer:

80%

Step-by-step explanation:

a dime is 10 cents and a dollar is 100 cents so 8times 10 is 80 so therefore 8 dimes is 80 cents so its 80% of a dollar

Answer:

If the mean amount of money that students from the university spend on a first date is $100, the probability is 0.026 that a randomly selected group of 32 students from the university would spend a mean of $92.23 or less on their most recent first dates.

Step-by-step explanation:

Therefore, she used 8 - 5 = 3 quarts of paint in the living room.

An English measurement of volume equal to one-quarter gallon is the quart. There are now three different types of quarts in use: the liquid quart, dry quart, and imperial quart of the British imperial system. One litre is about equivalent to each. It is split into four cups or two pints.

Legally, a US liquid gallon (sometimes just referred to as "gallon") is equal to 231 cubic inches, or precisely 3.785411784 litres.  Since a gallon contains 128 fluid ounces, it would require around 16 water bottles, each holding 8 ounces, to fill a gallon.

Here 2 gallons is equivalent to 8 quarts (2 gallons x 4 quarts/gallon = 8 quarts).

Mieko used 1/4 of the paint on her bedroom, which is 1/4 x 8 = 2 quarts.

She used 3 quarts on the hallway, so she used a total of 2 + 3 = 5 quarts on the bedroom and hallway.

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The exponential function of base e whose derivative at x=3 is 2y=e^(ax-b) is y = e^(4x - 5).

To create an exponential function of base e using the digits 1-6 at most one time each, with the condition that its derivative at x=3 is 2.

Given the function y = e^(ax-b), let's find its derivative:

1. Differentiate the function with respect to x: dy/dx = a * e^(ax-b)
2. Plug in x = 3 and set dy/dx = 2: 2 = a * e^(3a-b)

Now, we need to find the values of a and b using the digits 1-6, at most one time each. Let's use a = 4 and b = 5, as they seem reasonable and satisfy the single-use condition:

2 = 4 * e^(12 - 5)
2 = 4 * e^7

Now divide both sides by 4:
2/4 = e^7
1/2 = e^7

So, our exponential function using the digits 1-6 at most one time each and satisfying the given condition is                    y = e^(4x - 5).

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The equation that can be used to calculate Willy the whale's position relative to sea level.

Question Blank 1 of 4; -245, Question Blank 2 of 4; -83, Question Blank 3 of 4; +103, Question Blank 4 of 4; -225

The sea level is the level of the sea, which is taken as the zero mark on the number line.

The level of Willy the whale below sea level = 245 feet

The level Willy the whale descends = 83 feet

The level wheely the whale ascends = 103 feet

Therefore, the level to which Willy the whale starts = -245 feet

The level Willy the whale descends to = (-245 - 83) feet = -328 feet

The level to which Willy the whale then ascends to = -328 feet + 103 feet = -225 feet

The equation to calculate the position of Willy the whale is therefore;

-245 - 83 + 103 = -225

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To find the critical numbers of the function f(x) = x^3e^(-9x), we need to find the values of x where the derivative of the function is equal to zero or does not exist. These values correspond to the relative maxima, minima, or inflection points of the function.

To find the derivative of the function, we can use the product rule and the chain rule of differentiation. The derivative of f(x) is given by:

f'(x) = 3x^2e^(-9x) - 9x^3e^(-9x)

To find the critical numbers, we need to set f'(x) equal to zero and solve for x:

3x^2e^(-9x) - 9x^3e^(-9x) = 0

Factorizing out e^(-9x), we get:

3x^2 - 9x^3 = 0

Simplifying further, we get:

x^2(3 - 9x) = 0

Thus, the critical numbers of the function are x = 0 and x = 1/3. At x = 0, the function has a relative minimum, while at x = 1/3, the function has a relative maximum. To determine the nature of these critical points, we can use the second derivative test or examine the sign of the derivative in the intervals around the critical points.

Overall, finding the critical numbers of a function is an important step in analyzing its behavior and determining its extrema or points of inflection. By setting the derivative equal to zero and solving for x, we can identify the critical points and then use additional tests or analysis to determine their nature and significance.

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