Answers
The derivatives of the functions are listed below:
(a) [tex]f'(x) = -7\cdot x^{-\frac{9}{2} }- 2\cdot x + 4 - \frac{1}{5} - 5\cdot x^{-2}[/tex]
(b) [tex]f'(x) = \frac{1}{3}\cdot (x + 3)^{-\frac{2}{3} }\cdot (x+ 5)^{\frac{1}{3} } + \frac{1}{3} \cdot (x + 5)^{-\frac{2}{3} } \cdot (x + 3)^{\frac{1}{3} }[/tex]
(c) f'(x) = [(cos x + sin x) · (x² - 1) - (sin x - cos x) · (2 · x)] / (x² - 1)²
(d) f'(x) = (5ˣ · ㏑ 5) · ㏒₅ x + 5ˣ · [1 / (x · ㏑ 5)]
(e) f'(x) = 45 · (x⁻⁵ + √3)⁻⁸ · x⁻⁶
(f) [tex]f'(x) = (\ln x + 1)\cdot [7^{x\cdot \ln x \cdot \ln 7}+7\cdot (x\cdot \ln x)^{6}][/tex]
(g) [tex]f'(x) = -2\cdot \arccos x \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} \right) - \left(\frac{1}{1 + x} \right) \cdot \left(\frac{1}{2} \cdot x^{-\frac{1}{2} }\right)[/tex]
(h) f'(x) = cot x + cos (㏑ x) · (1 / x)
How to find the first derivative of a group of functions
In this question we must obtain the first derivatives of each expression by applying differentiation rules:
(a) [tex]f(x) = 2 \cdot x^{-\frac{7}{2} } - x^{2} + 4 \cdot x - \frac{x}{5} + \frac{5}{x} - \sqrt[11]{2022}[/tex]
[tex]f(x) = 2 \cdot x^{-\frac{7}{2} } - x^{2} + 4 \cdot x - \frac{x}{5} + \frac{5}{x} - \sqrt[11]{2022}[/tex] Given[tex]f(x) = 2 \cdot x^{-\frac{7}{2} } - x^{2} + 4\cdot x - \frac{x}{5} + 5 \cdot x^{-1} - \sqrt[11]{2022}[/tex] Definition of power[tex]f'(x) = -7\cdot x^{-\frac{9}{2} }- 2\cdot x + 4 - \frac{1}{5} - 5\cdot x^{-2}[/tex] Derivative of constant and power functions / Derivative of an addition of functions / Result(b) [tex]f(x) = \sqrt[3]{x + 3} \cdot \sqrt[3]{x + 5}[/tex]
[tex]f(x) = \sqrt[3]{x + 3} \cdot \sqrt[3]{x + 5}[/tex] Given[tex]f(x) = (x + 3)^{\frac{1}{3} }\cdot (x + 5)^{\frac{1}{3} }[/tex] Definition of power[tex]f'(x) = \frac{1}{3}\cdot (x + 3)^{-\frac{2}{3} }\cdot (x+ 5)^{\frac{1}{3} } + \frac{1}{3} \cdot (x + 5)^{-\frac{2}{3} } \cdot (x + 3)^{\frac{1}{3} }[/tex] Derivative of a product of functions / Derivative of power function / Rule of chain / Result(c) f(x) = (sin x - cos x) / (x² - 1)
f(x) = (sin x - cos x) / (x² - 1) Givenf'(x) = [(cos x + sin x) · (x² - 1) - (sin x - cos x) · (2 · x)] / (x² - 1)² Derivative of cosine / Derivative of sine / Derivative of power function / Derivative of a constant / Derivative of a division of functions / Result(d) f(x) = 5ˣ · ㏒₅ x
f(x) = 5ˣ · ㏒₅ x Givenf'(x) = (5ˣ · ㏑ 5) · ㏒₅ x + 5ˣ · [1 / (x · ㏑ 5)] Derivative of an exponential function / Derivative of a logarithmic function / Derivative of a product of functions / Result(e) f(x) = (x⁻⁵ + √3)⁻⁹
f(x) = (x⁻⁵ + √3)⁻⁹ Givenf'(x) = - 9 · (x⁻⁵ + √3)⁻⁸ · (- 5) · x⁻⁶ Rule of chain / Derivative of sum of functions / Derivative of power function / Derivative of constant functionf'(x) = 45 · (x⁻⁵ + √3)⁻⁸ · x⁻⁶ Associative and commutative properties / Definition of multiplication / Result(f) [tex]f(x) = 7^{x\cdot \ln x} + (x \cdot \ln x)^{7}[/tex]
[tex]f(x) = 7^{x\cdot \ln x} + (x \cdot \ln x)^{7}[/tex] Given[tex]f'(x) = 7^{x\cdot\ln x} \cdot \ln 7 \cdot (\ln x + 1) + 7\cdot (x\cdot \ln x)^{6}\cdot (\ln x + 1)[/tex] Rule of chain / Derivative of sum of functions / Derivative of multiplication of functions / Derivative of logarithmic functions / Derivative of potential functions [tex]f'(x) = (\ln x + 1)\cdot [7^{x\cdot \ln x \cdot \ln 7}+7\cdot (x\cdot \ln x)^{6}][/tex] Distributive property / Result(g) [tex]f(x) = \arccos^{2} x - \arctan (\sqrt{x})[/tex]
[tex]f(x) = \arccos^{2} x - \arctan (\sqrt{x})[/tex] Given[tex]f'(x) = -2\cdot \arccos x \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} \right) - \left(\frac{1}{1 + x} \right) \cdot \left(\frac{1}{2} \cdot x^{-\frac{1}{2} }\right)[/tex] Derivative of the subtraction of functions / Derivative of arccosine / Derivative of arctangent / Rule of chain / Derivative of power functions / Result(h) f(x) = ㏑ (sin x) + sin (㏑ x)
f(x) = ㏑ (sin x) + sin (㏑ x) Givenf'(x) = (1 / sin x) · cos x + cos (㏑ x) · (1 / x) Rule of chain / Derivative of sine / Derivative of natural logarithm /Derivative of addition of functions f'(x) = cot x + cos (㏑ x) · (1 / x) cot x = cos x / sin x / ResultTo learn more on derivatives: https://brainly.com/question/23847661
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Related Questions
How does the area below the mean compare to the area above the mean in a normal distribution?
Answers
The area below the mean compares to the area above the mean in a normal distribution as the areas are always equal regardless of the mean. Option A This is further explained below.
What is a normal distribution?Generally, The normal distribution, also known as the Gaussian distribution, is a kind of probability distribution that is symmetric around the mean. This means that it demonstrates that data that are closer to the mean are more likely to occur than data that are farther away from the mean. When represented graphically, the normal distribution takes the shape of a "bell curve."
In conclusion, In a normal distribution, the area below the mean is compared to the area above the mean since the areas are always equal regardless of the mean. This is true even if the mean is different.
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Complete Question
How does the area below the mean compare to the area above the mean in a normal distribution?
A. the areas are always equal regardless of the mean
B. the areas are sometimes equal depending upon the standard deviation of the distribution
C. the area above the mean is larger since the values are larger as you move above the mean
D. the areas are sometimes equal depending upon the value of the mean
Take 4x + 2 from 8x + 5.
4x + 3
4x - 3
-4x - 3
Answers
Answer:
4x + 3 just take away rhe values
Find P sophomore girl hint: p(a & b)/p(b)
Answers
Answer:
7/18
Step-by-step explanation:
P(sophomore girl) is 7 out of a total 30.
P(girl) is 18 out of 30.
(7/30) / (18/30) = 7/18
If you just look at the girls column you can see this immediately. A given is that we're looking for a girl (18 girls), then what is the chance that it's a sophomore. That is 7 out of 18.
Answer:
The answer would be 7 over 18 in fraction form fill in 7 then 18
Step-by-step explanation:
7/18
Which of the following statements is true?
0.8 <0.2 < 0.9 <0.801
0.2 <0.8 <0.9 <0.801
0.2 0.8 0.801 0.9
0.2 0.9 0.8 0.801
Answers
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
[tex]\: \sf \:0.2 < 0.8[/tex][tex] \: \sf \:0.8 < 0.801[/tex][tex] \: \sf \:0.801 < 0.9[/tex]Combining the above inequalities, we get :
[tex]\qquad❖ \: \sf \:0.2 < 0.8 < 0.801 < 0.9[/tex]
[tex] \qquad \large \sf {Conclusion} : [/tex]
Option C is correctIn a mathematics class, 24 students received an A on the third test, which is 150% of the students who received an A on the second test. How many students received an A on the second test
Answers
Answer: 16
Step-by-step explanation:
1.5x = 24
24/1.5 = 16 students.
Which of the following is a proportion?
Answers
The option that is a proportion is: B. 4/6 = 2/3.
What is a Proportion?A proportion can be defined as an equation whereby two ratios are set equal to each other, in such a way that the ratio on one side equals the ratio on the other side of the equation when simplified.
First Option is not a proportion because:
5/7 ≠ 10/12 (10/12 can be simplified further as 5/6, which is not equal to 5/7).
Second option is a proportion because:
4/6 = 2/3
8/12 = 2/3
Thus, 4/6 = 8/12.
Third Option is not a proportion because:
14/21 = 2/3
9/12 = 3/4
Therefore, 14/21 ≠ 9/12.
Fourth Option is not a proportion because:
9/15 = 3/5
12/18 = 2/3
Therefore, 9/15 ≠ 12/18.
In conclusion, the option that is a proportion is: B. 4/6 = 2/3.
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Quadrilateral WXYZ has coordinates W(−3, 2), X(1, 3), Y(2, −2), and Z(−1, −2). What are the coordinates of the vertices of D2(WXYZ)?
Answers
the coordinates of the vertices of D2(WXYZ) are;
W' ( 2, 3)
X' ( 3, -1)
Y' ( -2, -2)
Z'(−1, −2)
How to determine the coordinatesIt is important to note that the general formula for finding the transformation of the coordinates is;
M ( h, k) = M' ( k, -h)
Where
h is the first coordinatesk is the second coordinatesFor the quadrilateral WXYZ we have to find the transformed coordinates
For W, we have
W(−3, 2)
h = -3
k = 2
Substitute the values
W' ( 2, 3)
For X,
X(1, 3)
h = 1
k = 3
Substitute the values
X' ( 3, -1)
For Y
Y(2, −2)
h = 2
k = -2
Substitute the values
Y' ( -2, -2)
For Z
Z(−1, −2)
h = -1
k = -2
Z'(−1, −2)
Thus, the coordinates of the vertices of D2(WXYZ) are;
W' ( 2, 3)
X' ( 3, -1)
Y' ( -2, -2)
Z'(−1, −2)
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Convert ( 4 , 300 ∘ ) to rectangular form.
Answers
The rectangular representation of the polar point of (4 , 300) is (2,- 2√3)
According to the statement
we have given a coordinates of the rectangle and we have to find the polar coordinates.
So, For this purpose, we know that the
We Use the conversion formulas to convert from polar coordinates to rectangular coordinates which are
x = rcosθ
y = rsinθ
Substitute the given values in it then
x=(4)cos(300)
y=(4)sin(300)
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
x=(4)cos(60) -(1)
y= - (4)sin(60) -(2)
And then
x=(4)cos(60)
x=(4)(1/2)
x = 2 -(3)
and
y= - (4)sin(60)
y= - (4)(√3/2)
y= - 2√3 -(4)
Replace (3) with (1) and (4) with (2)
then it becomes
x = 2 and y= - 2√3
The rectangular representation of the polar point of (4 , 300) is (2,- 2√3)
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Hey guys I need some help with this question so if anyone could help that would be great THANK YOU!!
Answers
The left hand derivative of the given function comes out to be 3a² + 3ah + h².
Deducing the Left Derivative:
The given function is,
f(x) = x³ + 2
⇒ f(a) = a³ + 2
The left hand limit is the definition of the left-hand derivative of f: f′⁻(x) = [tex]lim_{h- > 0}[/tex]f(x+h)f(x)h. F is said to be left-hand differentiable at x if the left-hand derivative exists.
Now, the formula for the left derivative of a function is given as,
f'(a)⁻ = [ f(a+h) - f(a) ] / [ (a+h) - a]
f'(a)⁻ = [ ((a+h)³ + 2) - (a³+2) ] / h
f'(a)⁻ = (a³ + 3a²h + 3ah² + h³ + 2 - a³ - 2) / h
f'(a)⁻ = (3a²h + 3ah² + h³) / h
f'(a)⁻ = h(3a² + 3ah + h²) / h
f'(a)⁻ = 3a² + 3ah + h²
Hence, the left derivative is 3a² + 3ah + h².
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Find the value of z.
X
7
Y
3
Z
Z = √[?]
Answers
Answer:
z = [tex]\sqrt{30}[/tex]
Step-by-step explanation:
using the Altitude- on- Hypotenuse theorem
(leg of big Δ )² = (part of hypotenuse below it ) × (whole hypotenuse)
z² = 3 × (3 + 7) = 3 × 10 = 30 ( take square root of both sides )
z = [tex]\sqrt{30}[/tex]
PLEASE ANSWER ASAP! Tysm in advance
Answers
Answer:
Step-by-step explanation:
8 adults means 48 children
11 adults means 66 children
15 adults means 120 children
Teresa bought a new desktop computer. One side of the desktop screen is 14 inches and the other side is 18 inches. What is the length of the diagonal of the desktop screen
Answers
Given the width and length of Teresa's new desktop computer, the length of the diagonal of the desktop screen is approximately 22.8 inches.
What is the length of the diagonal of the desktop screen?If a diagonal line cuts through a rectangle, it forms two equal right triangles. the side lengths of this triangle can be easily determined using Pythagoras theorem. Pythagoras theorem is expressed as;
c² = a² + b²
Where c is the hypotenuse or diagonal, a is base length and b is perpendicular height.
Given the data in the question;
Perpendicular height b = 14inBase length a = 18inHypotenuse or Diagonal c = ?We substitute into the equation above.
c² = a² + b²
c² = (18in)² + (14in)²
c² = 324in² + 196in²
c² = 520in²
c = √( 520in² )
c = 22.8in
Given the width and length of Teresa's new desktop computer, the length of the diagonal of the desktop screen is approximately 22.8 inches.
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Which features are present
in this polar graph?
Answers
The features that exist present in this polar graph is that D. Symmetry about the polar axis θ = 0.
What is polar graph?A polar curve exists as a form constructed utilizing the polar coordinate system. Polar curves exists determined by points that exist at a variable distance from the origin (the pole) depending on the angle calculated off the positive x-axis.
The polar graph can be said to contain a symmetry about the polar axis which indicates that theta or θ exists equivalent to 0.
This exists because the graph is symmetric about the x-axis. This indicates that all the lines of symmetry can be seen intersecting the graph at x = 0. This demonstrates that there exists indeed symmetry in the polar graph.
Therefore, the correct answer is option D. symmetry about the plane θ = 0
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Part A: In complete sentences, explain the relationships between all pairs of special angles 1, 2, 3, and 4 created by transversal line b and parallel lines d and e.
Answers
The relationship between all pairs of special angles 1, 2, 3, and 4 created by transversal line b and parallel lines d and e include:
Angle 4 and 3 would be considered equal because they are alternative interior angles.Angle 1 and 2 are supplementary to each other i.e sum of their angle is 180 degrees.Angles 1 and 3 are vertical angles thereby making them equal.What is an Angle?These are usually formed when two straight lines meet at a common endpoint or vertex.
Angles 3 and 4 are equal due to them being alternative interior angles and other relationships are mentioned above.
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Numbers from 1 to 50.
Find the probability of choosing
numbers with last digit 6.
Answers
0.1 or 10%
Step-by-step explanation:The numbers between 1 and 50 that end in 6 are as follows:
6, 16, 26, 36, 46
There are therefore 5 numbers between 1 and 50 that end in 6.
This means there are 5 out of 50, or 5/50.
5/50 is 0.1 as a decimal, or 10% in percentage form.
A pair of shoes is reduced by 30% down to a price of £56
What was the original price of the shoes?
Answers
Answer:
To know the final result we have to make 30% of 56 dollars and add the result to the 56 dollars to know the original price.
Operations:
30% de 56 = (30x56)/100= 16,8 dollars.
56+16,8= 72,8 dollars (final result)
A pair of shoes is reduced by 30% down to a price of £56 then the original price of the pair of shoes was £80.
To find the original price of a pair of shoes using a mathematical approach. We are given that the shoes were reduced by 30% and their final price is £56. We will use algebraic equations to determine the original price of the shoes.
Let's denote the original price of the shoes as "P". Since the shoes were reduced by 30%, we can express this reduction as a decimal, which is 0.30 (30% = 30/100 = 0.30).
The amount of reduction is then given by 0.30 multiplied by the original price, which is 0.30P.
The final price of the shoes after the reduction is £56. We can set up an equation to represent this:
P - 0.30P = 56
Now, let's simplify the equation:
0.70P = 56
Next, we need to isolate "P" on one side of the equation. To do this, we divide both sides by 0.70:
[tex]\[ \frac{{0.70P}}{{0.70}} = \frac{{56}}{{0.70}} \][/tex]
This simplifies to:[tex]\[ P = \frac{{56}}{{0.70}} \][/tex]
Now, let's calculate the value of "P":
[tex]\[ P = \frac{{56}}{{0.70}} = 80 \][/tex]
So, the original price of the pair of shoes was £80.
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2×7×7-3×3×3×2+3×2×2-6×6
I need an answer
Answers
Answer:
98-54+12-36
110-54-36
56-36
20Ans
X
-5
Probability 17
-3
-2 0
0 2
13 33 16 11
3
.10
Find the probability that x <-3
Answers
The value of the probability is 0.30
How to determine the probability?Using the table of values, we have:
P(x <= -3) = P(x = -5) + P(x = -3)
From the table of values, we have:
P(x = -5) = 0.17
P(x = -3) = 0.13
Substitute the known values in the above equation
P(x <= -3) = 0.17 + 0.13
Evaluate the sum
P(x <= -3) = 0.30
Hence, the value of the probability is 0.30
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please help me figure this out
Answers
Answer:
-25
Step-by-step explanation:
[tex] \dfrac{(20 - 5^2)(16 + 2^2)}{-2^3 + (3 \times 2^2)} = [/tex]
First, do all exponents.
[tex] = \dfrac{(20 - 25)(16 + 4)}{-8 + (3 \times 4)} [/tex]
Now do each operation in parentheses.
[tex] = \dfrac{(-5)(20)}{-8 + 12} [/tex]
Multiply in the numerator. Add in the denominator.
[tex] = \dfrac{-100}{4} [/tex]
Divide the numerator by the denominator.
[tex] = -25 [/tex]
Answer:
-25
Step-by-step explanation:
PEMDAS
The PEMDAS rule is an acronym representing the order of operations in math:
ParenthesesExponentsMultiplication and Division (from left to right)Addition and Subtraction (from left to right)Given expression:
[tex]\sf \dfrac{(20-5^2)(16+2^2)}{-2^3+(3 \times 2^2)}[/tex]
As the given expression is a fraction, carry out the operations in the numerator and denominator first before finally dividing them.
Following PEMDAS, carry out the calculations inside the parentheses first, then carry out the rest of the calculations following the order of operations:
Parentheses
Calculate the exponents inside the parentheses:
[tex]\implies \sf \dfrac{(20-25)(16+4)}{-2^3+(3 \times 4)}[/tex]
Multiply:
[tex]\implies \sf \dfrac{(20-25)(16+4)}{-2^3+(12)}[/tex]
Add and subtract:
[tex]\implies \sf \dfrac{(-5)(20)}{-2^3+(12)}[/tex]
Exponents
Calculate the exponent:
[tex]\implies \sf \dfrac{(-5)(20)}{-8+(12)}[/tex]
Multiply and Divide
Multiply:
[tex]\implies \sf \dfrac{-100}{-8+(12)}[/tex]
Add and Subtract
Add:
[tex]\implies \sf \dfrac{-100}{4}[/tex]
Finally, divide the numerator by the denominator:
[tex]\implies \sf -25[/tex]
Twelve different video games showing drugs were observed. The duration times of drugs were recorded, with the times (seconds) listed below. Assume that these sample data are used with a 0.01 significance level in a test of the claim that the population mean is greater than 75 sec. If we want to construct a confidence interval to be used for testing that claim, what confidence level should be used for a confidence interval? If the confidence interval is found to be -34.1 sec < μ < 238.3 sec, what should we conclude about the claim?
88 15 537 53 0 52 197 40 182 0 2 59
1.) The confidence level should be _____%
2.) What should we conclude about the claim?
The given confidence interval __(contains / does not contain)___ the value of 75 sec, so there ___( is / is not )___ sufficient evidence to support the claim that the mean is greater than 75 sec.
_____________________________________________
NOTE: Please explain like I'm five. I'm not understanding why the confidence level should be anything but 90% and I don't know *why* we would conclude what we would conclude about this claim.
Answers
The answers to the questions are:
1. The confidence level is 99 percent.
2. We have to conclude that there is no sufficient evidence available to support this claim because the Confidence interval contains 75 sec.
How to solve for the confidence level1. The confidence level here should be
1- 0.01 = 0.99
= 99 percent
Given that, 99% confidence interval for population mean (μ) is (-34.1 sec u< u < 264.1 ) seconds.
We are to test the claim that the population mean is greater than 75 sec.
2.
The given confidence interval contains the value of 75 sec, so there is not sufficient evidence to support the claim that the mean is greater than 75 sec.
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Ritchie got a job at the movie theater. On the first day, he sold 5 adult tickets and 7 children tickets for a total of 104. On the second day he sold 7 adult tickets and 3 children tickets for a total of 98. What is the price for each ticket?
Answers
Answer:
A = $11; C = $7 tickets
Step-by-step explanation:
Using the given info, we can create a system of equations to find the price of adult and children tickets.
Thus, we have 5A + 7C = 104 and 7A + 3C = 98
The easiest method to solve would be elimination:
Answer:
adult: $11children: $7Step-by-step explanation:
The sales on the two days can be expressed using equations that can be solved for ticket prices.
SetupLet x and y represent the prices of adult and children's tickets, respectively. Then the sales revenue for the two days can be expressed in the equations ...
5x +7y = 1047x +3y = 98SolutionOne of the easiest solution methods is to use a graphing calculator. The first attachment shows the prices are $11 for an adult ticket; $7 for children tickets.
Using the matrix functions of a calculator, the augmented matrix of the equation coefficients can be reduced to row-echelon form. This, too, shows the solution to be (adult price, children price) = ($11, $7). See the second attachment. (The solution is the right-most column of the reduced matrix.)
Yet another solution method can use the coefficients from the equations written in general form:
5x +7y -104 = 07x +3y -98 = 0In this form, we define three products of "cross multiplication":
Δ1 = (5)(3) -(7)(7) = -34
Δ2 = (7)(-98) -(3)(-104) = -374
Δ3 = (-104)(7) -(-98)(5) = -238
Using those, we find the variable values to be ...
x = Δ2/Δ1 = -374/-34 = 11
y = Δ3/Δ1 = -238/-34 = 7
(adult price, children price) = ($11, $7)
__
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Live fund are selling their holdings at $5000 per share. How much would live fun receive if they sold half their holdings in Marx brothers?
Answers
Answer:
2500
Step-by-step explanation:
5000 ÷ 2 = 2500
Factor.
3x² +7x
I don’t know what to do
Answers
Answer:
Most you can do is factor out the x and turn it into x (3x + 7)
roots would be x = 0 and x = -7/3
Answer: x(3x+7)
Step-by-step explanation:
You would factor the x out of both of your values and put on the outside of the parenthesis. And you put the two numbers that you have left inside of the parenthesis. And that is as far down as this function can be factored.
helpppppp pleaseee asap
tysm
Answers
Answer:
(4,8)
hi hi hi hi hi hi hi hi
Which of the following represents all solutions to the equation
1/3x^2 +10 = 7
Answers
Answer:
1 +-3i
Step-by-step explanation:
Answer:
answer is 1) x=±3i
Step-by-step explanation:
An electrician plans to install solar panels on a rectangular section of roof with an area 180m2. This width of this section of roof is 7 1/5 m across. What is the length of this section of roof?
Answers
The length of the rectangular section of the roof is 25 m
Calculating areaFrom the question, we are to determine the length of the section of the roof
From the given information,
The area of the rectangular section = 180 m²
The width of the rectangular section = 7 1/5 m = 7.2 m
Using the formula for area of a rectangle
A = l × w
Where A is the area
l is the length
and w is the width
Then,
180 = l × 7.2
l = 180/7.2
l = 25 m
Hence, the length of the rectangular section of the roof is 25 m
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Answer:
25
Step-by-step explanation:
Just did it on Khan Academy.
Consider the statements below. Which are propositions? Mark all that apply.
2+2=7
A giant spider with hairy legs.
There are more men than women at BYU-Idaho.
2+2=4
There are more women than men at BYU-Idaho.
Ron hates spiders.
Are you tired today?
Answers
The following statements are considered to be propositions:
2 + 2 = 7There are more men than women at BYU-Idaho.2 + 2 = 4Ron hates spiders.What is deductive reasoning?Deductive reasoning can be defined as a type of logical reasoning that typically involves drawing conclusions based on a given set of rules and conditions or from one or more premises (factual statements) that are assumed to be generally (universally) true.
What is a proposition?A proposition can be defined as a type of statement (assertion) that is typically used to express an opinion or a judgement, with either a true or false answer.
This ultimately implies that, a proposition refers to a type of statement (assertion) that is either a true or false.
In this context, we can infer and logically deduce that the following statements are considered to be propositions:
2 + 2 = 7There are more men than women at BYU-Idaho.2 + 2 = 4Ron hates spiders.Read more on propositions here: https://brainly.com/question/24158168
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Jake invested his whole life savings today in an investment that pays 6% interest, compounded annually. In ten years, this investment will be worth $531,402. What is Jake's life savings today?
Answers
Answer:
$ 296 732 .10
Step-by-step explanation:
FV = PV ( 1 + i)^n FV = future value = 531402 PV = ?
i = decimal interest per period = .06
n = periods = 10
531 402 = PV ( 1.06)^10 Solve for PV = 296 732 .10
Z^4-5(1+2i)z^2+24-10i=0
Find the value of z.
Can someone please help me with this one?
Answers
Using the quadratic formula, we solve for [tex]z^2[/tex].
[tex]z^4 - 5(1+2i) z^2 + 24 - 10i = 0 \implies z^2 = \dfrac{5+10i \pm \sqrt{-171+140i}}2[/tex]
Taking square roots on both sides, we end up with
[tex]z = \pm \sqrt{\dfrac{5+10i \pm \sqrt{-171+140i}}2}[/tex]
Compute the square roots of -171 + 140i.
[tex]|-171+140i| = \sqrt{(-171)^2 + 140^2} = 221[/tex]
[tex]\arg(-171+140i) = \pi - \tan^{-1}\left(\dfrac{140}{171}\right)[/tex]
By de Moivre's theorem,
[tex]\sqrt{-171 + 140i} = \sqrt{221} \exp\left(i \left(\dfrac\pi2 - \dfrac12 \tan^{-1}\left(\dfrac{140}{171}\right)\right)\right) \\\\ ~~~~~~~~~~~~~~~~~~~~= \sqrt{221} i \left(\dfrac{14}{\sqrt{221}} + \dfrac5{\sqrt{221}}i\right) \\\\ ~~~~~~~~~~~~~~~~~~~~= 5+14i[/tex]
and the other root is its negative, -5 - 14i. We use the fact that (140, 171, 221) is a Pythagorean triple to quickly find
[tex]t = \tan^{-1}\left(\dfrac{140}{171}\right) \implies \cos(t) = \dfrac{171}{221}[/tex]
as well as the fact that
[tex]0<\tan(t)<1 \implies 0along with the half-angle identities to find
[tex]\cos\left(\dfrac t2\right) = \sqrt{\dfrac{1+\cos(t)}2} = \dfrac{14}{\sqrt{221}}[/tex]
[tex]\sin\left(\dfrac t2\right) = \sqrt{\dfrac{1-\cos(t)}2} = \dfrac5{\sqrt{221}}[/tex]
(whose signs are positive because of the domain of [tex]\frac t2[/tex]).
This leaves us with
[tex]z = \pm \sqrt{\dfrac{5+10i \pm (5 + 14i)}2} \implies z = \pm \sqrt{5 + 12i} \text{ or } z = \pm \sqrt{-2i}[/tex]
Compute the square roots of 5 + 12i.
[tex]|5 + 12i| = \sqrt{5^2 + 12^2} = 13[/tex]
[tex]\arg(5+12i) = \tan^{-1}\left(\dfrac{12}5\right)[/tex]
By de Moivre,
[tex]\sqrt{5 + 12i} = \sqrt{13} \exp\left(i \dfrac12 \tan^{-1}\left(\dfrac{12}5\right)\right) \\\\ ~~~~~~~~~~~~~= \sqrt{13} \left(\dfrac3{\sqrt{13}} + \dfrac2{\sqrt{13}}i\right) \\\\ ~~~~~~~~~~~~~= 3+2i[/tex]
and its negative, -3 - 2i. We use similar reasoning as before:
[tex]t = \tan^{-1}\left(\dfrac{12}5\right) \implies \cos(t) = \dfrac5{13}[/tex]
[tex]1 < \tan(t) < \infty \implies \dfrac\pi4 < t < \dfrac\pi2 \implies \dfrac\pi8 < \dfrac t2 < \dfrac\pi4[/tex]
[tex]\cos\left(\dfrac t2\right) = \dfrac3{\sqrt{13}}[/tex]
[tex]\sin\left(\dfrac t2\right) = \dfrac2{\sqrt{13}}[/tex]
Lastly, compute the roots of -2i.
[tex]|-2i| = 2[/tex]
[tex]\arg(-2i) = -\dfrac\pi2[/tex]
[tex]\implies \sqrt{-2i} = \sqrt2 \, \exp\left(-i\dfrac\pi4\right) = \sqrt2 \left(\dfrac1{\sqrt2} - \dfrac1{\sqrt2}i\right) = 1 - i[/tex]
as well as -1 + i.
So our simplified solutions to the quartic are
[tex]\boxed{z = 3+2i} \text{ or } \boxed{z = -3-2i} \text{ or } \boxed{z = 1-i} \text{ or } \boxed{z = -1+i}[/tex]
Someone please help me with this!!!
Answers
Answer:
279/1640
Step-by-step explanation:
The relations between the various trig functions can be used to find the necessary function values.
Other trig functionsSolving for cos(θ), we have ...
41 cos(θ) = 9
cos(θ) = 9/41 . . . . . divide by 41
The Pythagorean relation tells you ...
sin²(θ) +cos²(θ) = 1 ⇒ sin(θ) = √(1 -cos²(θ))
sin(θ) = √(1 -(9/41)²) = (√(41² -9²))/41 = 40/41
The tangent relation is ...
tan(θ) = sin(θ)/cos(θ) = (40/41)/(9/41) = 40/9
Expression value
The value of the given expression is ...
(sin(θ) -cos(θ))/tan(θ) = (40/41 -9/41)/(40/9) = (31/41)(9/40) = 279/1640
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Additional comment
This can also be figured by a suitable calculator or spreadsheet. Output formatted as a fraction is often an option. Here, the result is rational.