Substitution with Periodic Functions
Find ∫sin(t)/cos^3(t) dt
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The integral of periodic function is -(1/2)cos^(-2)(t) + C or -(1/2)(sec^2(t)) + C.
The integral of the periodic function sin(t)/cos^3(t) dt can be found as follows,
Performing substitution with the function u = cos(t). This means that du = -sin(t) dt.
Rewriting the integral in terms of u. The integral becomes -∫(1/u^3) du.
Integrating with respect to u. The result is -∫u^(-3) du = -(1/2)u^(-2) + C, where C is the constant of integration.
Replacing u with the original function, cos(t).
The final answer is -(1/2)cos^(-2)(t) + C or -(1/2)(sec^2(t)) + C.
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Related Questions
In terms of the water lily population change, the value 3.915 represents:
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Answer:
he initial number of water lily’s :the value 1.106 represents the population increase rateedge
Step-by-step explanation:
The value 3.915 represents the initial population in the exponential equation of water lilies.
What are exponential functions?When the expression of function is such that it involves the input to be present as exponent (power) of some constant, then such function is called exponential function.
There usual form is specified below. They are written in several such equivalent forms.
The standard exponential equation is given as; y = ab^x
where;a is the initial population b is the rate (growth or decline)
We have been Given the exponential equation; y = 3.915(1.106^x)
On comparison, we can see that,
a = 3. 915
Hence, This shows that the value 3.915 represents the initial population of water lilies.
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Given f(x) = 5x + 3, find f(5)
Answers
Answer:
Hey there!
f(x)=5x+3
f(5)=5(5)+3
f(5)=25+3
f(5)=28
Let me know if this helps :)
Answer:
f(5) =28
Step-by-step explanation:
\(f(x) = 5x + 3\\f(5) =?\\\\\)
Substitute 5 for x in the given function
\(f(5) = 5(5) +3\\f(5) = 25+3\\f(5) =28\)
Find the area of the parallelogram with the given vertices. k(1, 1, 3), l(1, 3, 5), m(6, 9, 5), n(6, 7, 3).
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The area of the parallelogram is \(\sqrt{132}\).
What is a parallelogram?A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. Also, the interior angles on the same side of the transversal are supplementary. Sum of all the interior angles equals 360 degrees.
Given that,
vertices k(1, 1, 3), l(1, 3, 5), m(6, 9, 5), and n(6, 7, 3)
Area of Parallelogram = |KL × KN|
KL = (1, 1, 3) - (1, 3, 5) = (0, 2, 2)
KN = (1, 2, 3) - (3, 7, 3) = (2, 5, 0)
Area of the parallelogram = cross product of two vectors, represented by the adjacent sides.
Area of Parallelogram = |(0, 2, 2) × (2, 5, 0)|
= |i(2 × 0 - 5 × 2) - j(0 × 0 - 2 × 2) + k(0 × 5 - 2 × 2)|
= |i(-10) - j(-4) + k(-4)|
= |-10i + 4j - 4k|
= \(\sqrt{(-10)^{2}+(4)^{2}+(-4)^{2} }\)
= \(\sqrt{100+16+16}\)
= \(\sqrt{132}\)
Hence, The area of the parallelogram is \(\sqrt{132}\).
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expand (2a +36) (2a-3b
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Find sin
Please help
Answers
============================================================
Explanation:
The side opposite angle A is BC = 8. This side is as far as you can get from angle A.
The hypotenuse is the longest side, and always opposite the 90 degree angle. The hypotenuse is AC = 10.
We can then say,
sin(angle) = opposite/hypotenuse
sin(A) = BC/AC
sin(A) = 8/10
sin(A) = (4*2)/(5*2)
sin(A) = 4/5 which points to choice B as our answer.
What single percentage change is equivalent to a 12% increase followed by a 14% decrease?
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Answer:
3.68% decrease
Step-by-step explanation:
12% increase = 112% of the original value
112% as a decimal = 112/100 = 1.12
14% decrease = 100% - 14% = 86% of the original value
86% as a decimal = 86/100 = 0.86
Therefore, if you increase the number n by 12% and then decrease by 14%:
n × 1.12 × 0.86 = n × 0.9632
0.9632 as a percentage = 0.9632 × 100 = 96.32%
Therefore, the overall percentage change is 100% - 96.32% = 3.68% decrease
A triangle has side lengths of (9v-7w)(9v−7w) centimeters, (7v+10x)(7v+10x) centimeters, and (8x+5w)(8x+5w) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?
Answers
Perimeter of triangle is P= 18x+16v-2w.
How do triangles work?
A triangle is a polygon with three vertices and three sides. The angles of the triangle are formed by the connection of the three sides end to end at a single point.
The triangle's three angles add up to a total of 180 degrees.
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side lengths of a triangle are (9v-7w),(7v+10x),(8x+5w) centimetres.
We know that perimeter of the triangle is sum of all the three sides.
So here perimeter p= addition of all sides of a triangle.
Therefore ,
P = ( 9v - 7w) + (7v + 10x ) + ( 8x + 5w)
= 9v - 7w + 7v + 10x + 8x + 8w
= 18x + 16v - 2w
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NEED HELP ASAP - Algebra 2
Create your own instructions on how to simplify radical expressions. To accompany your explanation, create your own problem to show the steps in an example.
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To simplify a radical expression, you can use a few different techniques. One common technique is to factor the expression under the radical sign (the radicand) as much as possible.
How to simplify radical expression?Another technique you can use is to use the properties of radicals. For example, you can use the product property of radicals to combine like terms under the radical sign
You can also use the quotient property of radicals to divide out a factor under the radical sign. Finally, some radicals are unsimplifiable by these methods. You can approximate a value of radical or try to get it in a form of fraction with radical.
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What is seventeen HUNDRETHS in standard form?
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Answer:
im pretty sure it is .17
Step-by-step explanation:
Answer:
0.17
Step-by-step explanation:
I'm not entierly sure if there's like an equation or what not to this, but 0.17 is 17 hundreths.
The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = ????????????-1x, the horizontal line y = 3 and the vertical line x = 1. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?
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The volume of the solid is 9 cubic units, which is determined by the base in the first quadrant bounded by the y-axis, the graph of y = x-1, the horizontal line y = 3, and the vertical line x = 1. As the cross sections perpendicular to the x-axis are squares, the volume can be calculated.
The volume of the solid can be calculated by finding the area of the base and then multiplying it by the height of the solid. The base of the solid is in the first quadrant and is bounded by the y-axis, the graph of y = x-1, the horizontal line y = 3, and the vertical line x = 1. This region forms a trapezoid, with an area of 6. As the cross sections perpendicular to the x-axis are squares, the height of the solid is 3 units. Thus, the volume of the solid is 6 multiplied by 3, which gives us a total volume of 9 cubic units. To calculate the volume, it is important to identify the base of the solid and the shape of the cross sections perpendicular to the x-axis. This will allow us to calculate the area of the base, and multiply it by the height of the solid to determine the total volume.
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if f(1) = 12, f ' is continuous, and 6 f '(x) dx 1 = 16, what is the value of f(6)
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To find the value of f(6), we can use the information given about the function f(x) and its derivative f'(x).The value of f(6) is 44/3.
Given that f'(x) is continuous, we can apply the Fundamental Theorem of Calculus. According to the theorem:
∫[a to b] f '(x) dx = f(b) - f(a)
In this case, we are given that:
∫[1 to 6] 6 f '(x) dx = 16
We can simplify the integral:
6 ∫[1 to 6] f '(x) dx = 16
Since f'(x) is the derivative of f(x), the integral of 6 f '(x) dx is equal to 6 f(x). Therefore, we have:
6 f(6) - 6 f(1) = 16
Substituting the given value f(1) = 12:
6 f(6) - 6(12) = 16
6 f(6) - 72 = 16
Next, we isolate the term with f(6):
6 f(6) = 16 + 72
6 f(6) = 88
Finally, we solve for f(6) by dividing both sides by 6:
f(6) = 88 / 6
f(6) = 44/3
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Question 1 help urgently!
Answers
This expression also cannot be simplified to x, which confirms that f and g are not inverses. The answer is False.
Describe Function?In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range) with the property that each input is related to exactly one output. A function is often represented by a formula, graph, or table, and can be used to model real-world situations or solve mathematical problems. Common types of functions include linear functions, quadratic functions, exponential functions, and trigonometric functions. Functions are essential in calculus, which is the study of rates of change and the accumulation of small changes over time.
To check whether the given functions g(x) and f(x) are inverses of each other, we need to verify whether the composition of g and f, and the composition of f and g, both give the identity function.
First, let's find the composition of g and f, which is g(f(x)):
g(f(x)) = g(\(\sqrt[5]{}\)(-x - 3)) = - (\(\sqrt[5]{}\)(-x - 3))⁵⁻³
This expression cannot be simplified to x, which means that g(f(x)) is not equal to the identity function, and therefore g and f are not inverses.
Alternatively, we can also check the composition of f and g, which is f(g(x)):
f(g(x)) = \(\sqrt[5]{}\)(-(-x⁵ - 3) - 3) = \(\sqrt[5]{}\)(-x⁵)
This expression also cannot be simplified to x, which confirms that f and g are not inverses.
Therefore, the answer is False.
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the snellen letter chart is commonly used to test for
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The snellen letter chart is commonly used to test for visual acuity or clarity of vision.
The Snellen letter chart, named after the Dutch ophthalmologist Herman Snellen, is a widely used tool to assess visual acuity or clarity of vision. It consists of rows of letters or optotypes that decrease in size from top to bottom. The chart is typically displayed at a standardized distance of 20 feet (or 6 meters) from the person being tested.
When an individual reads the Snellen chart, the process involves the following steps:
1. Light enters the eye through the cornea, the transparent front surface of the eye.
2. The cornea focuses the incoming light onto the lens, which further refines the light rays.
3. The lens adjusts its shape to ensure that the light is accurately focused onto the retina, the light-sensitive layer at the back of the eye.
4. The retina contains specialized cells called photoreceptors, which convert light into electrical signals.
5. Among the photoreceptor cells, cones are primarily responsible for color vision and visual acuity. When a person focuses on a specific optotype (such as the letter "E" on the Snellen chart), the light reflecting from that optotype enters the eye and forms an image on the retina.
6. The cones in the retina, specifically in the central area called the macula, respond to the light stimulus and generate electrical signals.
7. These electrical signals are then transmitted via the optic nerve to the visual cortex in the brain for further processing and interpretation.
8. The brain processes the received signals and translates them into the perception of the letter "E" or any other optotype being viewed.
The clarity of vision is assessed by determining the smallest line of letters that can be accurately identified. The Snellen chart assigns a visual acuity measurement based on the ratio of the distance at which the chart is viewed and the distance at which a person with normal vision can read the line accurately. For example, if a person can read the 20/20 line from a distance of 20 feet, it indicates that they have normal visual acuity.
The Snellen chart is an essential tool in optometry and ophthalmology for diagnosing and monitoring vision problems such as nearsightedness (myopia), farsightedness (hyperopia), astigmatism, and other visual impairments. It helps determine the level of visual acuity and guides the prescription of corrective lenses or other treatments to improve vision.
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Find the mode for the given data set.
A.) 6
B.) 34
C.) 37
Answers
Answer:
its b 34
Step-by-step explanation:
Answer:
the actual answer is 37. I have the assignement and 37 was correct.
Step-by-step explanation:
If a company's Cost of Goods Sold is $158,000 for the period, beginning and ending Inventory balances are $18,000 and $13,000, respectively, and the beginning and ending Accounts Payable balances are $19,000 and $7,500, respectively, what is the amount of the cash paid to suppliers?
Answers
Answer:
$164500
Step-by-step explanation:
From the given information:
Decrease in inventory = Beginning inventory - Ending inventory
Decrease in inventory = $18000 - $13000
Decrease in inventory = $5000
Decrease in account payable = $19,000 - $7,500
Decrease in account payable = $11500
The amount of cash paid to suppliers is computed as:
Cost of goods sold $158000
Less: Decrease in inventory - 5000
Add: decrease in accounts payable + 11500
The amount of cash paid back to suppliers: $164500
This question is based on the increment decrement of cost of goods sold.
Therefore, the amount of the cash paid to suppliers is $164500.
Given:
Cost of Goods Sold is $158,000. Beginning and ending Inventory balances are $18,000 and $13,000. The beginning and ending accounts payable balances are $19,000 and $7,500.
We have to calculated the amount of the cash paid to suppliers.
Now, we have to calculated the decrement in inventory.
Decrement in inventory = Beginning inventory - Ending inventory
Decrement in inventory = $18000 - $13000
Decrement in inventory = $5000
Now, we have to calculated the decrement in amount payable.
Decrement in Accounts Payable = Beginning Accounts Payable
- Ending Accounts Payable
Decrement in account payable = $19,000 - $7,500
Decrement in account payable = $11500
The amount of cash paid to suppliers is calculated as:
Amount of cash paid to suppliers = Cost of Goods Sold - Decrement in inventory + Decrement in account payable
Amount of cash paid to suppliers = $158,000 - $5000 + $11500
= $164500
Therefore, the amount of the cash paid to suppliers is $164500.
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In each of Problems 1 through 4, draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t → 0. If this behavior depends on the initial value of y at t = 0, describe the dependency.
A. y' = 3 – 2y B. y' = 2y – 3
C. y'= -1 -2y D. y' =1+2y
Answers
The collection of short line segments with a slope that should satisfy the specified differential equation at each point is referred to as the direction field.
The stated differential equation is :
y' = 2y – 3 => dy/dx = 2y – 3
Find equilibrium solution, at dy/dx = 0.
2y – 3 = 0
y = 3/2
At y = 3/2, the intervals are: ]3/2, ∞[
Put any value of y in this differential equation,
2y – 3 at y = 2
2*2 – 3 = 1 > 0
Thus, slope becomes positive for y > 3/2 .
As, the relation between t and the obtained solution y are increases.
Thus, y < 3/2 : intervals are: ]- ∞, 3/2[
Put any value of y in this differential equation,
2y – 3 at y = 1
2*1 – 3 = -1 < 0
Thus, slope becomes negative for y < 3/2 .
As, the relation between t and the obtained solution y in decreases.
Therefore, the direction field for the stated differential equation is obtained.
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the correct question is-
Draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t → 0. If this behavior depends on the initial value of y at t = 0, describe the dependency.
y' = 2y – 3
A computer monitor is 20 inches wide. The aspect ratio, which is the ratio of the width of the screen to the height of the screen, is
16:9
What is the length of the diagonal of the screen, to the nearest whole inch?
Enter your answer in the box.
Answers
Answer:
45/4 inches closer 11
Step-by-step explanation:
16/9 = 20/x
9/4= x/5
45/4 inches = x
11< x< 12
Can someone help me understand how to do this?
Answers
Answer:
The answer is
\(k = 48\)
The rectangular floor of a classroom is 30 feet in length and 20 feet in width. A scale drawing of the floor has a length of 15 inches. What is the area, in square inches, of the floor in the scale drawing?
HURRY PLEASEEEEEEEE
Answers
The area of the floor in the scale drawing will be equal to 150 in².
What is the area?An object's area is how much space it takes up in two dimensions. It is the measurement of the number of unit squares that completely encompass the surface of a compact figure. The squared unit, which is frequently expressed as inches, square feet, etc., is the accepted unit of area.
As per the given information in the question,
Length of the classroom, l = 30 feet and,
The width of the classroom, w = 20 feet.
As per the scale drawing, the length of the classroom, l' = 15 feet
Let the scale width, w' = x feet
As we can see that the original length is twice the scale drawing.
Then, the original width also is twice the scale drawing.
So, w' = 20/2 = 10 feet.
Now, the area of the floor in the scale drawing will be,
A = wl
A = (15)(10)
A = 150 in².
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How does g(x)=1.85x-6 change over a unit interval in x ? g(x) increases by a factor of 1.85 g(x) increases by 1.85 g(x) decreases by 85% g(x) increases by 85%
Answers
The function g(x) = 1.85x - 6 increases by a factor of 1.85 over a unit interval in x.
To determine how the function g(x) = 1.85x - 6 changes over a unit interval in x, we can evaluate the function at two consecutive points within that interval and compare the values.
Let's consider two points, x1 and x2, where x2 is one unit greater than x1. We can calculate the values of g(x) at these points:
g(x1) = 1.85x1 - 6
g(x2) = 1.85x2 - 6
The difference between g(x2) and g(x1) is:
g(x2) - g(x1) = (1.85x2 - 6) - (1.85x1 - 6)
= 1.85x2 - 1.85x1
= 1.85(x2 - x1)
From this calculation, we can see that the difference between g(x2) and g(x1) is equal to 1.85 multiplied by the difference between x2 and x1. This indicates that g(x) increases by a factor of 1.85 over a unit interval in x.
Therefore, the correct answer is that g(x) increases by a factor of 1.85 over a unit interval in x.
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Given that SR//TU, show that Angle SVR is similar Angle UVT.
Answers
Answer:
See below
Step-by-step explanation:
We are given that \(SR ||T U\). Therefore, \(\angle R \cong \angle T\) and \(\angle S \cong \angle U\) because alternate interior angles of parallel lines are congruent. Thus, \(\triangle SVR \sim \triangle UVT\) by the AA Similarity Postulate. Hope this helps!
what is the solution of the equation 6-2^(x)=1
Answers
Answer:
Step-by-step explanation:
6-2^x=1
-2^x+6=1
-2^x+6-6=1-6
-2^x=1-6
-2^x=-5
x=-5
HELP ME PLEASE WHAT IS THE ANSWER OF THE FRACTIONS
Answers
-2 1/5
at least I think so lol
Which expression is equivalent to (9x + 5y)²?
a) 9x² + 45xy + 5y²
b) 18x² + 45xy + 10y²
c) 45x² + 45xy + 45y²
d) 81x² + 90xy + 25y²
e) 0
Answers
If you don’t understand how then plz don’t hesitate to ask me in the comments.
he neighborhood of 1250 people has 30 city blocks. each block is $\frac{1}{16}$ mile long by $\frac{1}{8}$ mile wide. find the population density of the neighborhood in people per square mile.
Answers
The population density of the neighborhood of 1250 people in people per square mile is 1534 people per square mile
What is the population density?Population density is the number of people per unit area of a location.
The population density is the number of people per square mile in the neighborhood
The population of the neighborhood = 1,250 people
The number of city blocks in the neighborhood = 30
The length of each city block = (1/16) mile
The width of each neighborhood = (1/8) mile
The area of each neighborhood = (1/16) × (1/8) = 1/128 square miles
The area of the whole neighborhood = 30 × 1/128 = 15/64 square miles
The population density = 1250/(15/64) = 16,000/3 ≈ 5334 people per square mile (rounding up)
The number of people per square mile, the population density, therefore is 5333 people per square mile
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a psychologist is studying the self image of smokers, which she measures by the self-image (si) score from a personality inventory. she would like to estimate the mean si score, , for the population of all smokers. she plans to take a random sample of si scores for smokers and estimate via this sample. assuming that the standard deviation of si scores for the population of all smokers is , what is the minimum sample size needed for the psychologist to be confident that her estimate is within of ? carry your intermediate computations to at least three decimal places. write your answer as a whole number (and make sure that it is the minimum whole number that satisfies the requirements).
Answers
To calculate the minimum sample size needed, we can use the formula: n = (z * σ / E)^2
Where:
z = the z-score associated with the desired level of confidence (let's assume 95% confidence, so z = 1.96)
σ = the standard deviation of the population (given in the problem statement)
E = the maximum error margin (given in the problem statement)
Plugging in the values, we get:
n = (1.96 * σ / E)^2
To determine the value of σ/E, we need more information. Let's assume that the maximum error margin E is 0.5, which means that the psychologist wants to be within 0.5 points of the true population mean si score.
Now, let's say that σ = 10 (just as an example). Then, σ/E = 10/0.5 = 20.
Plugging this into the formula, we get:
n = (1.96 * 10 / 0.5)^2 = 384.16
Rounding up to the nearest whole number, the minimum sample size needed is 385.
Therefore, the psychologist would need to take a random sample of at least 385 si scores from smokers to be confident that her estimate of the population mean si score is within 0.5 points.
Learn more about Standard Deviation here:- brainly.com/question/475676
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What is the solution of x2+49 = x+5?
= x +
12
X=
ol
f
12
O X=--
x = -6 or x = -3
no solution
Answers
Answer:
x = 12/5
Step-by-step explanation:
take the square of both sides
(x^2+49) = (x+5)^2
x^2 + 49 = (x+5) (x+5)
x^2+49= x^2 + 10x + 25
10x + 25 = 49
10x = 24
x = 12/5
pleaseee help with this question!!
Answers
A professor must randomly select 4 students to participate in a mock debate.
There are 20 students in his class. In how many different ways can these
students be selected, if the order of selection does not matter?
Answers
Answer:
Step-by-step explanation:
We can use combinations
20 choose 4:
20!/(16!*4!)
= (20*19*18*17)/4!
= 4845
Please help!
(at the start it says "Albert's mom bought" but it was accidentally cropped out...sorry)
Answers
Answer:
Option D
Step-by-step explanation:
We convert 5 2/3 into an improper one:
17/3
There are 5 people in his family
17/3 * 1/5 = 17/15
So option D