Which of the following is the best linear approximation for f(x) = cos(x) near x= π/2
The best linear approximation for f(x) = cos(x) near x = π/2 is L(x) = π/2 - x. This conclusion is reached by finding the slope of the tangent line to the function at x = π/2 (which is the cosine derivative at this point), and then applying the formula for linear approximation.
Explanation:The linear approximation of a function at a certain point is the line tangent to the function at that point. In this case, we want the linear approximation for f(x) = cos(x) near x = π/2. The slope of the tangent line at a point is equal to the derivative of the function at that point, and the cosine derivative is -sin(x). So at x = π/2, the slope is -sin(π/2) = -1.
Now, we use the formula for the linear approximation, which is L(x) = f(a) + f'(a)(x-a). In this case, a is π/2, so we get L(x) = cos(π/2) - 1(x - π/2). Since cos(π/2)=0, the best linear approximation is L(x) = 0 - 1(x - π/2) = π/2 - x.
Note:We use the
cosine derivative
to find the slope of the tangent line and the
linear approximation formula
to find the equation of the line. We also assume a
small-angle approximation
since we are close to x = π/2.
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The equation of a circle is (x−2)2+(y−16)2=169 .
What is the circle's radius?
Enter your answer in the box.
units
The radius of the circle with the equation (x−2)^2+(y−16)^2=169 is 13 units.
The equation of the circle provided is (x−2)2+(y−16)2=169. In the standard form of a circle's equation, (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius, we see that 169 represents r2. Hence, to find the radius of the circle, we take the square root of 169.
The radius r is calculated as r = √169 = 13 units. Therefore, the circle's radius is 13 units.
The amounts of 8 charitable contributions are $100, $80, $250, $100, $500, $1000, $100, and $150.
The mean, median, and mode of the amounts are given below. mean = $285 median = $125 mode = $100
Which value describes the amount received most often?
On a rectangular soccer field, Sang is standing on the goal line 20 yards from the corner post. Jazmin is standing 99 yards from the same corner post on the nearest adjacent side of the field. What is the distance from Sang to Jazmin?
A.119 yards
B.101 yards
C.10,201 yards
D.1,980 yards
Ten vendors are selling at an event, and an attendance of 42,000 is projected. It is anticipated that 65 percent of the attendees will make purchases. Assuming that each vendor has an equal chance of having an attendee prefer their products, what is the average number of customers each will serve?
Answer:
2,730
Step-by-step explanation:
Complete the steps in order to write the inverse of f(x) = x + 5 1. = x + 5 2. = y + 5 3. = y 4. = x – 5
The inverse of the given function y =x+5 is y =x-5.
The given function is:
y =x+5
What is an invertible function?
A function is invertible only if each input has a unique output and each element of the codomain should be mapped.
Step1: check function is invertible i.e. bijective i.e. one-one and onto or not
f(x₁)=x₁+5
f(x₂) = x₂+5
f(x₁) = f(x₂)
x₁+5 =x₁+5
x₁=x₂
So, the function is one-one or injective.
For onto range and codomain should be same.
Step 2: interchange x and y
x=y+5
Step3: bring 5 to the left side of the equation.
x-5 = y
Step4: write in y=f(x) form
y = x-5
Therefore, the inverse of the given function y =x+5 is y =x-5.
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What is the number six million, thirty thousand, expressed in scientific notation? 6.03 × 10^6
6.3 × 10^5
6.3 × 10^6
6.03 × 10^4
Use the formula aequals upper p left parenthesis 1 plus r right parenthesis superscript t to find the rate r at which $3000 compounded annually grows to $4320 in 2 years.
The answer is 7,323 because i dont know i asked my friend
What is the total amount and the amount of interest earned on $6,500 at 6% for 25 years?
Final answer:
The total amount on a $6,500 investment at a 6% annual interest rate compounded yearly for 25 years is approximately $28,353.25, with the interest earned being about $21,853.25.
Explanation:
To calculate the total amount and the amount of interest earned on $6,500 at 6% for 25 years, we need to identify whether the interest is compounded annually, monthly, or if it's simple interest. Since the type of interest isn't specified, we'll assume it's compounded annually, which is common in savings accounts. The formula for compound interest is:
A = P(1 + r/n)nt
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount ($6,500)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the time the money is invested or borrowed for, in years (25 years)
For this example, since the interest is compounded annually, n = 1. First, convert the interest rate from a percentage to a decimal by dividing by 100: 6% / 100 = 0.06. Now, plug the values into the formula:
A = $6,500(1 + 0.06/1)1*25
A = $6,500(1 + 0.06)25
A = $6,500(1.06)25
Calculating this out, A, which is the total future amount, comes out to:
A = $28,353.25 approximately
To find out the total interest earned, subtract the principal from the final amount:
Total interest = A - P
Total interest = $28,353.25 - $6,500
Total interest = $21,853.25 approximately
The total amount is about $28,353.25 and the amount of interest earned is about $21,853.25.
A fence on a hill uses vertical posts L and M to hold parallel rails N and P. If m∠1 = 6x and m∠12 = 4x, then what is the measure of ∠5?
Answer;
D: 108 degrees
Answer:
Measure of ∠5 is 108°
Step-by-step explanation:
Given m∠1 = 6x and m∠12 = 4x, we have to find the measure of ∠5.
As ∠1 and ∠5 are alternate exterior angle hence equal i.e ∠1=∠5
Also, ∠5 and ∠12 are co-interior angles ∴these are supplementary
⇒ m∠5+m∠12=180°
⇒ 6x+4x=180°
⇒ 10x = 180° ⇒ x=18
∴ m∠5 = 6x = 6(18) = 108°
Hence, measure of ∠5 is 108°
In order to conduct a certain experiment, four students are randomly selected from a class of 20. how many different groups of four stdents are possible
A square sheet of paper is folded diagonally. What is the length of LN?
A) 10 in.
B) 20 in.
C) 10√2
D) 5√4
Use the guess-and-check method to solve the equation: 6(x + 3) + 1 = 31
If you round off 23 to nearest ten.what is the answer
The diagram shows the locations of John and Mark in relationship to the top of a tall building labeled A.
A) Describe < 4 as it relates to the situation
B) Describe < 3 as it relates to the situation
If the factors of a polynomial are x + 7 and x – 3, what values of x make that polynomial 0?
A. 7 and 3
B. –7 and –3
C. 7 and –3
D. –7 and 3
The values of x make that polynomial 0 is x=-7 and x=3. Therefore, option D is the correct answer.
Given that, the factors of a polynomial are x + 7 and x – 3.
We need to find what values of x make that polynomial 0.
What is polynomial?An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Now, x+7=0 and x-3=0
⇒x=-7 and x=3
The values of x make that polynomial 0 is x=-7 and x=3. Therefore, option D is the correct answer.
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Wo trains leave stations miles apart at the same time and travel toward each other. one train travels at miles per hour while the other travels at miles per hour. how long will it take for the two trains to meet?
An equation of the line passing through (6,−3) having slope −35 is
After the data was recorded, Mary gives her latest book a rating of 1 star. How will this new rating affect the mean rating of her books? The mean book rating will not change.
The weekly fee for staying at the Pleasant Lake Campground is $20 per vehicle and $10 per person. Last year, weekly fees were paid for v vehicles and p persons. Which of the following expressions gives the total amount, in dollars, collected for weekly fees last year?
Thomas bought a new backpack that was 30% off the original price. The sale resulted in a $15 discount. What was the original price of the backpack?
a.) $45
b.)$50
c.)$65
d.)$70
Final answer:
To calculate the original price of Thomas's backpack, we set up an equation with the information that a 30% discount resulted in a $15 savings, which, when solved, indicates the original price was $50. The correct option is b.)$50
Explanation:
Thomas bought a new backpack at a 30% discount, which saved him $15. To find the original price, we can set up an equation to solve for the original price. Let's denote the original price as 'P'. The equation would therefore be 0.30 * P = $15, meaning 30% of the original price is equal to the $15 discount he received.
First, divide both sides of the equation by 0.30 to isolate 'P':
P = $15 / 0.30
P = $50
Hence, Thomas's backpack originally cost $50 before the discount was applied. Therefore, the correct answer is b.) $50.
2x + 4y = 12 converted into slope intercept form (y=mx+b)??
To convert 2x + 4y = 12 into slope-intercept form, we must put it into y = mx + b form, where m represents the slope of the line and b is the y intercept of the equation. To begin, we should subtract 2x from both sides so that we can get the variable y alone on the left side of the equation.
2x + 4y = 12
4y = -2x + 12
Next, because there cannot be a coefficient on the variable y in this form, we must divide both sides of the equation by 4.
y = -2/4x + 3
Because 2 and 4 are both divisible by 2, we can divide both the numerator and the denominator by 2 to further simplify the equation.
y = -1/2x + 3
Therefore, the answer is y = -1/2x + 3.
Hope this helps!
To convert the equation 2x + 4y = 12 into slope-intercept form (y=mx+b), you subtract 2x from both sides and then divide by 4 to isolate y, resulting in y = -1/2x + 3, where the slope (m) is -1/2 and the y-intercept (b) is 3.
Explanation:To convert the equation 2x + 4y = 12 into slope-intercept form (y=mx+b), we need to solve for y. Here's how you can do it step by step:
Begin with the original equation: 2x + 4y = 12.Subtract 2x from both sides to isolate the y-term: 4y = -2x + 12.Divide each term by 4 to solve for y: y = -½x + 3.Now the equation is in slope-intercept form where m (the slope) is -1/2 and b (the y-intercept) is 3. The slope m is defined as the rise over the run of the straight line, and the y-intercept b is the point where the line crosses the vertical axis, which in this case is when x=0 and y=3.
Help please!!! I would appreciate it.
When figuring out how much concrete is needed to build a pool what do you multiply the price of the concrete by?
The first term of an arithmetic sequence is -5 , and the twelfth term is 1.7 .find the common difference
You are selling cookies for your organization. One box of cookies is sold for $3. Thirty percent of the sale price goes to the bakery to pay for the cookies. The rest of the sale price is split evenly between the regional headquarters for your organization, and your local chapter. How much does your local chapter earn for each box of cookies that are sold? How many boxes of cookies do you have to sell to earn $100 for your local chapter?
A jar contains 5 pink and 8 navy marbles. A marble is drawn at random. What is the theoretical probability of NOT drawing pink?
Give the next 3 numbers in the sequence: 3, 7, 11, 15, ___, ___, ___
The ratio of boys to girls is 4:3 .T here are 35 students in a class how many students are boys ?? Pleaseee help
I am wondering if the answer to this question is Table B. At x = 1, f'(x) = 1/2. At x = 2, f'(x) = 3. Does this mean that, between 1 and 2, the slope should be greater than 1/2 and less than 3? (e.g. graph B)?