Mathematics High School

Which of the following is the best linear approximation for f(x) = cos(x) near x= π/2

Answers

Answer 1
The local linear approximation of f near x = a is given by
                              f(x) ≈ f(a) + f'(a)(x-a)
Evaluating f at π/2
                          f(π/2) = cos(π/2) = 0                        
Since f(x) = cos(x), differentiating gets us
                           f'(x) = -sin(x)
                       f'(π/2) = -sin(π/2) = -1
So the local liner approximation is
                              f(x) ≈ 0+ -1(x-π/2)
                              f(x) ≈ -x+π/2
The answer to this question is -x+π/2
Answer 2
Final answer:

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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Related Questions

Jon performs 3 transformations on a pentagon. First, he reflects it over the x-axis. Then, he translates it 3 units to the left and 4 units up. Finally, he reflects the image over the line x = 1. Are the original and transformed images congruent? Justify your answer.

Answers

Reflection and translation do not change the dimensions (or angles) of the figure.

The original and transformed images are congruent.

yes; Reflections and translations are rigid motions. Rigid motions preserve the size of the original image.


Find the inverse function f^-1(x) for the given function f(x). f(x) =3x-8

Answers

[tex]f(x)=3x-8\\\\y=3x-8\ \ \ \ |+8\\\\3x=y+8\ \ \ \ |:3\\\\x=\dfrac{y+8}{3}[/tex]

Answer: [tex]y=\dfrac{x+8}{3}=\dfrac{1}{3}x+\dfrac{8}{3}[/tex]

Final answer:

To find the inverse of the function f(x) = 3x - 8, you swap x and y and solve for y, resulting in the inverse function f^-1(x) = (x + 8) / 3.

Explanation:

To find the inverse function f-1(x) for the given linear function f(x) = 3x - 8, we will follow these steps:

Replace f(x) with y: y = 3x - 8.

Swap x and y to make it x = 3y - 8.

Solve for y to get the inverse function.

Starting with x = 3y - 8, we add 8 to both sides to obtain x + 8 = 3y. Then, we divide both sides by 3 to isolate y, resulting in y = (x + 8) / 3. Therefore, the inverse function is f-1(x) = (x + 8) / 3.

does anyone know where to find the poly-graphical point of a circle?

Answers

it is the middle i think
x = cx + r * cos(a)
y = cy + r * sin(a)

How do I find the solution for this quadratic?

Answers

Hello,
Please, see the attached file.
Thanks.

Find the slope of the line

-4
-1/4
1/4
4

Answers

Find 2 "easy" points on the graph, and identify their coordinates.  One such point would be (-4,1) and another would be (4,2).

run = 8 (from -4 to +4)
rise = 2 (from 0 to 2)

              rise           2
slope = ------- = -------- = 1/4                   y-intercept is (0,1)
               run          8

equation of this line is   thus   y = mx + b, or   y = (1/4)x + 1  (answer)

Jonathan opened an auto repair shop. In the first month, 11 cars were serviced at his shop. The number of cars serviced at Jonathan's shop increased at 9 cars per month.

How many total cars were serviced at Jonathan's shop at the end of the 7th month?

329 cars
65 cars
266 cars
201 cars

Answers

The number of cars in first month = 11
Number of cars in second month = 11 + 9 =20
Number of cars in third month = 20 + 9 = 29

If we observe, the number of cars form an Arithmetic sequence with first term as 11 and common difference 7. We are to find the total cars received till the end of 7th term in the shop i.e. we are to find the sum of first 7 terms of the sequence.

So, using the formula of sum Arithmetic series we can write:

Cars  received in 7th months = [tex] \frac{7}{2}(2(11)+(7-1)(9)) = 266 [/tex] cars

Thus, 266 cars were received at the end of 7th month at Jonathan's auto repair shop.

 

Graph the function y = 6/x . Draw a chart to show the ordered pairs that are on the graph.
Answer:

x y
-3
-2
-1
1
2
3










Answers

Answer:

  see the attachment for the graph and chart

Step-by-step explanation:

A spreadsheet or graphing calculator is convenient for making a chart of ordered pairs.

The median is 28; 16,24,x,48

Answers

x = 32 

To find the media, you take the middle number. If there is no discrete middle number, take the sum of the two closest to the middle and divide by 2. 

(24 + x)/2 = 28

Multiply by 2 on both sides to get:

24 + x = 56

Subtract 24 from both sides to isolate the variable:

x = 56 - 24
x = 32

another geometry file upload

Answers

cross multiplication 13x -13 = 8x+12 
combining like terms 13x -8x =12 +13
5x= 25     x= 5

x=5

What is the equation of the circle with a diameter of 6 units and a center located at (2, -5) on the coordinate plane?

Answers

if the diameter is 6, then its radius is half that, or 3, thus,

[tex]\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{2}{ h},\stackrel{-5}{ k})\qquad \qquad radius=\stackrel{3}{ r} \\\\\\\ [x-2]^2+[y-(-5)]^2=3^2\implies (x-2)^2+(y+5)^2=9[/tex]
Final answer:

The equation of a circle centered at (2, -5) with a diameter of 6 units is (x-2)² + (y+5)² = 9.

Explanation:

The equation of a circle in the coordinate plane is given by (x-h)² + (y-k)² = r², where (h, k) are the coordinates of the center of the circle, and r is the radius. In this case, the center of the circle is at (2, -5) and the diameter is 6 units. Therefore, the radius (r) is half the diameter, which is 3 units.

Substituting h=2, k=-5, and r=3 into the equation, we get the equation of the circle: (x-2)² + (y+5)² = 9.

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He probability of choosing a vowel (a, e, i, o, or u) from a deck of cards containing the 26 letters of the alphabet is shown below. what is the probability of choosing the letters a and e one after the other without replacement?

Answers

The probability is 1/650.

There is 1 a out of 26 letters; P(a) = 1/26

There is 1 e out of 26 letters; however, since it comes after drawing the a and is done without replacement, it is out of 25: 1/25

Together, their probabilities are 1/26(1/25) = 1/650

10 POINTS!!! FULL ANSWER IN STEP BY STEP FORMAT!!

Answers

a) The signs of the first derivative (g') tell you the graph increases as you go left from x=4 and as you go right from x=-2. Since g(4) < g(-2), one absolute extreme is (4, g(4)) = (4, 1).

The sign of the first derivative changes at x=0, at which point the slope is undefined (the curve is vertical). The curve approaches +∞ at x=0 both from the left and from the right, so the other absolute extreme is (0, +∞).

b) The second derivative (g'') changes sign at x=2, so there is a point of inflection there.

c) There is a vertical asymptote at x=0 and a flat spot at x=2. The curve goes through the points (-2, 5) and (4, 1), is increasing to the left of x=0 and non-increasing to the right of x=0. The curve is concave upward on [-2, 0) and (0, 2) and concave downward on (2, 4]. A possible graph is shown, along with the first and second derivatives.

The price of a gym membership has a one-time sign-up fee and a monthly fee. The price can be modeled by the function y = 10x + 15, where x is the number of months. What is the y-intercept, and what does it represent?

10; it represents the monthly fee
10; it represents the one-time sign-up fee
15; it represents the monthly fee
15; it represents the one-time sign-up fee

Answers

Its C because you are trying to find the y-intercept so you have to substitute 0 for X and solve.
The y-intercept is 15, this represents the one-time sign-up fee. 10x represents the monthly fee where x is the number of months.

3 red pens, 9 blue pens, 8 green pens
What is the probability of picking a blue? In a decimal

Answers

There are a total of 20 pens. There are 9 blue pens out of 20 total pens. In decimal format, you have a chance of 0.45 chance. I hope this helps!

the probability of picking a blue is 0.45.

What is probability?

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

here, we have,

3 red pens, 9 blue pens, 8 green pens

now, we get,

There are a total of 20 pens.

There are 9 blue pens out of 20 total pens.

In decimal format,

i.e. 9/20

=0.45

you have a chance of 0.45 .

hence, the probability of picking a blue is 0.45.

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For the solids show, choose all the statements that will be true if we can infer the solids' congruence by Cavalieri's Principle. The radii are congruent. The bases are congruent. The altitudes are congruent. The bases are parallel.

Answers

I presume the two solids are similar to those in the picture attached.

Cavalieri's principle states that if two solids have equal height and if the cross-sections intersected by planes parallel to the bases and equally distant from them have always the same ratio, then the two volumes will have the same ratio.

Therefore, we can say that:
1) the radii are congruent (bases and cross-sections are congruent circles);
2) the bases are congruent;
3) the altitudes are congruent;
4) the bases are parallel.

Hence, all four statements are TRUE.

Answer:

All four are true :)

Step-by-step explanation:

Need help on this asap!!

Answers

Hello,
Please, see the attached file.
Thanks.

Multiply the polynomials.
(x – 7)(x2 + 3x – 3)
A. x3 – 4x2 – 3x + 21
B. x3 – 7x2 – 24x + 21
C. x3 – 4x2 – 24x + 21
D. x3 – 7x2 – 3x + 21

Answers

Answer:

C) x³ -4x² - 24x + 21.

Step-by-step explanation:

Given : (x – 7)(x² + 3x – 3).

To find : Multiply the polynomials.

Solution : We have given  (x – 7)(x² + 3x – 3).

Distributes x over (x² + 3x – 3) and  -7 over (x² + 3x – 3).

Then

x (x² + 3x – 3) -7(x² + 3x – 3).

x³ + 3x² -3x - 7x² - 21x +21.

Combine like terms

x³ + 3x² - 7x² -3x -21x + 21.

x³ -4x² - 24x + 21.

Therefore, C) x³ -4x² - 24x + 21.

Answer:

x³ -4x² - 24x + 21.

Step-by-step explanation:

Ape-x

A rectangular crate has the dimensions of 32.1 inches by 28.2 inches by 1 foot.What is the volume of the crate? Round your answer to the nearest tenth, if necessary
a)6.3 cubic feet
b)16. cubic feet
c)18.4 cubic feet
d)905.22 cubic feet

Answers

32.1×28.2×1=905.22
So the answer is D
Convert all to feet:
28.2 inches = 28.2 ÷ 12 = 2.35 feet
32.1 inches = 32.1 ÷ 12 = 2.675 feet

Volume = Length x Width x Height
Volume = 2.35 x 2.675 x 1 = 6.3 ft³ (nearest tenth)

Answer: 6.3 ft³

Chris plants a sunflower that is 6 inches tall. The sunflower is expected to grow at an average rate of 1.5 inches per day during the next month.


A. Create an equation that Chris can use to find the number of days, x, it will take the sunflower to grow to a height of 45 inches.


B. How many days will it take the sunflower to grow to a height of 45 inches?

Answers

A.  The equation that represents this scenario would be 6 + 1.5d = 45. The reason is that the plant starts out at 6 inches(6) and (+) grows 1.5 inches each day (1.5d).

B.  We will solve the equation to find out how many days it will take to get to 45 inches.

6 + 1.5d = 45
-6             -6
1.5d  = 39 
1.5       1.5
d = 26

The number of days it would take is 26.


PLEASE HELP! BRAINLIEST!!!!!!! PLEASE

Jade used candle molds, as shown, to make candles that were perfect cylinders and spheres:



What is the approximate difference in the amount of wax needed to make a candle from each of these molds? (Use π = 3.14.)

27.0 cubic inches
47.1 cubic inches
84.78 cubic inches
91.12 cubic inches

Answers

When finding the volume for a cylinder you use V=Bh or in this case...3.14xr squared. 3.14x3 squared by 2. This equals 28.26 when after rounding. Then for a sphere the formula is V=4/3x3.7x2 cubed by 3. This equals 220 after rounded. Subtract those and you have yourself your answer.

Hello!

First you have to find the volume of the cylinder

To find the volume of a cylinder you use the equation

[tex]V = \pi r^{2} h[/tex]

V is volume
r is radius
h is height

Put in the values you know and 3.14 for pi

[tex]V = 3.14 * 3^{2} * 7[/tex]

Square the number

V = 3.14 * 9 * 7 

Multiply

V = 197.82
------------------------------------------------------------------------------------------------------
Next you find the volume of the sphere

To find the volume of a sphere you use the equation

[tex]V = \frac{4}{3} \pi r^{3} [/tex]

V is volume
r is radius

Put in the values you know and 3.14 for pi

[tex]V = \frac{4}{3} * 3.14 * 3^{3} [/tex]

Cube the number

[tex]V = \frac{4}{3} * 3.14 * 27[/tex]

Multiply

V = 113.04
------------------------------------------------------------------------------------------------------
Finally you subtract the two volumes from each other

197.82 - 113.04 = 84.78
------------------------------------------------------------------------------------------------------
The answer is C)84.78

Hope this helps!

Suppose N represents a power of 10. what is the value of n when 31,100 is rounded to the nearest power of 10?

Answers

We get the approximate value of [n] as 5.

What is logarithm? What is a mathematical equation and expression?A quantity representing the power to which a fixed number (the base) must be raised to produce a given number. We can write -

        [tex]$\log_{b}({b^x})=x[/tex]

A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

We have 31,100 is rounded to the nearest power of 10 and [N] represents a power of 10.

We can write -

10ⁿ = 31100

10ⁿ = 311 x 10²

10ⁿ ⁻ ² = 311

log(10ⁿ ⁻ ²) = log(311)

(n - 2) log 10 = log (311)

n - 2 = log(311)/log(10)

n - 2 = 2.5

n = 4.5

n = 5 (approx.)

Therefore, we get the approximate value of [n] as 5.

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When rounding 31,100 to the nearest power of 10, we get the value of n is 4.

To find the value of n when 31,100 is rounded to the nearest power of 10, we first need to determine which powers of 10 are closest to 31,100. The nearest powers of 10 to 31,100 are 104 (10,000) and 105 (100,000). Since 31,100 is closer to 104 than to 105, we round it down to 104.

Therefore, when 31,100 is rounded to the nearest power of 10, the value of n is 4.

The process involves converting the number into scientific notation and then adjusting the exponent, which represents the number of places the decimal point moves. In our case, to convert 31,100 to a number between 1 and 10, we would express it as 3.11 × 104, confirming that the value of n is 4.

Please help with geometry

Answers

The formula of the area of the triangle:
[tex]A_\Delta=\dfrac{ah}{2}[/tex]
We have:
[tex]a=6.3cm;\ h=8cm[/tex]
therefore

[tex]A_\Delta=\dfrac{6.3\cdot8}{2}=25.2\ cm^2[/tex]
Answer: 25.2 cm².

[Geometry Basics] I'm just checking if this is correct:

~ Endpoint: (-9, -1), midpoint: (8,14)

Answer I got: (25, 31)


~ Endpoint: (10,12), midpoint: (6,9): (2,6)


I will give you brainiest if it's correct :) And 20 pts.

Answers

[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ x &,& y~) % (c,d) &&(~ -9 &,& -1~) \end{array}\qquad % coordinates of midpoint \left(\cfrac{ x_2 + x_1}{2}\quad ,\quad \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{-9+x}{2}~~,~~\cfrac{-1+y}{2} \right)=\stackrel{midpoint}{(8,14)}\implies \begin{cases} \cfrac{-9+x}{2}=8\\\\ -9+x=16\\ \boxed{x=25}\\ -------\\ \cfrac{-1+y}{2} =14\\\\ -1+y=28\\ \boxed{y=29} \end{cases}[/tex]

[tex]\bf -------------------------------\\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ x &,& y~) % (c,d) &&(~10 &,& 12~) \end{array}\qquad \\\\\\ \left( \cfrac{10+x}{2}~~,~~\cfrac{12+y}{2} \right)=\stackrel{midpoint}{(6,9)}\implies \begin{cases} \cfrac{10+x}{2}=6\\\\ 10+x=12\\ \boxed{x=2}\\ -------\\ \cfrac{12+y}{2} =9\\\\ 12+y=18\\ \boxed{y=6} \end{cases}[/tex]

Find the length of the hypotenuse of a right triangle whose legs are 5 and sqare root of 2

Answers

2² + (√5)² = c²
4 + 5 = c²
9 = c²
c = 3

Sorry this is so late and was never answered. I just saw it pop up in the related questions. 

Answer:

The length of the hypotenuse is 3√3 units.

Step-by-step explanation:

It is given that the length of the legs of a right triangle are 5 units and √2 units.

According to the Pythagoras theorem, in a right angled triangle

[tex](hypotenuse)^2=(leg_1)^2+(leg_2)^2[/tex]

Substitute leg₁=5 and leg₂=√2 in the above formula.

[tex](hypotenuse)^2=(5)^2+(\sqrt{2})^2[/tex]

[tex](hypotenuse)^2=25+2[/tex]

[tex](hypotenuse)^2=27[/tex]

Taking square root both the sides.

[tex]hypotenuse=\sqrt{27}[/tex]

[tex]hypotenuse=3\sqrt{3}[/tex]

Therefore the length of the hypotenuse is 3√3 units.

Find the quotient of 3/5 and 9/10 . Express your answer in simplest form.

Answers

Hello there, and thank you for asking your question here on brainly.

Short answer: 2/3

Why?

When dividing two fractions, you don't divide, you find the reciprocal of the second fraction (9/10) and multiply. So, it's 3*10/9*5. Simplify. {3 * 10 => 30} ~ {9 * 5 => 45} Now we have 30/45. This can be simplified. Simplified, 30/45 reduces down to 2/3.

Hope this helped you! ♥

The quotient of 3/5 and 9/10 is the value obtained by dividing both values, Hence, the quotient of the division is 2/3

The numerator = 3/5

The denominator = 9/10

The division operation can be expressed thus :

3/5 ÷ 9/10

Take the reciprocal of the denominator and change the sign to multiplication ;

Reciprocal of 9/10 = 10/9

Hence, we have ;

3/5 × 10/9 = 30/45

In simplest form ; 30/45 = 10/15 = 2/3

Therefore, the quotient is 2/3

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A 9 cm tall cone shaped paper cup can hold up to 58.9 cm cubed of water. What is the minimum amount of paper needed to make the paper cup, assuming no overlap in the paper? Use 3.14 for π.

Answers

Volume of a cone, V = πr^2h/3

r = Sqrt 3V/πh = Sqrt [(58.9*3)/(3.14*9)] = 2.5 cm, minimum

Surface area, A = πr [r+ Sqrt (h^2+r^2)] =  3.14*2.5*[2.5+ Sqrt (9^2+2.5^2)] = 92.95 cm^2

Therefore, minimum amount of paper required is 92.95 cm^2


What is the exact area of a circle with a diameter of 20 cm?

400πcm^2

10πcm^2

100πcm^2

20πcm^2

Answers

Hi there! The answer is 100πcm^2

The radius of a circle can be found by dividing the diameter by 2. In other words : the radius is half of the diameter.
Therefore, the radius of a circle with a diameter of 20 cm is 10 cm.

The exact area of a circle can be found using the following formula:
[tex]a = \pi \times \: {r}^{2} \\ with \\ a = area \\ r = radius[/tex]

In order to find our answer, we need to plug in the data from the question.
[tex]a = \pi \: \times {10}^{2} = 100\pi[/tex]

Hence, the answer is 100πcm^2.
Final answer:

The exact area of a circle with a diameter of 20 cm is 100π cm². This is found using the formula Area = πr², with 'r' being the radius of the circle.

Explanation:

The subject of this question falls under the category of Mathematics, specifically geometry. In order to find the area of a circle, we use the formula: Area = πr², where r is the radius of the circle.

The diameter of the circle is given as 20 cm, so the radius, which is half the diameter, would be 10 cm. Substituting the radius value into the formula gives us: Area = π(10 cm)² = 100π cm². Therefore, the exact area of the circle with a diameter of 20 cm is 100π cm².

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Please help me with this question

Answers

The graph shows the maximum residual is ...
  ◉  Data point (10, 10); Residual = 4.50

_____
Apparently, this question makes no use of the line of best fit.

The midpoint of is m(–8, 1). one endpoint is k(–6, 5). find the coordinates of the other endpoint l.

Answers

the correct question is
The midpoint of kl is m(–8, 1). one endpoint is k(–6, 5). find the coordinates of the other endpoint l.

we know that
the formula of midpoint is
Xm=(x1+x2)/2----> 2*Xm=x1+x2------> x2=2*Xm-x1
Ym=(y1+y2)/2----> 2*Ym=y1+y2------> y2=2*Ym-y1

let
(x1,y1)-------> (–6, 5).
(Xm,Ym)-----> (-8,1)

find (x2,y2)
x2=2*Xm-x1-----> 2*(-8)-(-6)----> -10
 y2=2*Ym-y1----> 2*(1)-5-----> -3

the point l is (-10,-3)

Please help, I need to turn in today

Answers

she will use 24 yards of trim.
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