The derivate of a function f(x) is determinated as:
[tex]f^{\prime}(x)=\lim _{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]For the function
[tex]f(x)=x^2-2x[/tex]First we have to determine de f ( x + h ) as follow:
[tex]f(x+h)=(x+h)^2-2(x+h)[/tex][tex]f(x+h)=x^2+2xh+h^2-2x-h^{}[/tex]Then we calculate and simplify the coeficient in the first formula
[tex]\frac{f(x+h)-f(x)}{h}[/tex][tex]\frac{(x^2+2xh+h^2-2x-h^{})-(x^2-2x)}{h}[/tex][tex]\frac{x^2+2xh+h^2-2x-h^{}-x^2+2x}{h}=\frac{h^2+2xh-h}{h}[/tex][tex]\frac{h^2+2xh-h}{h}\text{ = }\frac{h(h+2x-1)}{h}=h+2x-1[/tex]So the derivate is:[tex]f^{\prime}(x)=\lim _{h\to0}\frac{f(x+h)-f(x)}{h}=\lim _{h\to0}h+2x-1[/tex][tex]f^{\prime}(x)=0+2x-1[/tex][tex]f^{\prime}(x)=2x-1[/tex]------------------------------------------------------------------------------------The equation of the tangent:
First you need to know that the derivate of a function is equal to the slope (m) of the tangent of this function
And the equation to thist tangent in a specific point will be find using the next formula:
[tex]y-f(x_0)=m(x-x_0)_{}[/tex]We have to calculate the slope in the point x =4 using the derivate:
[tex]m=2x-1[/tex][tex]m=2(4)-1=7[/tex]In the point x=4
Calculate the value of f(x0) substituting in the function the given point x:
[tex]f(4)=4^2-2(4)\text{ = }8[/tex]Knowing that we put the value of m and f(x0) in the equation of the tangent:
[tex]y-8=7(x-4)_{}[/tex][tex]y-8=7x-28[/tex][tex]y=7x-28+8[/tex]So the equation of the tangent in x= 4 is:[tex]y=7x-20[/tex]---------------------------------------------------------------------------
The normal line
The slope of the normal line is the opposite of the slope of theu tangent in an espesific point:
[tex]m_n=-\frac{1}{m_t}[/tex]So in this situation is:
[tex]m_n=-\frac{1}{7}[/tex]The equation of the normal line is given by the next formula:
[tex]y-f(x_0)=m_n(x-x_0)_{}[/tex]Replacing the data we obtain:
[tex]y-8=-\frac{1}{7}(x-4)_{}[/tex][tex]y-8=-\frac{1}{7}x+\frac{4}{7}[/tex]So the equation of the normal line is:[tex]y=-\frac{1}{7}x+\frac{60}{7}[/tex]Rachel is making premade 3-inch by 5-inch tile rectangles for her driveway. If it takes her 6 minutes to position each rectangle, how many hours will it take her to create a 2-foot by 4-foot pattern drive? Write your answer as a decimal. (ASAP)
The time taken to create the driveway is 7 hours 42 minutes
The dimensions of the rectangle tile = 3 inch by 5 inches
The area of the rectangle tile = Length × width
= 3×5
= 15 square inches
Time taken to position each rectangle = 6 minutes
The dimensions of the driveway = 2 foot by 4 foot
Convert the dimensions to inches
2 foot by 4 foot = 2×12 inches by 4×12 inches
= 24 inches by 48 inches
The area = Length × Width
= 24×48
= 1152 square inches
Number of tiles needed = Area of the driveway/ Area of the tile
= 1152/15
= 76.8
≈77 tiles
Time taken to create the drive way = Number of tile× Time taken to position each rectangle
= 77×6
= 462 minutes
= 7 hours 42 minutes
Hence, the time taken to create the driveway is 7 hours 42 minutes.
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Angela’s bedroom has an area of (2x2-7x-15) square feet. The length of the room is (2x+3) feet. What is the width of her bedroom?
The width of her bedroom is x - 5
How to determine the widthIt is important to note that the formula for area of a rectangle is expressed as;
Area = lw
Where;
l is the length of the roomw is the width of the roomNow, let's substitute the values into the formula, we have;
2x²-7x-15 = (2x+3)w
Divide both sides by the coefficient of w
w = 2x²-7x-15/(2x+3)
Factorize the numerator
w = 2x²- 10x + 3x - 15/(2x+3)
Factor the common terms of the numerator
w = 2x(x - 10) + 3(x - 5)/(2x+3)
enclose the expressions
w = (2x + 3)(x-5)/(2x+3)
Divide the common terms
w = x - 5 feet
Hence, the value is x - 5
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HELP!!!!! It's in the photo.
After estimation the area of the composite figure is equal to about 22 square units.
As given in the question,
For the given composite figure:
Let us estimate the area of 1 square = 1 square units
Total number of full squares = 12
Total number of more than half = 6
Consider area of more than half as 1 square units.
Total number of half squares = 8
Area of less than half is negligible.
Total estimated area of given composite figure
= 12 + 6 + [(1/2) ×8]
=12 + 6 +4
=22 square units
Therefore, after estimation the area of the composite figure is equal to about 22 square units.
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b) The nearest-known exoplanet from earth is 4.25 light-years away. About how many miles is this? Give your answer in standard form. Please Help
4.25 light years is 2.4984x10¹³ miles
What is light years?
A light-year, sometimes spelt light year, is a huge measure of length used in astronomy that is comparable to approximately 9.46 trillion kilometers (9.461012 kph) or 5.88 trillion miles (5.881012 mi). A light-year, as defined by the International Astronomical Union (IAU), is the distance that light travels in a vacuum in one Julian year (365.25 days). The phrase light-year is frequently misconstrued as a unit of time since it contains the time-measurement word "year." The light-year is most commonly used in non-specialist contexts and popular scientific publications to indicate distances to stars and other cosmic distances.
1 light years is = 5.8786x10¹² miles
4.25 light years = 5.8786x10¹²x4.25 = 2.4984x10¹³ miles
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Tim buys a new computer for $1200. After four years, he sells the computer to a friend for $450.
Let y represent the current value of the computer, and x represent the number of years. Write a
linear equation in slope-intercept form to model the value of the computer over time. Find the value
of the computer after 6 years.
A linear equation in slope-intercept form for computer; y = -187.5(x) + 1200.
The value of the computer after 6 years will be $75.
What is defined as the slope-intercept form?An intercept is a y-axis point whereby the slope of a line passes. It is the y-coordinate of a point on the y-axis at which a straight line or a curve intersects. This is represented by the equation for a straight line, y = mx+c, where m is the slope and c is really the y-intercept.Tim spends $1200 on a new computer.
He sells this same computer to that of a friend for $450 after four years.
Let y represent the computer's current value and x depict the number of years.
Create a linear equation in slope-intercept pattern to model the computer's value over time.
y = mx + b
y = 1200,
when x (time) is 0, thus b = 1200
when x = 4, y = 450,
Calculate the slope m.
4m + 1200 = 450
4m = 450 - 1200
m = -750/4
m = -187.5
Put the value of m in linear equation;
y = -187.5x + 1200
Noe, the value of computer after 6 years, put x = 6.
y = -187.5(6) + 1200
y = 75
Thus, the value of the computer after 6 years will be $75.
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Can you check these for my please it would be much appreciated! And yes I will mark brainliest
ΔDEF ≅ Δ RPQ by ASA rule and line segment RF ≅ line segment HR.
In triangle DEF and triangle PQR,
We can see that:
DF = RQ = 5cm
0∠D = ∠Q = 60°
∠F = ∠R = 80°
So, we get that:
ΔDEF ≅ Δ RPQ ( ASA rule.)
Also, we are given that:
quadrilateral RAUF ≅ Quadrilateral HBQR
This means that:
line segment RF ≅ line segment HR.
Therefore we get that, ΔDEF ≅ Δ RPQ by ASA rule and line segment RF ≅ line segment HR.
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( Please help )
Answer the question in photo.
Answer: 5/3
Step-by-step explanation:
The scale factor is the same as the ratio of the length of a side of the image to the length of the corresponding side in the preimage.
We know that N'Q'=10 and NQ=6, so the scale factor is 10/6 = 5/3.
prove that,
b) (a + b)2 = (−a − b)2
c) (a − b)2 = a2 − 2ab + b2
We will prove both of the identities by expanding the exponents, only in the first one we need to use a little trick that is multiplying by (-1)^2 = 1.
How to prove the identities?
Let's start with the first one:
(a + b)^2 = (- a - b)^2
We can start with the left side and write:
(a + b)^2 = (a + b)*(a + b)
Now, remember the rule of signs:
(-1)*(-1) = 1
So we can multiply by one the expression above and get:
(a + b)*(a + b) = (a + b)*(a + b)*1 = (a + b)*(a + b)*(-1)*(-1)
Now we can reorder the product to get:
(a + b)*(a + b)*(-1)*(-1) = (-1)*(a + b)*(-1)*(a + b) = (-a - b)*(-a - b) = (-a - b)^2
So we proved the first part, now the second one:
(a - b)^2
Let's just expand this:
(a - b)^2 = (a - b)*(a - b) = a*a + a*(-b) + (-b)*a + (-b)*(-b) = a^2 - 2ab - b^2
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The sum of two consecutive odd number is 72.
PLEASE ANSWER PLEASE PLEASE I NEEDTHE ANSWER FAST PLEASE
Which equation represents the scenario above.
Responses
A 2x + 3 = 72
B 2x + 2 = 72
C 2x + 1 = 72
D 2x = 70
Answer: B
Step-by-step explanation:
Answer:
B
Step-by-step explanation:
let the 2 consecutive odd numbers be x and x + 2 , then the sum is
x + x + 2 = 72 , that is
2x + 2 = 72 → B
subtract 2 from both sides
2x = 70 ( divide both sides by 2 )
x = 35
x + 2 = 35 + 2 = 37
the 2 consecutive odd numbers are 35 and 37
WILL MARK BRAINLEIST!!
Given f(x) and g(x) = f(x + k), use the graph to determine the value of k.
-4
-5
5
4
Answer:
k = 5
Step-by-step explanation:
f(x) = x + 4
g(x) = x + 9 = (x + 5) + 4 = f(x + 5)
So k = 5
85 cubed in all three forms exponential form expanded form and you add evaluate
The cube of the number five is written in these following forms:
Exponential: 5³.Expanded form: 5 x 5 x 5.Evaluation: 125.How to calculate the cube of a number?The cube of a number n is equivalent to multiplying the number n by itself three times.
Hence, in expanded form, the cube of a number n is written as follows:
n x n x n.
In exponential form, the cube of a number n is written as follows:
n³.
In this problem, the number of interest is the number five, hence:
In exponential form, the cube of 5 is given by: 5³.In expanded form, the cube of 5 is given by: 5 x 5 x 5.The result of the expression, which is the evaluation, is given by:
5 x 5 x 5 = 25 x 5 = 125.
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write a paragraph proof for the following
Proved that ∠ 2 = 107° because these two angles are supplementary angles .
What are supplementary angles ?Angles that add up to 180 degrees are referred to as supplementary angles. For instance, angle 130° and angle 50° are complementary angles since the sum of these two angles is 180°. If the two additional angles are combined, they create a straight line and an angle.
However, it should be emphasized that the two complementary angles do not necessarily need to be close to one another. As a result, any two angles can be supplementary if their sum is 180 degrees. However, it should be emphasized that the two complementary angles do not necessarily need to be close to one another. So, if two angles are complementary, they can be any two angles.
Given : ∠ 1 and ∠ 2 are supplementary angles
that is ∠ 1 + ∠2 = 180 °
Given ∠ 1 = 73°
thus when we substitute this value in the equation we get ∠ 2 = 180 - 73
∠ 2 = 107 °
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Please answer it with WORKING
Answer:
5 bottles of hand sanitizer
2 thermometers
4 packets of masks
Step-by-step explanation:
We start by factoring each of the numbers to find the common factor which will give us the number of bags used to distribute.
1155 = 5 x 231
462 = 2 x 231
924 = 4 x 231
We see that 231 is the common factor for all 3 numbers. It is also the greatest common factor.
Therefore the greatest number of bags that can be packed = 231
Each bag will contain:
5 bottles of hand sanitizer
2 thermometers
4 packets of masks
Mrs. Lahti bought a pair of shoes for $50.80 and used a coupon for 20% off. What was her discount?
In a case whereby Mrs. Lahti bought a pair of shoes for $50.80 and used a coupon for 20% off then her discount will be $12.7.
What is a discount?A discount can be described as as the decreases in price of a product which can be given to the consumer from the seller.
The given discount given in the question is 20%
Then the price she bought the pair of shoe is $50.80
then the market price will be 100/80 * 50.80 = $63.5
Then the discount she got is ( $63.5 - $50.80) = $12.7
Therefore, discount she got is $12.7
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Here is a data set:
1 2 3 3 4 4 4 4 5 5 6 7
A. For the data set with the value of 70, why would the median be a better choice for the measure of
center than the mean?
For the given data set, median is better choice for measure than mean because:
Median is usually a better measure of the center when there are extreme values as it is remains unaffected by the precise numerical value. The most common measure of the center is mean.
What is median and mean?Adding all numbers in the data and dividing it by the number of values in the set gives mean of a data. Whereas, median is the middle value in a data set, implying that 50% of data points can have a value smaller or equal to the median and 50% of data points can have a value higher or equal to the median.
For a small data set, firstly, count the number of data points and arrange them in increasing order. If the number of data is odd, then add 1 to the number and divide the result by 2 to obtain the rank of the data whose value is the median.
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Determine which integer will make the inequality x − 4 < 16 true.
S:{16}
S:{20}
S:{21}
S:{32}
Answer:
x-4<16
x<16+4
x<20
S:[20] TRUE
The width of a rectangle is 3 more than half the length of the perimeter is 174 inches what is the length of the rectangle
(66 inches) because i don't know
but my answer is correct
HELP PLEASE!!
Write the slope intercept form of the equation for each line
Answer:
y = -5/9x
Step-by-step explanation:
Two points on the line; (4,-2) & (-5,3)
Finding the slope (m):
m = (3-(-2))/(-5-4)
= 5/-9
= -5/9
y = -5/9x + c
Since it passes through the origin, c = 0.
Hence, y = -5/9x
Hope this helps and feel free to mark this as brainliest :)
y is inversely proportional to x².
When x = 2, y = 8.
Find y in terms of x.
Answer: y=32/x^2
Step-by-step explanation: Please mark brainilest. Since y and x^2 are inversely proportional, x^2(y)=k. K is some constant. Since they gave us an example of x and y, we can calculate k as 4*8=32. So our updated equation is y(x^2)=32. Dividing both sides by x^2 we get y=32/x^2, so we have y in terms of x
Answer:
y = 32x
Step-by-step explanation:
y = k/x^2
8 = k/2^2
8 = k/4
8×4 = 32
k = 32
y = 32x
Domain and Range, PLEASE HELP!!
The domain for the given graph would be [-5,10] and range for the graph would be [0.4,5.6]. in both the case whether increasing or decreasing graph.
What is Domain?The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). The entire range of independent variable values is the domain of a function.
What is range?The range of a function is the set of outputs the function achieves when it is applied to its whole set of outputs. In the function machine metaphor, the range is the set of objects that actually come out of the machine when you feed it all the inputs.
How to find the domain and range in graph?Graphs can also be used to determine the domain and range of functions. The domain of a graph consists of all the input values shown on the x-axis because the domain refers to the set of possible input values. The range represents the set of possible output values shown on the y-axis.
Here, The domain would be [-5,10] as the graph is depicting.
The range would be[0.4,5.6].
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If the lengths of the sides of triangle abc are 4, 5, and 7 feet, what is the perimeter in feet of similar triangle def whose shortest side is 12 feet?
Given,
ΔABC and ΔDEF are similar triangles.
AB = 6 feet, BC = 4 feet, AC = 7 feet
Let the shortest side of ΔDEF be EF = 12 feet.
Let DE = x and DF = y
Since, ΔABC and ΔDEF are similar,
[tex]\frac{BC}{EF}[/tex] = [tex]\frac{AB}{DE} = \frac{AC}{DF}[/tex]
⇒ [tex]\frac{4}{12} = \frac{6}{x} =\frac{5}{y}[/tex]
On Solving,
x = 18 feet and y = 15 feet
We know,
Perimeter of a triangle = sum of all three sides
Hence, perimeter of ΔDEF = DE + EF + DF = 18 + 15 + 12 = 45 feet.
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r+ 5 < 1 OR r − 1 ≥ 1
Answer:
[tex]\boxed{\sf{r < -4\quad r\geq 2}}[/tex]
Step-by-step explanation:
Isolate the term of r.
1. r+5<1First, subtract by 5 from both sides.
r+5-5<1-5
Solve.
r<-4
2. r-1≥1Add by 1 from both sides.
r-1+1≥1+1
Solve.
Add the numbers from left to right.
r≥2
Therefore, the correct answer is r<-4 and r≥2.
I hope this helps, let me know if you have any questions.
2,3,1,4,0 solution i need this sequence please?
Answer:
01234
Step-by-step explanation:
Sequence is smallest to largest therefore the sequence is 01234
Answer:01234
Step-by-step explanation:
Hope this helps.
What is the quotient of (−72) ÷ (−9) ÷ 2? 16 4 −4 −16
Hi
[tex]\star\sim\star\sim\star\sim\star\sim\star\sim[/tex]
Q-u-e-s-t-i-o-n:what is -72:(-9):2A-n-s-w-e-r:4
bc a negative over a negative gives us a positive , so -72:(-9) is the same as 72:9 which is 8. Then we divide that by 2 and arrive at 4
[tex]\star\sim\star\sim\star\sim\star\sim\star\sim[/tex]
If possible, use the Law of Detachment to make a conclusion. If not possible to make a conclusion, explain why.
If there is lighting, then it is not safe to be out in the open.
Marla sees lighting from the soccer field.
Using the Law of Detachment, the conclusion is that it is not safe to be out in the open, as the conclusion is true, since the conditional and the hypothesis are true.
What is the Law of Detachment?The Law of Detachment is a logic axiom which states that if a conditional and the hypothesis are true, then the conclusion is also true.
Supposing a conditional statement p -> q, we have that:
p is the hypothesis.q is the conclusion.For this problem, the sentence is:
"If there is lighting, then it is not safe to be out in the open."
Hence p and q are given as follows:
p: presence of lightning.q: it is not safe to be out in the open.Marla sees lighting from the soccer field, hence the statement p is correct, and we know that the conditional is true, hence the conclusion that it is not safe to be out in the open can be reached.
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Help with this equation asap
Answer:
x = -3
Step-by-step explanation:
6x + 4 = 4x - 2
-4x -4x
--------------------
2x + 4 = -2
-4 -4
--------------------
2x = -6
÷2 ÷2
-------------
x = -3
I hope this helps!
6x + 4 = 4x -2
2x = -6
x = -3
Thomas wants to display leaf specimens for a science fair. thomas has 98 leaves, which is 3 fewer than the total needed. will the science display have a total of 111 leaves, 102 leaves, 101 leaves, or 95 leaves?
The total number of leaves that the science display have is 101 leaves.
As per the question, Thomas wants to display leaf specimens for a science fair. Thomas has 98 leaves.
Total number of leaves Thomas have = 98
Thomas has 98 leaves, which is 3 fewer than the total needed. To calculate the total number of leaves Thomas need in the science display , we need to add 3 leaves in the total number of leaves Thomas have. So, we can write this mathematically as,
Total number of leaves that the science display have = Total number of leaves Thomas have + 3
Total number of leaves that the science display have = 98 + 3 = 101
So, the science display have a total of 101 leaves.
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Help me please with this math question
Answer:
40°
Step-by-step explanation:
By the exterior angle theorem, [tex]m\angle 1=100^{\circ}[/tex]. Applying the theorem again, we get [tex]m\angle 2=40^{\circ}[/tex]
Her wages vary directly as the time she works. If her wages for 20 hours are $150, what are her wages for 38 hours
Answer:
285
Step-by-step explanation:
The mean of 10,31,36,36,38,42,47,48,49,52
Step-by-step explanation:
Mean means the average
So to find the average, add all the numbers then divide by how many numbers there are.
Total=389
Divide by 10 = 38.9