The result g'(y) = cos^2 g(y) for any real y.
To show that f(x) = tan x is 1-1 and differentiable on (-pi/2, pi/2), we can use the fact that the derivative of tan x is sec^2 x, which is continuous and positive on (-pi/2, pi/2).
This means that f(x) is increasing and never constant on this interval, thus satisfying the 1-1 condition. Furthermore, since sec^2 x is continuous on this interval, f(x) is also differentiable.
To find the inverse function g'(y), we can use the inverse function theorem, which states that if f is differentiable and 1-1 in an open interval containing x and if f'(x) is not equal to 0, then its inverse function g is differentiable at y = f(x) and g'(y) = 1/f'(x). Applying this theorem to f(x) = tan x, we have:
f'(x) = sec^2 x
f'(g(y)) = sec^2 g(y)
g'(y) = 1/f'(g(y)) = 1/sec^2 g(y) = cos^2 g(y)
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g'(y) = cos^2(g(y)) for any real y. This formula gives us the derivative of the inverse function g(x) of f(x) = tan(x).
a) To show that f(x) = tan(x) is one-to-one (1-1) on the interval (-π/2, π/2), we need to demonstrate that for any two distinct values of x in the interval, their corresponding function values are also distinct.
Let x1 and x2 be two distinct values in (-π/2, π/2), such that x1 ≠ x2. We then have:
f(x1) = tan(x1) and f(x2) = tan(x2)
To prove that f is 1-1, we need to show that if f(x1) = f(x2), then x1 = x2. Taking the contrapositive, if x1 ≠ x2, then f(x1) ≠ f(x2).
Assume x1 ≠ x2. We know that the tangent function has a period of π, so the values of tan(x) repeat after every π units. However, since x1 and x2 are both in the interval (-π/2, π/2), their corresponding tangent values will be distinct. Therefore, f(x1) ≠ f(x2), and we have shown that f is 1-1 on (-π/2, π/2).
To show that f is differentiable on (-π/2, π/2), we can demonstrate that the derivative of f(x) = tan(x) exists and is continuous on the interval. The derivative of tan(x) is sec^2(x), which is defined and continuous on (-π/2, π/2). Hence, f(x) = tan(x) is differentiable on (-π/2, π/2).
b) Since f(x) = tan(x) is 1-1 and differentiable on (-π/2, π/2), it has a differentiable inverse denoted as g(x).
According to the inverse function theorem, if f is differentiable and 1-1 on an interval I, and if f'(x) ≠ 0 for all x in I, then g'(y) = 1 / f'(g(y)).
In this case, f(x) = tan(x), which has a derivative of f'(x) = sec^2(x). Since f'(x) ≠ 0 for all x in (-π/2, π/2), we can use the inverse function theorem to find g'(y) for any real y.
Using the formula g'(y) = 1 / f'(g(y)), we substitute f(x) = tan(x) and solve for g'(y):
g'(y) = 1 / f'(g(y))
g'(y) = 1 / sec^2(g(y))
g'(y) = cos^2(g(y))
Therefore, g'(y) = cos^2(g(y)) for any real y. This formula gives us the derivative of the inverse function g(x) of f(x) = tan(x).
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Whose value is less than 1 is called?
A fraction whose value is less than 1 is called a proper fraction. The numerator is less than the denominator.
What is fraction?A fraction represents the part of a set of objects. It has two parts separated by a dividing line, generally written a/b, where a is numerator and b is denominator.
There are 3 types of fraction. They are
1. Proper fraction
A proper fraction is a fraction whose numerator is less than the denominator. It makes the value is less than 1.
Example: 1/4, 3/5.
2. Improper fraction
An improper fraction is a fraction whose numerator is more than the denominator. The value is more than one.
Example: 6/5, 7/4.
3. Mixed fraction
A mixed fraction is a combination of a whole number and a proper fraction. It is the quotient and the remainder of improper fraction.
Example:
7/5 = 1 2/57/2 = 3 1/2From the explanation, we get a fraction that the value is less than one is a proper fraction.
Your question is incomplete, but most probably your full question was
A fraction whose value is less than 1 is called ...
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Lorenzo is running the 100-meter dash at his school’s track meet. After 67 meters his right shoe falls off he can’t stop to pick it up so how many meters does Lorenzo have to run with 1 shoe
Answer:
Step-by-step explanation:
Lorenzo is running the 100-meter dash and after 67 meters his right shoe falls off. So the remaining distance that Lorenzo has to run with 1 shoe is:
100-meter dash - 67 meters = 33 meters
So, Lorenzo has to run 33 meters with 1 shoe.
Answer:
Step-by-step explanation:
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At the end of a snow storm, Nolan saw there was a lot of snow on his front lawn. The temperature increased and the snow began to melt at a steady rate. The depth of snow on Nolan's lawn, in inches, can be modeled by the equation S=-1.75t+14,S=−1.75t+14, where tt is the time, in hours, after the snow stopped falling. What is the yy-intercept of the equation and what is its interpretation in the context of the problem?
Answer:
r
Step-by-step explanation:
e
Find the length of the third side, if necessary, write in simplest radical form.
Answer:
The length of the third side is 7.
Step-by-step explanation:
We can use the Pythagorean theorem to evaluate the length of the third side.
[tex]a^2+b^2=c^2[/tex]
Let the third side be [tex]a[/tex].
Lets solve for [tex]a.[/tex]
Subtract [tex]b^2[/tex] from both sides.
[tex]a^2=c^2-b^2[/tex]
Take the square root of both sides.
[tex]a=\sqrt{c^2-b^2}[/tex]
We are given
[tex]c=\sqrt{58} \\b=3[/tex]
Lets evaluate [tex]a[/tex].
[tex]a=\sqrt{(\sqrt{58} )^2-3^2}[/tex]
Simplify.
[tex]a=\sqrt{58-9}\\a=\sqrt{49} \\a=7[/tex]
A student writes an incorrect step while checking if the sum of the measures of the two remote interior angles of triangle ABC below is equal to the measure of the exterior angle.
Step 1: m∠x + m∠y + m∠z = 180 degrees (sum of angles of a triangle)
Step 2: m∠w − m∠z = 90 degrees (corresponding angles)
Step 3: Therefore, m∠x + m∠y + m∠z = m∠w + m∠z
Step 4: So, m∠x + m∠y = m∠w
In which step did the student first make a mistake and how can it be corrected? (5 points)
a
Step 1; it should be m∠x + m∠y + m∠z = 180 degrees (sum of corresponding angles)
b
Step 2; it should be m∠w + m∠z = 180 degrees (supplementary angles)
c
Step 1; it should be m∠x + m∠y + m∠z = 90 degrees (corresponding angles)
d
Step 2; it should be m∠w + m∠z = 90 degrees (alternate exterior angle)
Step 2 is the incorrect step and hence d is the correct answer.
What are Adjacent Angles?The two angles are said to be adjacent angles when they share the common vertex and side. The endpoint of the rays, forming the sides of an angle, is called the vertex of an angle. Adjacent angles can be a complementary angle or supplementary angle when they share the common vertex and side.
Given here: m∠w + m∠z = 90 but this is not true because they are adjacent angles and they lie on a line so they must supplement each other.
m∠w + m∠z = 180
Hence, Step 2 is the incorrect step and hence d is the correct answer.
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Determine the measure of DGF?
So on solving the provided question, we can say that - in the circle angle DGF = 59 .
What is circle?Every point in the plane that is a certain distance away from a certain point forms a circle (center). It is, thus, a curve formed by points moving in the plane at a fixed distance from a point. At every angle, it is also rotationally symmetric about the center. A circle is a closed two-dimensional object where every pair of points in the plane are equally spaced out from the "center." A line that goes through the circle creates a specular symmetry line. At every angle, it is also rotationally symmetric about the center.
here,
in the circle
angle AGC = angle DGF
so angle DGF = 59
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Joy is purchasing sweat shirts for her Taekwondo studio she will pay 25 for each shirt plus one time fee of 18.25 for the design
Answer:
Not sure if this is what you are looking for because there isn’t much information given but the equation for this problem would be
25x+18.25
Step-by-step explanation:
Hope this helps
Is there a factor on this pls help Questiob : 5x ^ 2 - 8x + 9
Answer:
(5x - 3)(x - 3)
Step-by-step explanation:
Yes, the expression "5x^2 - 8x + 9" can be factored.
One way to factor it is to find the two numbers that multiply to give 9 (the constant term) and add to give -8 (the coefficient of x). These two numbers are 3 and -3.
We can use these two numbers to factor out the common factor (x - 3) from the expression:
5x^2 - 8x + 9 = (5x^2 - 8x) + 9
= (5x^2 - 8x + 9) - 9 + 9
= (5x^2 - 8x + 9) - (3x - 3x)
= (5x - 3)(x - 3)
So the fully factored form of the expression is (5x - 3)(x - 3)
So, (5x - 3) and (x - 3) are the factors of the expression "5x^2 - 8x + 9"
Write the following expression in its
simplest form.
2x²y
xy
-
3x + 12x.xy
To write the expression in its simplest form, we need to simplify each term and then combine like terms.
The first term of the expression is 2x²y. This term is already in its simplest form and doesn't require any further simplification.
The second term of the expression is xy. This term is also in its simplest form and doesn't require any further simplification.
The third term of the expression is -3x. This term is also in its simplest form and doesn't require any further simplification.
The fourth term of the expression is 12x.xy. To simplify this term, we need to distribute the 12 over the x and y. So, we get 12x.y.
Now that all terms are simplified, we can combine like terms. Since the terms xy and 12x.y are the same, we can add them together to get (12+1)xy = 13xy.
So, the expression in its simplest form is 2x²y + 13xy - 3x.
We can further simplify the above expression by combining x²y and xy together as (x²+1)xy = x³y
So, the final expression in its simplest form is x³y + 13xy - 3x.
After kicking, one coach likes to play practice games with 6 players on each
team. She doesn’t like any team to have more or fewer than 6 players and
she doesn’t like anyone to be left off a team. What are the possible numbers
of players who must show up to practice so that the coach can make these
teams?
Note that the question is about numbers in multiples of 6 or which can be divided equally by 6. Thus the likely number of people who must show up to practice so that the coach can make these teams are 6, 12, 18, 24, 30, 36, 48, and so on. The bottom line is that the number must be a common multiple of 6.
What is the proof that each of the numbers above will create teams of x players each?One way to show that each of the numbers above will equal an exact team of 6 without anyone left off is by dividing each by 6. If we get an integer, then we are correct. Thus:
6/6 = 1 Team12/6 = 2 Teams18/6 = 3 Teams24/6 = 4 Teams30/6 = 5 Teams36/6 = 6 teams48/6 = 8 Teams.Hence, the above Common Multiples are correct.
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A magician asks a volunteer to think of two different positive integers without telling her what they are. She then asks him to calculate x, the sum of the larger number with the square of the smaller, and y, the difference between the numbers. The volunteer tells her that x = 9 and y = 3. Find the original numbers.
Answer: 2 and 5.
Step-by-step explanation: We can use algebra to solve for the original numbers. Let's call the smaller number "a" and the larger number "b". From the information given, we know that:
x = b + a^2
y = b - a
The volunteer tells the magician that x = 9 and y = 3. So we can use these values to create a system of equations:
b + a^2 = 9
b - a = 3
To solve for the original numbers, we can first add the two equations together to eliminate b:
a^2 + 2a + 2 = 12
a^2 + 2a = 10
a^2 + 2a - 10 = 0
(a + 5)(a - 2) = 0
This tells us that the two solutions for a are -5 and 2. But since a is a positive integer, we know that it must be 2.
We can now substitute this value of "a" back into the original equations to find "b":
b + a^2 = 9
b + 2^2 = 9
b + 4 = 9
b = 5
So the original numbers are 2 and 5.
need help with math tyyy
Answer:
The answer is D!
Makena deposited the following
amounts in her savings account last
month: $6.74, $5.21, $5.53, and $3.52.
Divide the sum by 30 to find the average
amount she saved per day for the month.
Show your work.
Makena deposited $0.70 per day.
What is Average?In mathematics, the middle value—which is determined by dividing the sum of all the values by the total number of values—is the average value in a set of numbers. To calculate the average of a set of data, add up all the values and divide the result by the total number of values.
Given:
The savings account last month: $6.74, $5.21, $5.53, and $3.52.
Makena deposited $6.74,$5.21,$5.53 and $3.52 in her saving account.
To find the average amount she saves per day.
We add all the deposited savings
= ( 6.74 + 5.21 + 5.53 + 3.52)
= $21.031
Now, Makena Saved per day for the month = $21.031/30
=$0.70
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I’ll give brainlist!!What is the restricted domain of the quadratic(y=x^2 +4) that would ensure that the inverse is a function? Justify the answer
We can restricted domain of the quadratic(y=x^2 +4) is {-∞,0} or {0,∞}.
What is Quadratic Equation?
Having the form ax^2 + bx + c = 0, quadratic equations are second-degree algebraic expressions. From the term "quad," which meaning square, comes the word "quadratic." A quadratic equation is a "equation of degree 2," to put it another way. A quadratic equation is employed in numerous situations. In addition, there are many uses for quadratic equations in physics, engineering, astronomy, etc.
There are a maximum of two solutions for x in the second-degree quadratic equations. These two solutions for x are denoted as (, ) and are also known as the roots of the quadratic equations.
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). A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.
It cannot have an inverse function because the largest exponent is even (2), unless the domain is constrained.
We know that the vertex is at x = 0 since our equation is already in vertex form;
a(x - h)^2 + k, or 1(x - 0) + 4
In order to limit the domain to only one side of x = 0
either {-∞,0} or {0,∞}
Therefore, We can restricted domain of the quadratic(y=x^2 +4) is {-∞,0} or {0,∞}.
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Write the rule for the linear function shown in the table I’ll give brainliest
Martin is starting a new job. During training, he will earn $16 per hour. Once he starts, Martin will earn $20 per hour. What is the percent of change in Martin's pay from training to actually working?
Answer:
25%
Step-by-step explanation:
percent change = (new value - old value)/(old value) × 100%
percent change = (20 - 16)/(16) × 100%
percent change = 25%
2. State the rule for the following translation:
Answer:
Step-by-step explanation:
7 left 3 up
(x-7)(y+3)
Why is base e used in exponential functions?
Base e used in exponential functions because base e is the base rate of growth shared by all continually growing processes which lets us take a simple growth rate.
What does the base represent in an exponential function?In an exponential function, the base b is a constant. The exponent x is the independent variable in which the domain is the set of real numbers. There are two types of exponential functions, those are exponential growth and exponential decay. In the function f (x) = b^x when b > 1, the function represents exponential growth.
Base e allows us to take a simple growth rate, where all change occurs at the end of the year, and find the influence of compound, continuous growth, where every nanosecond or faster, we are growing slightly.
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A vase is in the shape of a cone. The height is 12 inches and the diameter is 4.4 inches.
What is the lateral surface area to the nearest tenth of a square inch
The lateral surface area of the cone is:
A = 84.3 in²
How to get the lateral surface area of the vase?We know that for a cone of height H and a radius R, the lateral surface area is given by the formula:
A = pi*R*( √(H² + R²))
Where pi = 3.14
Here the height is 12 inches, and the diameter is 4.4 inches, then the radius is:
R = 4.4i/2 = 2.2 in
Then the lateral surface area is:
A = 3.14*2.2in*( √((12in)² + (2.2in)²))
A = 84.3 in²
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SOMEONE HELPP PLSSS
The figure shows a line graph and two shaded triangles that are similar:
image below
The slope is 1/2 throughout the entire line.
What is a line graph?A line graph, also known as a line chart or a line plot, is commonly drawn to show information that changes over time. You can plot it by using several points linked by straight lines. It comprises two axes called the “x-axis” and the “y-axis”. The horizontal axis is called the x-axis.In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G.Line graphs are used to track changes over short and long periods of time For the second triangle, the rise is 1 and the slope is 2. This gives us a slope of 1/2.Any linear function will have the same slope throughout the entire line.me. When smaller changes exist, line graphs are better to use than bar graphs. Line graphs can also be used to compare changes over the same period of time for more than one groupThe slope is 1/2 throughout the entire line.Looking at the first triangle, the rise is 2 and the run is 4. That gives us a slope of 2/4 = 1/2.
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how do i graph this in the image attached?
The value of the equation is P ( 4 , 4 ) , where P is the point of intersection of the equations
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the first equation be represented as A
Now , the value of A is
4x - 2y = 8
Adding 2y on both sides of the equation , we get
2y + 8 = 4x
Subtracting 8 on both sides of the equation , we get
2y = 4x - 8 be equation (1)
Now , let the second equation be represented as B
Now , the value of B is
y = ( 3/2 )x - 2
On simplifying the equation , we get
2y = 3x - 4 be equation (2)
Now , subtracting equation (2) from equation (1) , we get
4x - 8 - ( 3x - 4 ) = 0
4x - 8 - 3x + 4 = 0
x - 4 = 0
Adding 4 on both sides of the equation , we get
x = 4
Substitute the value of x = 4 in equation ( 1 ) , we get
2y = 8
y = 4
So , the value of x = 4 and y = 4
The equations are plotted on the graph and the point of intersection is the solution to the equation , P ( 4 , 4 )
Hence , the equations are solved
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NEED HELP ASAP - Algebra 2 - 30 points
What is the measure of 90,and measureof the inscribe angle
Answer: 45 degrees
Step-by-step explanation:
By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the intercepted arc. The measure of the central angle ∠POR of the intercepted arc ⌢PR is 90°. Therefore, m∠PQR=12m∠POR =12(90°) =45°.
Write the Expression “ x - 9” in words
Answer:
"x-9" can also be written as "x minus nine" or "nine subtracted from x"
Which equation can she use as statement 5? (3x + 24):3x = 85:51 (3x + 24):85 = 51:3x (3x + 24):51 = 3x:85 34:24 = 3x:51.
The equation that she can use as statement 5 is c) (3x + 24): 51 = 3x: 85.
To prove that x = 12, Dora must first prove that the segments ST and RQ are parallel. This can be done by showing that the angles formed by their intersection are congruent.
Since the angles, QRT and STP are congruent (statement 2) and the angles SPT and QPR are congruent (statement 3), the Triangle-Angle Angle Similarity Postulate can be applied to show that triangle SPT is congruent to triangle QPR (statement 4).
Now, the corresponding sides of the two triangles must be in proportion in order to prove that x = 12. For this, Dora can use the equation (3x + 24) : 51 = 3x : 85 (statement 5). This equation states that the ratio of the lengths of the corresponding sides of the two triangles is equal to the ratio of 3x to 51 and 3x to 85, respectively.
Since the ratio of the sides of the two triangles must be equal, then 3x + 24 must be equal to 85 and 3x must be equal to 51. Solving for x, we get x = 12, proving that if segment ST is parallel to segment RQ, then x = 12.
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The complete question
Look at the figure shown below. Dora is writing statements as shown to prove that if segment ST is parallel to segment RQ, then x = 12.
1.Segment ST is parallel to segment QR
Given
2.Angle QRT is congruent to angle STP
Corresponding angles formed by parallel lines and their transversal are congruent.
3.Angle SPT is congruent to angle QPR
Reflexive property of angles.
4.Triangle SPT is congruent to triangle QPR
Angle-Angle Similarity Postulate
5.?Corresponding sides of similar triangles are in proportion.
Which equation can she use as statement 5?
(3x + 24) : 3x = 85 : 51
(3x + 24) : 81 = 3x : 51
(3x + 24) : 51 = 3x : 85
34 : 24 = 3x : 51
If triangles are similar what’s the scale factor
Answer:
1.5 on edg
Step-by-step explanation:
When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangles. In Figure 1, Δ ABC∼ Δ DEF. Figure 1 Similar triangles whose scale factor is 2 : 1. The ratios of corresponding sides are 6/3, 8/4, 10/5.
What is the greatest common
factor (GCF) of 24 and 56?
Answer:
8
I wish you luck in your assignments.
Somebody help please I don’t understand!!
Answer:
1) y=0.56x
2)y=5x
3)y= 0.25x
Step-by-step explanation:
1) 2.24/4 = $0.56 (Cost per carton)
2) 15/3=5 ( miles ran per hour)
3) 3/12 = 0.25 (cost per ounce)
Write in point-slope form an equation of the line that passes through the point (1,7) with slope
3|5
The equation of the line that passes through the point (1,7) with slope
3|5 is 5y=3x+32
What is the equation of a line given two points?Finding the slope between two points on a line and solving for the y-intercept in the slope-intercept equation y=mx+b allows us to formulate an equation for that line.
How do I find the equation of a linear line?To find an equation of a line given the slope and a point.
Identify the slope.
Identify the point.
Substitute the values into the point-slope form, y − y 1 = m ( x − x 1 ) . y − y 1 = m ( x − x 1 ) .
Write the equation in slope-intercept form.
According to the given equation
y=mx+c
where m=3/5
To find c we substitute the values of x as 1 and y as 7
y=3/5x+c
7=3/5+c
c=32/5
Hence substituting c in the equation we get
5y=3x+32
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Use the origin as the center of dilation and the given scale factor to find the coordinates of the vertices of the image of the polygon. K = 1/2
The new coordinates of the polygon are J'(-5/2, 3/2), K'(1, 3/2), L'(1, -3/2), M'(-5/2, -3/2).
What is transformation?Transformation is the movement of a point in the coordinate plane. Types of transformation are reflection, rotation, translation and dilation.
Dilation is the enlargement or reduction in the size of a figure by a scale factor.
Given the coordinates of polygon J(-5, 3), K(2, 3), L(2, -3), M(-5, -3). For a dilation with a scale factor of 1/2, the new coordinates are:
J'(-5/2, 3/2), K'(1, 3/2), L'(1, -3/2), M'(-5/2, -3/2).
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