Given:
[tex]\lim _{x\to0}\frac{\tan 2x-\sin 2x}{x^3}[/tex]Solve:
[tex]\lim _{x\to0}\frac{\tan 2x-\sin 2x}{x^3}[/tex]Use l'hopital's rule:
[tex]\begin{gathered} =\lim _{x\to0}\frac{\frac{d}{dx}(-\sin 2x+\tan 2x)}{\frac{d}{dx}(x^3)} \\ =\lim _{x\to0}\frac{-2\cos (2x)+2\tan ^2(2x)+2}{3x^2} \end{gathered}[/tex]Simplify:
[tex]\begin{gathered} =\lim _{x\to0}\frac{-2\cos (2x)+2\tan ^2(2x)+2}{3x^2} \\ =\lim _{x\to0}\frac{2(-\cos (2x)+\tan ^2(2x)+1)}{3x^2} \end{gathered}[/tex]Apply the constant multiple rule:
[tex]\begin{gathered} \lim _{x\to0}cf(x)=c\lim _{x\to0}f(x) \\ \text{With c=}\frac{2}{3} \\ f(x)=\frac{-\cos (2x)+\tan ^2(2x)+1}{x^2} \end{gathered}[/tex][tex]\begin{gathered} =\frac{2\lim _{x\to0}\frac{-\cos (2x)+\tan ^2(2x)+1}{x^2}}{3} \\ =\frac{2\lim _{x\rightarrow0}\frac{(4\tan ^2(2x)+4)\tan (2x)+2\sin (2x)}{2x}}{3} \end{gathered}[/tex]Similary :
[tex]\begin{gathered} =\frac{2\lim _{x\to0}(2\cos (2x)+12\tan ^4(2x)+16\tan ^2(2x)+4)}{3} \\ =\frac{2(6)}{3} \\ =4 \end{gathered}[/tex]Jason jumped off of a cliff into the ocean in Acapulco while vacationing with some friends. His height above ocean measured in feet as a function of time could be modeled by the function h(t) = -16t2 + 16t + 672, where t is the time in seconds and h is the height in feet. After how many seconds did Jason hit the water
You have the following function for the height respect to the water, Jason has after time t he jumped:
[tex]h(t)=-16t^2-16t+672[/tex]In order to determine the time Jason takes to hit the water, equal h = 0. Thus, you have a quadratic equation and it is necessary to find the zeros of the polynomial:
[tex]0=-16t^2-16t+672[/tex]use the following quadratic formula to find the zeros:
[tex]t=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]where a,b and c are the coefficients of the polynomial:
a = -16
b = -16
c = 672
replace these values into the formula:
[tex]\begin{gathered} t=\frac{-(-16)\pm\sqrt[]{(-16)^2-4(-16)(672)}}{2(-16)} \\ t=\frac{16\pm\sqrt[]{43264}}{-32} \\ t=\frac{16\pm208}{-32} \\ t=-0.5\pm(-6.5) \end{gathered}[/tex]Then, you obtain the following two values:
[tex]\begin{gathered} t=-7 \\ t=6 \end{gathered}[/tex]Select the positive values because negative time does not have physical meaning.
Hence, Jason takes 6 seconds to hit the water.
g(x) = 4x + 4
f(x) = x3 − 1
Find (g ◦ f)(x)
Answer:
4x³
Step-by-step explanation:
substitute x = f(x) into g(x)
(g ￮ f)(x)
= g(f(x))
= g(x³ - 1)
= 4(x³ - 1) + 4
= 4x³ - 4 + 4
= 4x³
Factor the expression over the complex numbers. x^2+25 Enter your answer in the box.
please help
The complex number factors are (x+5i) and (x-5i).
The term a + bi, where a and b are real numbers, can be used to represent any complex number.
A complex number is a component of a number system that includes an element with the symbol I, sometimes known as the imaginary unit, and that extends the real numbers by satisfying the equation i² = -1. Since no real number can fulfil the conditions in the previous equation, i was referred to as an imaginary number. The complex number a + bi is known as having real and imaginary parts, respectively, a and b. The group of complex numbers is represented by the letter C.The given expression is x² + 25
Now this can be written as :
x² - [(i)²×(5)²] as we know that i² = -1.
Now we use the algebraic form of a²-b² =(a + b)(a - b) to separate it into complex factors.
or , (x+5i)(x-5i)
Therefore the complex number factors are (x+5i) and (x-5i).
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Which Similarity Theorem is described by the text below?If the lengths of three sides of one triangle are proportional to the lengths of three sides ofanother, then the triangles are similar.A. AAB. ASAC. SASD. SSS
Answer:
SSS.
Step-by-step explanation:
The text above described the Side-Side-Side Theorem, which states that two triangles are congruent if three sides of one are equal or proportional respectively to three sides of the other (SSS=SSS).
Question 12 of 47Solve kx - 2 = 7 for x.
ANSWER:
D. x = 9/k
STEP-BY-STEP EXPLANATION:
We have the following equation:
[tex]kx-2=7[/tex]We solving for x:
[tex]\begin{gathered} kx=7+2 \\ \\ x=\frac{9}{k} \end{gathered}[/tex]The correct answer is D. x = 9/k
Pls help i give crowns
Answer:
1. 27
2. [tex]\frac{27}{7}[/tex]
Step-by-step explanation:
9 / 7/3
9 * 3/7
9/1 * 3/7
27/7
Answer:
The answer to the first question is:
[tex]27[/tex] groups.
And the second answer is:
[tex]\frac{27}{7}[/tex]
Step-by-step explanation:
Step 1: Analyze the number line
As we can see from the number line, each interval equals to [tex]\frac{1}{3}[/tex].
We can prove this by the fact that at the seventh interval, the value is [tex]\frac{7}{3}[/tex].
So, the value between each intervals would be:
[tex]\frac{7}{3}\div 7\\=\frac{1}{3}[/tex]
Step 2: Solving the first question:
We can see that the number [tex]9[/tex] lies on the [tex]27\text{th}[/tex] interval.
Since the value between each interval is [tex]\frac{1}{3}[/tex], it would take [tex]27[/tex] groups of [tex]\frac{1}{3}[/tex] to equal to [tex]9[/tex].
Step 3: Solving the second question:
From the number line, we can see that it takes four groups of [tex]\frac{7}{3}[/tex] to equal to [tex]\frac{28}{3}[/tex].
And since we are taking [tex]\frac{7}{3}[/tex] as a single group here, each interval in a group of [tex]\frac{7}{3}[/tex] would be [tex]\frac{1}{7}[/tex] of the group.
So, the answer of the division would be: [tex]4[/tex] (the number of groups of [tex]\frac{7}{3}[/tex]needed to reach [tex]\frac{28}{3}[/tex]) minus [tex]\frac{1}{7}[/tex] (the amount of the group that when subtracted, brings the entire answer to [tex]9[/tex]):
[tex]4-\frac{1}{7}\\\\\frac{4\times 7}{7}-\frac{1}{7}\\\\=\frac{28}{7}-\frac{1}{7}\\\\=\frac{28-1}{7}\\\\=\frac{27}{7}[/tex]
Plot (−1 3/4, 4 1/2) on the coordinate plane. Keyboard Instructions Initial graph state The horizontal axis goes from -5 to 5 with ticks spaced every 1 unit(s). The vertical axis goes from -5 to 5 with ticks spaced every 1 unit(s).
The point [tex](-1\frac{3}{4},4\frac{1}{2})[/tex] has been plotted on the coordinate plane.
Given the point [tex](-1\frac{3}{4},4\frac{1}{2})[/tex]
The coordinate plane is used to graph the points, lines and other objects.
First we have to convert the mixed fraction to the simple fraction
[tex]-1\frac{3}{4}[/tex] = -7/4
[tex]4\frac{1}{2}[/tex] = 9/2
Now we have to convert the simple fraction to decimal form
-7/4 = -1.75
9/2 = 4.5
The point is (-1.75,4.5)
Here the condition of axis is given, the horizontal axis is goes from -5 to 5, that is -5 < x < 5, and the vertical axis goes from -5 to 5 that is -5 < y < 8
Plot the points on the coordinate plane
Hence, the point [tex](-1\frac{3}{4},4\frac{1}{2})[/tex] has been plotted on the coordinate plane.
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Which set of values belong to the domain and range of a relation?
Domain is x values of the set and range is y values of the set.
What is domain and range?
Domain is the set of all first numbers of the ordered pairs in a relation and range is the set of second numbers of the ordered pair in relation.
The domain is the set of independent values, meaning the set of x-coordinates of the ordered pair as x is an independent variable.
The range is set of dependent values, meaning the set of y-coordinates of the ordered pair as y is an independent variable.
For example- we have a set
(1,2) (0,6) (2,3) (4,5)
Domain = {1,0,2,4}
Range = {2,0,3,5}
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Without using a calculator, order the
following expressions from least to greatest.
29 2
3
64
The expressions from least to greatest is (√3)² < 8² < 29².
How to compare expressions with indices?
Expressions with indices are of three types:
(1) when bases are equal and powers are different;
(2) when powers are equal but bases are different;
(3) when both powers and bases are unequal.
To solve an expression of type (1), we compare the powers for the given expressions only to determine the greater number efficiently, no heed needs to be given to the base.
To solve an expression of type (2), we compare the bases for the given expressions only to determine the greater number efficiently, no heed needs to be given to the powers.
To solve an expression of type (3), we reduce the power of an expression until it matches the other, so that for comparison only bases are to be taken into consideration without any heed to powers.
Given the expressions are 29², 3 and 64
The given expressions can be re-written as: 29², (√3)² and 8².
On comparing the expressions with the above literature, they match type (2), thus following the assertions we should compare the bases only to find the greatest number.
Thus, the correct order for expressions is: (√3)² < 8² < 29²
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Calculate Sample Variance for the following data collection: 10, 11,12, 13, 14, 18.
To calculate the variance follow these steps:
1. Work out the Mean
2. Then for each number: subtract the Mean and square the result .
3. Then work out the average of those squared differences.
The data are 10, 11, 12, 13, 14, 18
The first step find the mean
The mean = sum of data/number of data
The sum = 10 + 11 + 12+ 13 +14 + 18 = 78
The number =
Evaluate (m³ + 4) -n² if m= -4 and n=-6. Write your answer as an integer.
Answer:
-96
Step-by-step explanation:
m = -4
n = -6
(m³ + 4) - n² =
= [(-4)³ + 4] - (-6)²
= -64 + 4 - 36
= -60 - 36
= -96
Is the ordered pair a solution of the equation? 4x+5y=-1; (1,-1) And 2x-3y=7; (8,-3)
The given equations are:
4x + 5y = - 1
2x - 3y = 7
For the ordered pairs to be a solution of the equation, the equation must be true when the points are substituted into the equation. That is, the right hand side must equal the left hand side.
For 4x + 5y = -1
To test if (1, -1) is a solution, substitute x = 1, and y = -1 into the equation.
4(1) + 5(-1) = -1
4 - 5 = -1 (True)
Therefore, (1, -1) is a solution for the equation 4x + 5y = -1
For 2x - 3y = 7
To test if (8, -3) is a solution, substitute x = 8, and y = -3 into the equation
2(8) - 3(-3) = 7
16 + 9 = 7 ( False)
Therefore, (8, -3) is not a solution for 2x - 3y = 7
The volume of this cylinder is approximately 12,468.94 cubic feet.The radius is ___ feet.Use π = 3.14.
we have that
the volume of the cylinder is
V=pi(r^2)h
the radius is given
r=57 cm
Remember that
1 ft=30.48 cm
so
57 cm=57/30.48=1.87 ft
the radius is 1.87 ftWhat's the correct letter name for this below?
The correct letter name for ∠1 is ∠ABC.
How to name angles?There are various ways of naming angles . One can name an angle by its vertex, by the three points of the angle (the middle point must be the vertex), or by a letter or number written within the opening of the angle.
The method adopted to name the angle below is the three points of the angle.
Therefore, using the three points, the ∠1 can be named as follows:
∠1 = ∠ABC
The middle point is usually the vertex.
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Describe a series of transformations Matt can perform to decide if the two windows are congruent
Congruent shapes are produced by combining rotations, reflections, and translations. In fact, any pair of congruent shapes can be matched to one another by combining one or more of these three transformations.
Define transformations.A point, line, or geometric figure can be transformed into one of four different shapes or locations. Pre-Image refers to the shape of the object prior to transformation, whereas Image refers to the location and shape of the object after transformation.
Given information
We now know that the size and shape of the figures are preserved during stiff transformations (reflections, translations, and rotations). The pre-image and the actual image concur completely.
The following Matt's transformative skills:
Reflection
Because the The line of reflection remains the same distance away from comparable points from the pre-image to the image.
As a Congruence Transformation - rotations
Rotating a figure causes it to twist. Even though the figure is the same size and shape as before, it appears to have toppled over. A clock is a fantastic example of how the earth actually rotates. Every hour or every day, the clock's connecting arms revolve around their axis. A rotation's degree determines what it is, and common rotations include 90, 180, and 270 degrees. Before going back to its original position, the figure rotates a full 360 degrees. the direction of a rotation, whether it is counterclockwise or clockwise. It is possible to determine the degree, amount, and and the direction of a revolution.
Translational congruence transformation
When an object or shape is transported from one location to another without altering its size, shape, or orientation, the movement is referred to as a translation. During a translation, also known as a slide occasionally, every point on an object or shape is moved by the same amount and in the same direction.
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2. Calculate each surface area to the nearest tenth of a square centimetre ) r = 10 cm b) a) d=7 cm 30 cm 22 cm 3. bbelow 1
The formula to find the surface area of a cylinder is:
[tex]\begin{gathered} S.A.=2\pi r^2+2\pi rh \\ \text{ Where} \\ S.A\text{. is the surface area,} \\ r\text{ is the radius, and} \\ h\text{ is the height of the cylinder} \end{gathered}[/tex]So, in this case, we have:
[tex]\begin{gathered} r=10\operatorname{cm} \\ h=22\operatorname{cm} \\ S.A.=2\pi r^2+2\pi rh \\ S.A.=2\pi(10cm)^2+2\pi(10cm)(22cm) \\ S.A.=2\pi(10cm)(10cm)+2\pi(10cm)(22cm) \\ S.A.=628.3cm^2+1382.3cm^2 \\ S.A.=2010.6cm^2 \end{gathered}[/tex]Therefore, the surface area of the given cylinder rounded to the nearest square centimetre is 2010.6 cm².
What is the equal number for 2 tens and 15 ones
Sam has a piece of rope that is 3 1/2 feet long. He cuts the rope so that one piece is 1 1/3 feet long what is the length of the other half?
The length of the other half is:
[tex]\frac{7}{2}-\frac{4}{3}=\frac{7\cdot3-4\cdot2}{2\cdot3}=\frac{21-8}{6}=\frac{13}{6}[/tex][tex]\frac{13}{6}=\frac{12+1}{6}=\frac{12}{6}+\frac{1}{6}=2+\frac{1}{6}=2\frac{1}{6}\text{ fe}et[/tex]samuel carries three sacks of gravel to his motorcycle which weigh 800 cm³ cube how many liters did he carry?Plss answer it now I need help
Explanation:
To be able to determine the number of liters Samuel carried, let's convert first the 800 cm³ to liters.
The conversion formula is 1 cm³ = 0.001 liters.
So, to convert 800 cm³ to liters, let's multiply 800 by 0.001.
[tex]800cm^3\times\frac{0.001L}{1cm^3}=0.8L[/tex]Answer:
Determine the area of the shaded triangle inside the square explain your strategy
To find the area of the shaded region, we have to find the area of the square first.
[tex]A_{\text{square}}=l^2=(10in)^2=100in^2[/tex]Then, we find the area of each yellow triangle inside (up and down). The base of each triangle is 10 inches, and the height of each of them is 7 inches.
[tex]\begin{gathered} A_{\text{triangle}}=\frac{1}{2}bh=\frac{1}{2}\cdot10in\cdot7in=35in^2 \\ A_{\text{triangle}}=\frac{1}{2}\cdot10in\cdot7in=35in^2 \end{gathered}[/tex]Add the area of both triangles.
[tex]A_t=35in^2+35in^2=70in^2[/tex]At last, subtract the area of the triangles from the area of the square.
[tex]A_{\text{shaded}}=100in^2-70in^2=30in^2[/tex]Therefore, the area of the shaded region is 30 in^2.The strategy is to subtract the area of the yellow triangles (up and down) from the area of the square.
The answers are 0, 1, 3, and 3.5. I think the answer is 3 but I'm not sure
The y-intercept of any function is the point where the line intersect with the y-axis, at this point the value of x is zero.
If you look at the graph, the line intersects the y-axis at point (0,3), so the y-coordinate of the y-intercept is 3
I need help finding the vortex angle with these three sides. I don’t really know how to do this sort of thing with an isosceles triangle. Once I know the vortex angle I can figure out the base angles.
Answer:
[tex]\begin{gathered} \text{ Smalles angle in the triangle}\colon \\ D=14.36\degree \\ \text{ The measure of the two congruent angles:} \\ Step-by-step explanation:To find the angle D, use the law of cosines, which is represented as:
[tex]\begin{gathered} d^2=e^2+f^2-2(e)(f)\cos D \\ 3^2=12^2+12^2-2(12)(12)\cos D \\ 9=144+144-288\cos D \\ -279==-288\cos D \\ \frac{-279}{-288}=\cos D \\ D=\cos ^{-1}(\frac{-279}{-288}) \\ D=14.36\degree \end{gathered}[/tex]Then, since the base angles are equal in measure:
[tex]\begin{gathered}Write an equation of the line passing through point P(3,8) that is parallel to the line y\ ={1}{5}(x+4) NEEP HELP ASAP
Answer:
y = (1/5)(x + 37)Step-by-step explanation:
Slope-intercept form:
y = mx + bGiven line has a slope of m = 1/5.
Parallel lines have equal slopes, so the line we are looking for is:
y = (1/5)x + b and passing through point P(3,8).Substitute the coordinates and slope and find b:
8 = (1/5)*3 + bb = 8 - 3/5b = 7 2/5The line is:
y = 1/5x + 7 2/5 ory = (1/5)(x + 37)Hey I need help I got stuck I am struggling
According to the trigonometric relations in a right triangle, the following relation is valid:
[tex]\begin{gathered} \sin x\degree=\frac{\text{opposite leg}}{hypotenuse}=\frac{4.9}{7.5} \\ \sin x\degree=0.6533\ldots \\ x\degree=\arcsin 0.6533\ldots \\ x\degree\approx40.8\degree \end{gathered}[/tex]The triangle below will be translated 2 units up and 5 units to the right.3АA22-4-3045- 1Bс2What will be the coordinates of point A'?O (4,4)O (1,7)O (-3,-3)O (-6,0)
In general, a translation:
• in the x-direction by n units is given by: (x, y) → (x + n, y),
,• in the y-direction by m units is given by: (x, y) → (x, y + m),
,• the combination of both translations is: (x, y) → (x + n, y + m).
In this problem, we want to translate a triangle 2 units up and 5 units to the right, so we want to translate the points of the triangle in the following way:
[tex](x,y)\rightarrow(x+5,y+2)[/tex]From the graph we see that the coordinates of point A are:
[tex](x_A,y_A)=(-1,2)[/tex]Making the translation we get:
[tex](x_A+5,y_A+2)=(-1+5,2+2)=(4,4)[/tex]Answer
(4,4)
question 11, rewrite the formulaThe area of a triangle is A-a. Rewrite the formula for b.b. Rewrite the formula for h.
Given the formula:
[tex]A\text{ = }\frac{1}{2}bh[/tex]a) Rewrite for b:
First step: Multiply 2 to both sides
[tex]\begin{gathered} 2A\text{ = }\frac{1}{2}bh\text{ }\ast2 \\ \\ 2A\text{ = bh} \end{gathered}[/tex]Second step: divide both sides by h
[tex]\frac{2A}{h}\text{ = b}[/tex][tex]\begin{gathered} \text{Thus,} \\ b\text{ = }\frac{2A}{h} \end{gathered}[/tex]B) Rewrite for h:
First step: Multiply 2 to both sides
[tex]\begin{gathered} 2A\text{ = }\frac{1}{2}bh\ast2 \\ \\ 2A\text{ = bh} \end{gathered}[/tex]Second step: divide both sides by b
[tex]\begin{gathered} \frac{2A}{b}\text{ = }\frac{bh}{b} \\ \\ \frac{2A}{b}\text{ = h} \end{gathered}[/tex][tex]h\text{ = }\frac{2A}{b}[/tex]How would you write the phrase, “I do not want fries with my burger” in spanish?
The phrase, “I do not want fries with my burger” in Spanish is “No quiero papas fritas con mi hamburguesa".
What is a phrase?A phrase is a group of words or a single word that functions as a grammatical unit in syntax and grammar. The English phrase "the extremely happy squirrel," for instance, is a noun phrase that also incorporates the adjective phrase "very happy." A phrase may be made up of a single word or a whole sentence.
A phrase is typically a group of words with a specific idiomatic meaning or other importance, such as "all rights reserved," "economical with the truth," "kick the bucket," and the like. It could be a euphemism, a proverb, a permanent word, a figure of speech, or anything else. These are referred to as phrasemes in linguistics.
A phrase is a group of words that together convey one or more speech elements. They are crucial because they enhance your ability to communicate verbally and in writing. In this case, the phrase has been translated into Spanish.
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A certain species of fish require 1.5 cubic feet of water per fish to maintain a healthy environment. Find the maximum number of fish you could put in a tank measuring 5 feet by 3 feet by 4 feet.
Answer:
40
Step-by-step explanation:
Given information:
1.5 ft³ = Volume of water required per fish.Dimensions of the tank = 5 ft × 3 ft × 4 ftModel the tank as a rectangular prism.
[tex]\begin{aligned}\textsf{Volume of a rectangular prism}&=\sf width \times length \times height\\\\\implies \textsf{Volume of tank}&=\sf 5 \; ft \times 3\; ft \times 4 \; ft\\& = \sf 15\;ft^2 \times 4\;ft\\& = \sf 60 \; ft^3\end{aligned}[/tex]
To find the maximum number of fish that can be put in the tank, divide the found volume of the tank by the given volume of water required per fish:
[tex]\begin{aligned}\textsf{Maximum number of fish}&=\textsf{Volume of tank} \div \textsf{Water per fish}\\& = \sf 60\;ft^3 \div 1.5 \; ft^3\\& = \sf 60 \div 1.5\\& = \sf 40\end{aligned}[/tex]
Therefore, the maximum number of fish that can be put in the tank is 40.
The deer population in the united states can be predicted by the expression 523(1.099)^t where t is the number of years since 1990 . what does the value 523 represent?
The exponential growth or decay function is a function that grows or shrinks at a constant percent growth rate. The equation can be written in the following form:
[tex]f(x)=a(1+r)^x[/tex]or
[tex]f(x)=ab^x\text{ where b= 1+r}[/tex]here
a is the initial or starting value of the function or initial population.
r is the percent growth or decay rate, written as a decimal
and
b is the growth factor or growth multiplier.
Now, consider the following exponential model:
[tex]f(t)=523\left(1.099\right)^t[/tex]notice that in this case:
a = 523
according to the definition of exponential growth, we can conclude that the correct answer is:
Answer:523 is the initial population of deers.5. If y = 1 and x = then in terms of x, y = (A)+1 (B) (C) (D)
The question states:
Notice that the angle for which its measure is arctan(a/b), is the angle that has "a" as the oposite side, and "b" as the adjacent side , so it is the angle on the bottom right of the triangle.
Then the cosine of that angle, is defined as:
cos(angle) = adjacent/hypotenuse = b / squareroot( a^2+b^2)
[tex]\cos (\theta)=\frac{b}{\sqrt[]{a^2+b^2}}[/tex]which agrees with answer D in the list of possible answers given.