Answer:
[tex]\dfrac{\text{d}}{\text{d}x} \tan^{-1}x=\dfrac{1}{1+x^2}[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6.5 cm}\underline{Trigonometric Identity}\\\\$\tan^{-1}(A)-\tan^{-1}(B) \equiv \tan^{-1}\left(\dfrac{A-B}{1+AB}\right)$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{3 cm}$\displaystyle \lim_{h \to 0} \left[\dfrac{\tan^{-1} \theta}{\theta} \right]=1$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Differentiating from First Principles}\\\\$\text{f}\:'(x)=\displaystyle \lim_{h \to 0} \left[\dfrac{\text{f}(x+h)-\text{f}(x)}{(x+h)-x}\right]$\\\end{minipage}}[/tex]
Given function:
[tex]\text{f}(x)=\tan^{-1}x[/tex]
[tex]\implies \text{f}(x+h)=\tan^{-1}(x+h)[/tex]
Differentiating from first principles:
[tex]\begin{aligned}\text{f}\:'(x) & =\displaystyle \lim_{h \to 0} \left[\dfrac{\text{f}(x+h)-\text{f}(x)}{(x+h)-x}\right]\\\\& =\lim_{h \to 0} \left[\dfrac{\tan^{-1}(x+h)-\tan^{-1}x}{(x+h)-x}\right]\end{aligned}[/tex]
Using the trigonometric identity to rewrite the numerator:
[tex]\begin{aligned}& =\lim_{h \to 0} \left[\dfrac{\tan^{-1}\left(\dfrac{x+h-x}{1+x(x+h)}\right)}{(x+h)-x}\right]\\\\& =\lim_{h \to 0} \left[\dfrac{\tan^{-1}\left(\dfrac{h}{1+x^2+xh)}\right)}{h}\right]\end{aligned}[/tex]
[tex]\textsf{Multiply the denominator by }\dfrac{1+x^2+xh}{1+x^2+xh}:[/tex]
[tex]= \displaystyle \lim_{h \to 0} \left[\dfrac{\tan^{-1}\left(\dfrac{h}{1+x^2+xh)}\right)}{\dfrac{h(1+x^2+xh)}{(1+x^2+xh)}}\right][/tex]
Separate:
[tex]= \displaystyle \lim_{h \to 0} \left[\dfrac{\tan^{-1}\left(\dfrac{h}{1+x^2+xh)}\right)}{\dfrac{h}{(1+x^2+xh)}} \right] \cdot \displaystyle \lim_{h \to 0} \left[\dfrac{1}{1+x^2+xh}\right][/tex]
[tex]\textsf{Use }\displaystyle \lim_{h \to 0} \left[\dfrac{\tan^{-1} \theta}{\theta} \right]=1:[/tex]
[tex]= 1 \cdot \displaystyle \lim_{h \to 0} \left[\dfrac{1}{1+x^2+xh}\right][/tex]
As h gets close to zero:
[tex]= 1 \cdot \left[\dfrac{1}{1+x^2}\right][/tex]
Simplify:
[tex]=\dfrac{1}{1+x^2}[/tex]
Answer:
To find the derivative of [tex]\tan^{-1} x[/tex] using the first principle of derivative, we need to use the definition of the derivative:
f'(x) = lim(h->0) [(f(x + h) - f(x)) / h]
where f(x) = [tex]\tan^{-1} x[/tex].
Substituting f(x) into the definition of the derivative, we get:
f'(x) = lim(h->0) [([tex]\tan^{-1}[/tex](x + h) - [tex]\tan^{-1}/tex) / h]
To simplify this expression, we can use the formula for the inverse tangent of a sum:
[tex]\tan^{-1}[/tex](a + b) = [tex]\tan^{-1}[/tex]a + [tex]\tan^{-1}[/tex]b - [tex]\pi[/tex]/2
Using this formula, we can rewrite the numerator of the expression above as:
([tex]\tan^{-1}[/tex](x + h) - [tex]\tan^{-1}/tex) = [tex]\tan^{-1}[/tex]((x + h) / (1 + (x + h)^2)) - [tex]\tan^{-1}[/tex](x / (1 + x^2))
Now, substituting this expression back into the definition of the derivative, we get:
f'(x) = lim(h->0) [[tex]\tan^{-1}[/tex]((x + h) / (1 + (x + h)^2)) - [tex]\tan^{-1}[/tex](x / (1 + x^2))] / h
We can simplify this expression using algebra and trigonometry, and we get:
f'(x) = lim(h->0) [h / (1 + x^2 + hx + h^2 + x^2h + xh^2)] / h
f'(x) = lim(h->0) 1 / (1 + x^2 + hx + h^2 + x^2h + xh^2)
Now we can simplify this expression by dropping the terms that contain h^2 or higher powers of h, since they will approach zero faster than h as h approaches zero. We also drop the term containing x^2h, since it is a second-order term and will also approach zero faster than h. This leaves us with:
f'(x) = lim(h->0) 1 / (1 + x^2 + hx)
Now we can evaluate the limit as h approaches zero:
f'(x) = 1 / (1 + x^2)
Therefore, the derivative of [tex]\tan^{-1} x[/tex] by first principle of derivative is:
[tex]\frac{d}{dx}[/tex][tex]\tan^{-1} x[/tex] = 1 / (1 + x^2)
What is the value of 3-(-2)?
Answer:
5
Step-by-step explanation:
2 negatives will make a positive so 3-(-2) is equal to 3+2 and the answer is 5.
hope it helps
Please help with this question!
Hi Student!
Looking at this problem, it's fairly simple and just asks us to determine whether x = 1 is a solution to the expression that was provided. The first step that we must take to solving this is to plug in the value 1 for each of the places that we have x. Then we simplify and look at the final expression and if it evaluates to true than x = 1 is a solution to the expression.
Plug in the values
[tex]x^3 + 3 \ge 8 - 4[/tex][tex](1)^3 + 3 \ge 8 - 4[/tex]Simplify both sides
[tex]1 + 3 \ge 8 - 4[/tex][tex]4 \ge 4[/tex]Looking at the final expression that is shown, we can see that it states that 4 is greater than or equal to 4 which evaluates to true since 4 is equal to 4. Therefore, x = 1 would be considered a solution of the expression that was provided in this problem.
Write an expression to represent the given statement. Use n for the variable.
help please!! i need to turn this in
Answer:
a. [tex]x^{-2}[/tex]
b. [tex]x^{-6}[/tex]
c. 1
d. 1
e. [tex]x^{-6}[/tex]
f. [tex]x^{15}[/tex]
Step-by-step explanation:
find the modulus if the complex number 6-8i
[tex]|a+bi|=\sqrt{a^2+b^2}[/tex]
[tex]|6-8i|=\sqrt{6^2+(-8)^2}=\sqrt{36+64}=\sqrt{100}=10[/tex]
factorize 12f-42 thank you
Nora buys milk and onions at the store. She pays a total of $35.67. She pays a total of $5.91 for the milk. She buys 6 bags of onions that each cost the same amount. How much does each bag of onions cost?
Solve:
(√x + 2)(√x - 2) = 0
x = ?
Answer: x = 4
Step-by-step explanation:
(√x + 2)(√x - 2) = 0
√x*√x - 2√x + 2√x -4 = 0
x - 0 - 4 = 0
x = 4
Answer:
[tex]x=4[/tex]
Step-by-step explanation:
(√x + 2)(√x - 2) = 0
√x × √x - √x · 2 + 2 · √x - 2 · 2 = 0
√xx - 2 × √x + 2 · √x - 4 = 0
√x^1+1 - 2 · √x + 2 × √x - 4= 0
√x² - 2 · √x + 2· √x - 4 = 0
x^2/2 - 2 × √x + 2 · √x - 4 = 0
x - 2· √x + 2 · √x - 4 = 0
x - 4 = 0
(x - 4) + 4 = 0
x - 4 + 4 = 0
x = 4
Your history book cost $43 dollars in 1999. The earlier edition sold for $37 in 1992. Comparing the converted costs, which book is more expensive? (Assume the 1999 CPI is 166.3 and the 1992 CPI is 140.3)
When the converted costs are compared, the earlier edition of the history book is more expensive.
Which book is more expensive?The consumer price index measures the changes in price of a basket of good. It is used to measure inflation.
Converted price of the earlier edition = (CPI in 1992 / CPI in 1999) x later edition of the book
(140.3 / 166.3) x $43 = $36.28
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Criangle has side lengths measuring 2x + 2 ft, x + 3 ft,
dn ft.
11 12
13 14 15
Which expression represents the possible values of n,
in feet? Express your answer in simplest terms.
x-1
n = 3x + 5
n=x-1
3x+5
Please Help I'm not really good at math
The true statements about the triangles RST and DEF are: (a), (d) and (e)
How to determine the true statements?The statement ΔRST ≅ ΔDEF means that the triangles RST and DEF are congruent.
This above implies that:
The triangles can be mapped onto each other by rigid transformations such as reflection, translation and rotationThe transformation does not include dilationCorresponding sides are congruentThe above means that the possible true statements are: (a), (d) and (e)
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Which relation describes the graph?
The relation that describes the graph {(-1, 3), (-2, 1), (-3, -3), (-4, -5)}
What is graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
The coordinates of given graph:
(-1, 3)(-2, 1)(-3, -3)(-4, 5)Hence, the relation is {(-1, 3), (-2, 1), (-3, -3), (-4, 5)}.
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identify the equivalent value of 2.4494897…
Squares are the results of multiplying a value by itself. The value of 2.4494897… is equivalent to √6.
What is square root?Squares are the results of multiplying a value by itself. Whereas the square root of a number is a value that when multiplied by itself yields the original value. As a result, both are vice versa approaches. For example, the square of 2 is 4 and the square root of 4 is 2.
The value of 2.4494897… is equivalent to √6.
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At 3:40 P.M, the hour hand and the minute hand of a clock form an angle of:
Answer:
130 degrees
Step-by-step explanation:
3:40, it has traveled 6 * 40 = 240 degrees which is 110 degrees from vertical The difference is 240 - 110 = 130 degrees, which is the final answer.
Hope This Helped
Find the first five terms of the sequence for which:
a₁ =1/3 and d = 1/2
(Write your answers as fractions)
Which of the following matches a quadrilateral with the listed characteristics
below?
1. Figure has 4 right angles
2. Figure has 4 congruent sides
3. Both pairs of opposite sides parallel
A. Rectangle
B. Parallelogram
C. Trapezoid
D. Square
The quadrilateral that matches the characteristics is a square.
What is a quadrilateral?A quadrilateral is a 2 dimensional figure with 4 sides. Example of quadrilaterals are rectangle, rhombuses, square, parallelograms and many more.
Therefore, the quadrilateral that matches the characteristics is a square.
Properties of a square
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A con is tossed 400 times what is the standard deviation
Answer:
deviation is 10
this will help you
Find The reciprocal for
2/5
6/13
21/80
Answer:
5/2 13/6 80/21
Step-by-step explanation:
the reciprocal is just flipping the fraction over, if they asked you to simplify its
4 1/2, 2 1/6, and 3 4/5
Help!!!!! Meeee!! Asappppp
Reason:
Refer to the diagram below. The tiny square angle markers tell us we have a 90 degree angle, meaning those lines are perpendicular.
You can think of the parallel lines being the metal part of train tracks, while the perpendicular line connecting the metal rails is the wooden part of the train tracks.
Express 17 as a ratio of 68
Answer:
is it 1:4 or am I missing something
Step-by-step explanation:
One number is 3 more than 2 times another. The sum of the numbers is 15 . Find the two numbers.
Answer:
4 and 11
Step-by-step explanation:
See attached image
A blimp, suspended in the air at a height of 600 feet, lies directly over a line from a sports stadium to a planetarium. If an angle of depression from the blimp to the stadium is 37 degrees and from the blimp to the planetarium is 29 degrees, find the distance between the sports stadium and the planetarium.
Step-by-step explanation:
so, this sounds like the blimp is located between the stadium and the planetarium.
we have a triangle :
the ground distance between the stadium and the planetarium is the baseline.
and the 2 lines of sight from the blimp on one side to the stadium and on the other side to the planetarium are the 2 legs.
we know the height of this triangle is 600 ft.
the angle of depression down to the stadium is 37°. which makes the inner triangle angle at the ground point at the stadium also 37°.
and the angle of depression down to the planetarium is 29°. which makes the inner triangle angle at the ground point at the planetarium also 29°.
and because the sum of all angles in a triangle is always 180°, we know the angle at the blimp is
180 - 37 - 29 = 114°
in order to solve this triangle, we need to split it into 2 right-angled triangles by using the height of the main triangle as delimiter.
we get a stadium side and a planetarium side triangle.
the baselines (Hypotenuses) of the 2 triangles are the corresponding lines of sight from the blimp.
the height of the large triangle is also a height and a leg in each small triangle.
and the stadium side part of the large baseline (between ground point and intersection with the height) is the second leg for the stadium side triangle.
and correspondingly, the planetarium side part of the large baseline (between ground point and intersection with the height) is the second leg for the planetarium side triangle.
the inner blimp angle of the stadium side triangle is
180 - 37 - 90 = 53°
and the inner blimp angle of the planetarium side triangle is
180 - 29 - 90 = 61°
now we can use the law of sine to get the lengths of the 2 parts of the baseline of the large triangle.
and when we add these 2 numbers we get the distance between stadium and planetarium.
law of sine is
a/sin(A) = b/sin(B) = c/sin(C)
with the sides being opposite of the associated angles.
for the stadium side triangle we get
part1/sin(53) = 600/sin(37)
part1 = 600×sin(53)/sin(37) = 796.226893... ft
for the planetarium side triangle we get
part2/sin(61) = 600/sin(29)
part2 = 600×sin(61)/sin(29) = 1,082.428653... ft
the distance between the stadium and the planetarium is
part1 + part2 = 1,878.655546... ft
Solve |2x + 1| = 10?
[tex]|2x+1|=10\\2x+1=10 \vee 2x+1=-10\\2x=9 \vee 2x=-11\\x=4.5 \vee x=-5.5[/tex]
Answer:
[tex]x= \dfrac 92~~\textbf{or}~~x = -\dfrac{11}2\\\\[/tex]
Step-by-step explanation:
[tex]\underline {\textbf{Absolute value rule: }}\\\\\textbf{If}~ |a| = b~ \textbf{then} ~a =b~ \textbf{or}~ a =-b \\\\\underline{\textbf{Solution;}}\\\\\textbf{Given that,}\\\\~~~~~~~|2x+1| = 10\\\\\implies 2x +1 = 10~~~~~~\textbf{or} ~~~~~~~~~~~2x+1 = -10\\\\\implies 2x = 10-1~~~~~~\textbf{or}~~~~~~~~~~~2x = -10-1\\\\\implies 2x = 9~~~~~~~~~~~~~\textbf{or}~~~~~~~~~~~2x = -11\\\\\implies x= \dfrac 92~~~~~~~~~~~~~~\textbf{or}~~~~~~~~~~~x = -\dfrac{11}2\\\\[/tex]
Which of these is a correct expansion of (4x-2)(2x² + 3)?
OA. 4x 2x2 + (-2) 2x²+2x².3+(-2) 3
OB. 4x 2x² + 4x
3+2·2x²+2.3
OC. 4x 2x² + 4x 3+ (-2) 2x² + (-2) 3
HEY JUST WANTED TO GIVE THE AWNSER ITS C
The correct expansion of (4x-2)(2x² + 3) is 4x 2x² + 4x 3+ (-2) 2x² + (-2) 3.
The correct option is (C).
What is expression?An expression is a set of terms combined using the operations +, – , x or ,/.
Given expression: (4x-2)(2x² + 3)
= 4x (2x² + 3) -2 (2x² + 3)
= 4x* 2x² + 4x * 3- 2* 2x² - 2*3
= 4x* 2x² + 4x *3+ (-2) *2x² + (-2)* 3
Hence, the correct expansion is 4x 2x² + 4x 3+ (-2) 2x² + (-2) 3.
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Find the probability that a randomly
selected point within the circle falls
in the red shaded area.
r = 4 cm
[?]%
Round to the nearest tenth of a percent.
Enter
Answer:
[tex]31.8\%[/tex]
Step-by-step explanation:
The area of the circle is [tex]A=\pi r^2=\pi(4)^2=16\pi[/tex]
The area of the triangle is [tex]A=\frac{bh}{2}=\frac{8*4}{2}=\frac{32}{2}=16[/tex]
Hence, the probability of a randomly selected point within the circle falls in the red shaded area is [tex]\frac{16}{16\pi}=\frac{1}{\pi}\approx0.318\approx31.8\%[/tex]
how to find the length of sides using pythagorean theorem
Answer:
Pythagorean theorem is mainly used for finding the length of the third side of a right angled triangle if the lengths of other two sides are known.
For this, it is more useful if the theorem is stated in terms of the length of the sides of the triangle.
Consider a right angled triangle with sides of lengths p,b and h, where p,b and h are the lengths of perpendicular,base and hypotenuse respectively.
Draw squares on each of these sides.These squares have areas p²,b² and h².
For any right angled triangle with sides of length p,b and h;
[tex] \: \: \: \: \: \: \: {h}^{2} = {p}^{2} + {b}^{2} ....................(i)[/tex]
[tex] \therefore \: h = \sqrt{ {p}^{2} + {b}^{2} } [/tex]
From (i); p²=h²-b²
[tex] \therefore \: p = \sqrt{ {h}^{2} - {b}^{2} } [/tex]
Similarly,
[tex]b = \sqrt{ {h}^{2} - {p}^{2} } [/tex]
Step-by-step explanation:
[tex] \blue{ \frak{seolle_{aphr.odite} }}[/tex]
Multiple Choice Is the quadrilateral a parallelogram? If so, select the reason. W Z A) One angle is supplementary to both consecutive angles B) Not enough info or not a parallelogram C) Both diagonals bisect each other D) Both pairs of opposite angles are congruent E) Both pairs of opposite sides are congruent Skip Question 1 of 13 Submit
Answer: Both diagonals bisect each other.
please help! needs to be submitted in 15 minutes
6 / 2 = 3
Every 3 feet of shadow would be 1 foot of height of the tower.
171 feet/ 3 = 57
Answer: The tower is 57 feet tall.
In the figure below, what is the value of xº?
Answer:
62 degree
Step-by-step explanation:
A triangle has 180 degree
180 - (43 + 75) = 62
Which of the following are properties of the incenter of a triangle? Check all
that apply.
A. The incenter is equidistant from each side of the triangle.
B. The incenter of a triangle is always inside it.
C. The incenter is where all of the bisectors of the angles of the
triangle meet.
D. The incenter of an obtuse triangle lies on the outside of the
triangle.
The incenter of a triangle is the point at which the triangle's three internal angle bisectors connect. The correct options are A, B, and, C.
What is the incenter of a triangle?The incenter of a triangle is the point at which the triangle's three internal angle bisectors connect. In other words, it is the point at which the internal angle bisectors of the triangle intersect.
The properties that are true about the incenter of a triangle are:
A. The incenter is equidistant from each side of the triangle.
B. The incenter of a triangle is always inside it.
C. The incenter is where all of the bisectors of the angles of the triangle meet.
Hence, the correct options are A, B, and, C.
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