Answers
a) The value of [tex]k'(0)[/tex] is [tex]\frac{3\sqrt{3}}{2}[/tex].
b) The value of [tex]m'(5)[/tex] is approximately -0.034.
c) The value of [tex]x[/tex] is approximately 0.622.
a) [tex]f(x)[/tex] is a piecewise function formed by two linear functions, whose form is defined by the following definition:
[tex]f(x) = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} \cdot x + b[/tex] (1)
Where:
[tex](x_{1}, y_{1})[/tex], [tex](x_{2}, y_{2})[/tex] - Two distinct points of the line.[tex]x[/tex] - Independent variable.[tex]f(x)[/tex] - Dependent variable.[tex]b[/tex] - x-InterceptNow we proceed to determine the linear functions:
Line 1: [tex]x \in [-1, 2)[/tex]
[tex](x_{1}, y_{1}) = (0, 3)[/tex], [tex](x_{2}, y_{2}) = (1, 5)[/tex], [tex]b = 3[/tex]
[tex]f(x) = \frac{5-3}{1-0}\cdot x + 3[/tex]
[tex]f(x) = 2\cdot x + 3[/tex]
Line 2: [tex]x \in [2, 8][/tex]
[tex](x_{1}, y_{1}) = (2, 7)[/tex], [tex](x_{2}, y_{2}) = (8, 3)[/tex]
First, we determine the slope of function:
[tex]m = \frac{3-7}{8-2}[/tex]
[tex]m = -\frac{2}{3}[/tex]
Now we proceed to determine the intercept of the linear function:
[tex]7 = -\frac{2}{3}\cdot 2 + b[/tex]
[tex]b = \frac{25}{3}[/tex]
[tex]f(x) = -\frac{2}{3}\cdot x +\frac{25}{3}[/tex]
The first derivative of a linear function is its slope, and the first derivative of a product of functions is defined by:
[tex]k'(x) = f'(x)\cdot g(x) + f(x)\cdot g'(x)[/tex]
If we know that [tex]f(x) = 2\cdot x + 3[/tex], [tex]f'(x) = 2[/tex], [tex]g(x) = \sqrt{x^{2}-x+3}[/tex], [tex]g'(x) = \frac{2\cdot x - 1}{2\cdot \sqrt{x^{2}-x+3}}[/tex] and [tex]x = 0[/tex], then:
[tex]k'(x) = 2\cdot \sqrt{x^{2}-x+3}+\frac{(2\cdot x +3)\cdot (2\cdot x - 1)}{2\cdot \sqrt{x^{2}-x+3}}[/tex]
[tex]k'(0) = 2\sqrt{3}-\frac{3}{2\sqrt{3}}[/tex]
[tex]k'(0) = \frac{12-3}{2\sqrt{3}}[/tex]
[tex]k'(0) = \frac{9}{2\sqrt{3}}[/tex]
[tex]k'(0) = \frac{3\sqrt{3}}{2}[/tex]
The value of [tex]k'(0)[/tex] is [tex]\frac{3\sqrt{3}}{2}[/tex].
b) The derivative is found by means of the formulas for the derivative of the product of a function and a constant and the derivative of a division between two functions:
[tex]m'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{2\cdot [g(x)]^{2}}[/tex] (2)
If we know that [tex]f(x) = -\frac{2}{3}\cdot x +\frac{25}{3}[/tex], [tex]f'(x) = -\frac{2}{3}[/tex], [tex]g(x) = \sqrt{x^{2}-x+3}[/tex], [tex]g'(x) = \frac{2\cdot x - 1}{2\cdot \sqrt{x^{2}-x+3}}[/tex] and [tex]x = 5[/tex], then:
[tex]f(5) = 5[/tex]
[tex]f'(5) = -\frac{2}{3}[/tex]
[tex]g(5) = \sqrt{23}[/tex]
[tex]g'(5) = \frac{9\sqrt{23}}{46}[/tex]
[tex]m'(5) = \frac{\left(-\frac{2}{3} \right)\cdot \sqrt{23}-\left(-\frac{2}{3} \right)\left(\frac{9\sqrt{23}}{46} \right)}{3\cdot 25}[/tex]
[tex]m'(5) \approx -0.034[/tex]
The value of [tex]m'(5)[/tex] is approximately -0.034.
c) In this case we must find a value of [tex]x[/tex], so that [tex]h'(x) = 2[/tex]. Hence, we have the following formula below:
[tex]5\cdot e^{x}-9\cdot \cos x = 2[/tex]
A quick approach is using a graphing tool a locate a point so that [tex]5\cdot e^{x}-9\cdot \cos x = 2[/tex]. According to this, the value of [tex]x[/tex] is approximately 0.622.
To learn more on derivatives, we kindly invite to check this verified question: https://brainly.com/question/21202620
Related Questions
can somebody help solve 75 - 6x = 4x - 15
Answers
Answer:
x = 9
Step-by-step explanation:
75 - 6x = 4x - 15
We want to arrive at an answer that looks like
x = a number,
so we want x on the left side and numbers on the right side.
Remember that whichever operation we do to one side of the equation, we must do to both sides.
Since there is a 75 on the left side, and it is being added to -6x, subtract 75 from both sides.
75 - 75 - 6x = 4x - 15 - 75
Simplify both sides.
-6x = 4x - 90
Since there is a 4x on the right side, and it is being added to -90, subtract 4x from both sides.
-6x - 4x = 4x - 4x - 90
Simplify both sides.
-10x = -90
We want x, but x is being multiplied by -10, so we divide both sides by -10.
[tex] \dfrac{-10x}{-10} = \dfrac{-90}{-10} [/tex]
Simplify both sides.
x = 9
HELPPP PLS :)
find the value of x.
Answers
Answer:
9
Step-by-step explanation:
4x - 2 = 34 measure of two legs of isosceles triangle are equal
4x = 36
x = 9
Find the x- and y-intercepts of the parabola y=x2+4x−6.
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What is 40÷1/2 please help me
Answers
Answer: Pretty sure it's 80
Step-by-step explanation:
Explanation: when dividing by fractions here’s what I do. Instead of having a divide sign change it to a multiplication sign. Next flip your numerator and denominator of your fraction. In this case instead of 1/2 switch it to 2/1. You don’t see it but 40 is also 40/1. So multiply across aka numerator times numerator and denominator time denominator. 40 times 2 and 1 times 1. You’re left with 80/1 aka 80.
How do I write 435 in expanded from
Answers
Answer : 400 + 30 + 5
Step - by - step explanation : To write something in expanded form means to write the number individually. " To see the math value of individual digits ".
4 --> Since it's in the hundreds place that means the number is 400 !
3 --> Since it's in the tenths place that means the number is 30 !
5 --> Since it's the first digit that means it stays as the number 5 !
Hope this helped ! <33
Take care happy holidays ! Contact me if you need more help on this ! ( If you want )
Jordan made this design from three pieces of square shaped cloth. What is the total area of design Jordan made? EXPLAIN how u found your answer
Answers
Applying the area of a square, the total area of the design is: 116 cm²
Area of SquareA square has equal side lengths, s, this, the area of a square is given as: A = s².
The three pieces of square shaped cloth that Jordan made the design from is shown in the image attached below.
To find the total area of the design, find the area of each square shaped cloth and add all together.
Thus, applying the area of a square, the total area would be:
Total Area = 4² + 6² + 8²
Total Area = 116 cm²
Learn more about area of square on:
https://brainly.com/question/16758129
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If x²+y²=125 and xy=22, find the value of (x+y)².
Answers
[tex]\text{Given that,}~x^2 +y^2 = 125 ~ \text{and}~ xy =22\\\\(x+y)^2 = x^2 +y^2 +2xy = 125 +2(22) = 169[/tex]
[tex]\huge \bf༆ Answer ༄[/tex]
Here's the solution ~
[tex] \sf(x + y) {}^{2} [/tex][tex] \sf {x}^{2} + {y}^{2} + 2xy[/tex][ by using formula ]
[tex] \sf125 + (2 \times 22)[/tex][tex] \sf125 + 44[/tex][tex] \sf169[/tex]I hope it helps ~
Please help, very urgent!!!! 50 points!!
Answers
Answer:
f = 14
Step-by-step explanation:
it is because skephalo and hot and poggers science and dnf
Multiply and write the result in Scientific notation
(3 x 10^5) (2 x 10^2)
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Answers
Curtis has started to bike every day. His goal is to ride 16.5 kilometers each day. He bikes of that distance on Monday. He bikes 3.5 fewer
kilometers on Tuesday than he did on Monday.
How many kilometers did Curtis bike on Monday and Tuesday combined? Enter the answer in the box.
kilometers
help
Answers
Answer:29.5 kilometers
Step-by-step explanation: 16.5-3.5=13 then you add 13 and 16.5
The solution is: Eric biked total 1.875 Km.
What is unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given:
On Monday, Eric biked = 1 Km 125 m
= 1.125 Km
On Tuesday, Eric biked = 1.125 Km - 375 m
= 1.125 - 0.375 Km
= 0.75 Km
So, Eric biked on both day
=1.125 + 0.75
= 1.875 Km
Hence, Eric biked total 1.875 Km.
Learn more about unitary method here:
brainly.com/question/22056199
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complete question:
Eric biked 1 kilometer 125 meters on Monday. on Tuesday, he biked 375 meters less than on Monday how far did he bike both days
Given that
P
=
x
−
y
.
Find
P
when:
x
=
15
and
y
=
20
Answers
Answer:
the answer would be -5
Step-by-step explanation:
p=15-20 p=-5
A bookstore had 25 copies of a magazine. Yesterday the bookstore sold 3/5 of these coples. How many copies were sold yesterday?
Answers
Answer:
15
Step-by-step explanation:
25/5 = 5.
3 x 5 / 5+5+5 = 15
15/25 = 3/5
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Answers
Answer:
C is the answer
Step-by-step explanation:
Answer: The answer is C, -3/4.
Explanation: First graph the points, then add the straight line through them and to find the slope you do rise/run, formula being y2-y1/x2-x1. This means putting the second y value minus the first y value and dividing that by the second x value minus the first x value. Then simplify.
Chang works as a tutor for $11 an hour and as a waiter for $13 an hour. This month, he worked a combined total of 90 hours at his two jobs. Let t be the number of hours Chang worked as a tutor this month. Write an expression for the combined total dollar amount he earned this month.
Answers
Can you show more information please? I need more.
T = 90t + 90w
Step-by-step explanation:
90t = 90x11 = 990
90w = 90 x 13 = 1170
T = 90t + 90w
T = 990 + 1170
T = 2160
Elevator 1 in a building moved from ground position to a final position of +13 feet. Elevator 2 in the same building moved from ground to a final position of −10 feet. Which statement best describes the final positions of these two elevators? (5 points)
Question 4 options:
1)
Elevator 1 is 13 feet above ground level, and Elevator 2 is 10 feet below the position of Elevator 1.
2)
Elevator 1 is 13 feet below ground level, and Elevator 2 is 10 feet above the position of Elevator 1.
3)
Elevator 1 is 13 feet below ground level, and Elevator 2 is 10 feet above ground level.
4)
Elevator 1 is 13 feet above ground level, and Elevator 2 is 10 feet below ground level.
Answers
Answer:
Elevator 1 is 12 feet above ground level, and Elevator 2 is 15 feet below the position of Elevator 1. We are given that Elevator 1 is "+12 feet" and Elevator 2 is "-15 feet" meaning 15 feet below that of Elevator 1 (because they said it was where it was at Elevator 1 ground level prior), so this should be your answer "A".
Step-by-step explanation:
1 a
1985 x 99 I already know the answer: 196515
Answers
Answer:
the answer is 196515
Step-by-step explanation:
If u already knew the why u put upped the question?
plss help me.......
Answers
Answer:
The slope
Step-by-step explanation:
The area of a square is s2, where s is the side length of the square. Jack has a square-shaped flower garden in his yard. Each side of the garden is 8 feet. What is the area of the flower garden?
Answers
Answer:
64 square feet
Step-by-step explanation:
A = s² = 8² = 8*8 = 64
find the difderence of 3/4 and 1/28
Answers
Answer:
5/7
Step-by-step explanation:
3/4-1/28
21/28-1/28
20/28
simplify
5/7
Need answer to the question
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Answer:
B) Symmetric property of congruenceStep-by-step explanation:
The angles 1 and 2 are marked as congruent
If a ≅ b, then b ≅ a, this is called the symmetric property of congruence:
The symmetric property states that if one figure is congruent to another, then the second figure is also congruent to the first.Correct choice is B
Answer:
Symmetric property of similarity
Step-by-step explanation:
If a=a then that's reflexive propertyIf a=b and b=c then a=c, transitive propertyGiven the system: 3x-3y=9 -9x+12y= -30
What number would you multiply the TOP equation by to eliminate your "x" values with addition?
Answers
Answer:
you would multiply by 3.
Step-by-step explanation:
Hi there!
In order to solve this system of equations through elimination, the sum of either the x or y values will have to equal 0. To eliminate x, you would have to find what, when added, will make the sum 0. In this case, a positive 9 will cancel out the -9 and give us 0. All we have to do is multiply the top equation by 3 so that the x value is equal to 9x!
I hope this helps!
Everything in the store was at 30% discount . A dress was originally for 65$ . What was the discount price ?
Answers
Answer: $45.50
Step-by-step explanation:
30% discount means 30% off the original price
1. find 30% of 65$
=.3*65
=19.5
2. subtract that from 65$
=65-19.5
=45.5
the discount price was $45.50
can someone help me with this math problem?
Answers
Answer:
∠ B ≅ ∠ E
Step-by-step explanation:
Since the triangles are congruent then corresponding angles are congruent
ABC ≅ DEF
The corresponding angles are B and E
Answer:
The answer is [tex]\angle B\cong \angle E[/tex]
Step-by-step explanation:
Given that
[tex]\Delta ABC \cong \Delta DE[/tex][tex]F[/tex]
Hence corresponding angles are congruent
[tex]\angle B\cong \angle E[/tex]
After a storm damages the community center, Shanika and her friends hold fundraising events to help pay for repairs. After the first event, they raise $240, which is 10% of the total amount that they want to raise
Answers
Answer:
Step-by-step explanation:
The total they want to raise is $2,400 if that is what you're asking?
-2 (y - 3) = y + 24 HELPPP PLEASEE ITS DUEEE IN 5MINN
Answers
Answer:
y = -6
Step-by-step explanation:
-2(y - 3) = y + 24
-2y + 6 = y +24
-18 = 3y
-6 = y
What is the answer to this?
Answers
Answer:
2x+1=9
Step-by-step explanation:
[tex]\sqrt{2x+1}+5=2[/tex]
[tex]\sqrt{2x+1}=-3[/tex]
[tex]2x+1=9[/tex]
If they want you to solve, then you can go farther.
2x=8
x=4
Check for extraneous solutions:
[tex]\sqrt{2(4)+1} +5 = 2[/tex]
Leora is finding the length of a horizontal line segment on a coordinate grid She subtracted the x coordinates and got
-17.5 What is her next step?
Add 17.5
Take the absolute value of -17.5
Add the y.coordinate values.
Round up to -17
Answers
Answer:
Take the absolute value of -17.5
Step-by-step explanation:
Answer:
2. Take the absolute value of ' - 17. 5 '
Step by Step Explanation:
The distance/length is always positive, so make sure to take the absolute value of -17.5
Ellie and Isaiah are helping their friend Nora move. If Ellie can move 4 boxes for
every 16 boxes that Isaiah moves, then how many boxes can Isaiah move for every
box that Ellie moves?
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Answer:
Isaiah can move 4 boxes for every box Ellie moves.
Pls give brainliest. :D
2, 1, 2, 4, 5, 4
What is the mean
Answers
[tex] \sf \: mean = \frac{Sum \: \: of \: \: Observation} {Total \: \: number \: \: of \: \: Observations} \\ \sf = \frac{2 + 1 + 2 + 4 + 5 + 4}{6} \\ = \sf \frac{18}{6} = 3[/tex]
Answer:
3
Hope you could get an idea from here.
Doubt clarification - use comment section.
A line with a slope of 2 passes through the point (6,4). What is its equation in
slope-intercept form?
Answers
Answer:
y=2x-8
Step-by-step explanation:
y-y1=m(x-x1)
y-4=2(x-6)
y=2x-12+4
y=2x-8