Find the derivative of each function. Simplify each derivative and express all exponents as positive values.
Answers
Answer:
[tex]\boxed{f^{\prime}(x)=x^2^{}-\frac{1}{2}}[/tex]Explanation:
Step 1. The function we have is:
[tex]f(x)=\frac{x^3}{3}-\frac{x}{2}[/tex]And we are asked to find the derivative of the function. The rule to find the derivative for this type of function is:
[tex]\begin{gathered} \text{for a function }of\text{ the form} \\ f(x)=ax^n \\ \text{The derivative is:} \\ f^{\prime}(x)=a(n)x^{n-1} \end{gathered}[/tex]Step 2. Before we apply the derivative rule, remember the following:
[tex]\begin{gathered} \text{for a function } \\ f(x)=g(x)+h(x) \\ \text{The derivative is:} \\ f^{\prime}(x)=g^{\prime}(x)+h^{\prime}(x) \end{gathered}[/tex]This means that we need to derivate each part or term of the function and combine them for the total derivative.
Step 3. Apply the derivative rule from step 1 to the given function.
First we rewrite the function as follows:
[tex]\begin{gathered} f(x)=\frac{x^3}{3}-\frac{x}{2} \\ \downarrow \\ f(x)=\frac{1}{3}x^3-\frac{1}{2}x^1 \end{gathered}[/tex]Apply the derivative rule:
[tex]f^{\prime}(x)=\frac{1}{3}(3)x^{3-1}-\frac{1}{2}(1)x^{1-1}[/tex]Step 4. The last step is to simplify the expression:
[tex]\begin{gathered} f^{\prime}(x)=\frac{1}{3}(3)x^{3-1}-\frac{1}{2}(1)x^{1-1} \\ \downarrow \\ f^{\prime}(x)=\frac{1}{3}(3)x^2-\frac{1}{2}(1)x^0 \\ f^{\prime}(x)=x^2-\frac{1}{2}x^{^0} \\ \sin ce^{} \\ x^0=1 \\ \downarrow\text{ The result is }\downarrow \\ f^{\prime}(x)=x^2-\frac{1}{2} \end{gathered}[/tex]Answer:
[tex]\boxed{f^{\prime}(x)=x^2^{}-\frac{1}{2}}[/tex]Related Questions
dilate triangle LMN by a scale factor of four using the origin as the center of dilation label the image triangle LMN and give the coordinates for each of the vertices
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Let's begin by listing out the information given to us:
please help me out thanks
Answers
According to the solving the matrix will be BF = [tex]\left[\begin{array}{ccc}4&0&-3\\15&9&6\end{array}\right][/tex]
What is the matrix?A matrix is a rectangular array as well as table of characters, numbers, or expressions that is set up in rows and columns to represent a mathematical object or a characteristic of one. As an illustration, consider a matrix containing two rows as well as three columns..
What does matrix actually mean?In the field of optics, matrices are employed to model refraction and reflection. Quantum mechanics, electrical circuits, and resistor conversion all benefit from the usage of matrices. In electric circuits, matrices are utilized to solve equations relating to the AC network.
According to the given data:B = [tex]\left[\begin{array}{ccc}2&-1&2\\3&-1&-2\end{array}\right][/tex]
F = [tex]\left[\begin{array}{ccc}3&1&1\\-2&-2&1\\-2&-2&-2\end{array}\right][/tex]
Now solving for BF:
B = [tex]\left[\begin{array}{ccc}2&-1&2\\3&-1&-2\end{array}\right][/tex] x F = [tex]\left[\begin{array}{ccc}3&1&1\\-2&-2&1\\-2&-2&-2\end{array}\right][/tex]
BF = [tex]\left[\begin{array}{ccc}6+2-4&2+2-4&2-1-4\\9+2+4&3+2+4&3-1+4\end{array}\right][/tex]
BF = [tex]\left[\begin{array}{ccc}4&0&-3\\15&9&6\end{array}\right][/tex]
According to the solving the matrix will be BF = [tex]\left[\begin{array}{ccc}4&0&-3\\15&9&6\end{array}\right][/tex].
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Find the formula round your answer to two decimal places if necessary
Answers
Explanation
Give that
[tex]\begin{gathered} f(x)=\frac{1}{x} \\ \\ g(x)=\frac{x+1}{3} \end{gathered}[/tex]To get the value of
[tex](fog)(x)[/tex]we will have
[tex](f\text{ o g})(x)=f\left(\frac{x+1}{3}\right)=\frac{3}{x+1}[/tex]The value (f o g) (x) is
[tex]\left(fog\right)x=\frac{3}{x+1}[/tex][tex]The\:domain\:of\:a\:function\:is\:the\:set\:of\:input\:or\:argument\:values\:for\:which\:the\:function\:is\:real\:and\:defined[/tex]The graph is
Thus, the Domain is
[tex]\begin{bmatrix}\mathrm{Solution:}\:&\:x<-1\quad \mathrm{or}\quad \:x>-1\:\\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:-1\right)\cup \left(-1,\:\infty \:\right)\end{bmatrix}[/tex]H-E-B buys cases of almonds and walnuts. Almonds are packaged 15 bags per case and walnuts are packaged 17 bags per case. H-E-B orders no more than 200 bags of almonds and walnuts at a time. H-E-B pays $24 per case of almonds and $27 per case of walnuts, but will not order more than $300 total at any one time. H-E-B makes a profit of $8 per case on almonds and $9 per case on walnuts.a)Express the constraints in terms of x and y, where x is case of almonds and y is case of walnuts.b)Graph and find the feasible region, identify the vertices.c)Find the Objective Functiond)Using the objective function and the vertices of the feasible region, determine how many cases of almonds and how many cases of walnuts H-E-B should sell to maximize its profit.
Answers
x is case of almonds and y is case of walnuts.
Almonds are packaged 15 bags per case and walnuts are packaged 17 bags per case.
H-E-B orders no more than 200 bags of almonds and walnuts at a time.
So,
x + y < 200
where x and y refers to the number of bags
=======================================================
H-E-B pays $24 per case of almonds and $27 per case of walnuts, but will not order more than $300 total at any one time.
But keep in mind that : Almonds are packaged 15 bags per case and walnuts are packaged 17 bags per case.
So,
24 * (x/15) + 27 * (y/17) < 300
===========================================================
The constraints are:
x + y < 200
(24/15) x + (27/17) y < 300
So, the graph of the previous constraints is as following :
What is the relationship between 7.0/10^2 and 7.0/10^3
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[tex]7/10^2[/tex] is ten times [tex]7/10^3[/tex] or [tex]7/10^3[/tex] is one-tenth of [tex]7/10^2[/tex].
[tex]10^2[/tex] and [tex]10^3[/tex] are written in the exponent form.
Exponent
Exponent is defined as the method of expressing large numbers in terms of powers. That means, exponent refers to how many times a number multiplied by itself.
We are given two numbers [tex]7/10^2[/tex] and [tex]7/10^3[/tex].
We have to find the relationship between the two given numbers.
We know that,
[tex]7/10^2[/tex] can be written as 7/100 and [tex]7/10^3[/tex] can be written as 7/1000.
We can write,
7/1000 = (7/100) *(1/10)
Hence, 7/1000 is one-tenth of 7/100.
Also, we can write,
7/100 = (7/1000)*10
Hence, 7/100 is ten times 7/1000.
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3 What is the length of the missing leg? 15 m I meters
Answers
Answer
The length of the missing leg = 12 m
Explanation
The triangle presented is a right angled triangle because one of its angles is equal to 90 degrees.
The Pythagorean Theorem is used for right angled triangle, that is, triangles that have one of their angles equal to 90 degrees.
The side of the triangle that is directly opposite the right angle or 90 degrees is called the hypotenuse. It is normally the longest side of the right angle triangle.
The Pythagoras theorem thus states that the sum of the squares of each of the respective other sides of a right angled triangle is equal to the square of the hypotenuse. In mathematical terms, if the two other sides are a and b respectively,
a² + b² = (hyp)²
For this question,
Hypotenuse side = 15 m
a = 9 m
b = d = ?
a² + b² = (hyp)²
9² + d² = (15)²
81 + d² = 225
d² = 225 - 81
d² = 144
d = √144
d = 12 m
Hope this Helps!!!
Answer:
We'd use the pythagorean theorem to solve for the other leg's length. As the image suggests, we'd use the theorem,
[tex] {a}^{2} + {b}^{2} = {c}^{2} [/tex]
If one leg is 9 , that's our a value.
If the hypotenuse is 15 , that's our c value.
Now we just have to find the b value.
Plugging in the variables,
[tex] {9}^{2} + {b}^{2} = {15}^{2} [/tex]
[tex] {9}^{2} = 81 \\ {15}^{2} = 225 \\ 81 + {b}^{2} = 225[/tex]
Subtract 81 from both sides.
[tex] {b}^{2} = 144[/tex]
Square root both sides.
[tex] \sqrt{ {b}^{2} } = b \: and \: \sqrt{144 = 12 } [/tex]
so b = 12
The temperature fell from 0°F 6 3/5 to or below O°F in 2 1/5hours. What was the temperature change per hour?
Answers
The temperature fell from 0°F to [tex]-6\frac{3}{5}[/tex].or below O°F in [tex]2\frac{1}{5}[/tex]hours. the temperature change per hour is -3 degree
What is temperature difference ?When two independent temperature readings are subtracted, or when calculating the amount of temperature rise or fall, the result is a temperature difference, also referred to as a delta temperature (T).
We are aware that throughout the course of hours, the temperature dropped from 0 to= [tex]-6\frac{3}{5}[/tex].
Therefore, we will divide the change in temperature by the number of hours to determine the rate of change of the temperature.
In order to solve for that, we shall convert the mixed number into improper fractions.
[tex]-6\frac{3}{5}=\frac{-33}{5}degrees\\ 2\frac{1}{5} =\frac{11}{5}hours\\[/tex]
Hourly temperature change =[tex]\frac{\frac{-33}{5} }{\frac{11}{5} }[/tex]
=-3
As a result, there was a -3 degree shift in temperature every hour.
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The patient recovery time from a particular surgical procedure is normally distributed with a mean of 4 days and a standard deviation of 1.6 days. Let X be the recovery time for a randomly selected patient. Round all answers to 4 decimal places where possible.a. What is the distribution of X? X ~ N([],[])b. What is the median recovery time?daysc. What is the Z-score for a patient that took 5.7 days to recover? []d. What is the probability of spending more than 4.5 days in recovery? []e. What is the probability of spending between 3.5 and 4.3 days in recovery? []f. The 85th percentile for recovery times is []
Answers
Part B
What is the median recovery time?
Remember that
For a normal distribution, mean = median = mode
so
The median is 4 days
Part C
What is the Z-score for a patient that took 5.7 days to recover?
Remember that
z =(x - μ)/σ
where
μ=4 days
σ=1.6 days
X=5.7 days
substitute
z=(5.7-4)/1.6
z=1.0625
Part D
What is the probability of spending more than 4.5 days in recovery?
z =(x - μ)/σ
μ=4 days
σ=1.6 days
x=4.5 days
Find out z
z=(4.5-4)/1.6
z=0.3125
using a z-score table values
P(X>4.5)=0.3773
Part E
What is the probability of spending between 3.5 and 4.3 days in recovery?
For x=3.5 -------> z=(3.5-4)/1.6=-0.3125
For x=4.3 -----> z=(4.3-4)/1.6=0.1875
using a z-score table values
P(3.5 < X < 4.3)=0.1970
Part F
Remember that
If your score is in the 85th percentile, it means that 85% of the scores are below your score and 15% are above your score
using a z-score table values
the value of Z=1.036
Find out the value of x
1.036=(x-4)/1.6
solve for x
x=5.6576 days
Refer to this diagram for the next five statements (Questions 1–5). Write the correct postulate, theorem, property, or definition that justifies the statement below the diagram.
Given: ∠MRS ≅ ∠MRO
m∠MRO + m∠MRS = m∠SRO
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This statement is justified by the angle addition postulate.
Using the value 22/7 for π, Calculate the area of a flat ring whose inside and outside diameters are 4cm and 10cm respectively
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The area of a flat ring whose inside and outside diameters are 4cm and 10cm is 66cm²
What is the area of the circle?The region encircled by a circle with radius r is known as πr² in geometry. Here, the Greek letter stands for the constant proportion of a circle's circumference to its diameter, which is roughly equivalent to 3.14159.
Circular ring's inner diameter (d) is equal to 4 cm.
Circular ring's inner radius (r) is 2 cm.
The circumference of the ring (D) is 10 cm.
Circular ring's (R) outer radius is 5 cm.
To locate
the circumference of the ring.
Solution:
The following formula can be used to calculate the area of a circular ring whose inner and outer diameters are given:
Area of a circle equals (R2 - r2).
= 22(5^2-2^2) / 7
= 66cm²
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Drag and drop the answers into the boxes to correctly complete the statement.
A sequence of transformations that maps ABC to A'B'C' is a ______ followed by a _______
reflection across the x-axis
rotation of 180 degrees about the origin
translation 2 units right
rotation of 90 degrees counterclockwise about the origin
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A sequence of transformations that maps ABC to A'B'C' is a Rotation of 90° counterclockwise about the origin followed by a Translation of 2 units right.
This is further explained below.
What is Translation?Generally, The purpose of translation is to convey the meaning of a text written in one language into another language using texts written in those respective languages.
In conclusion, A rotation of 90 degrees anticlockwise around the origin, followed by a translation of 2 units right, is the sequence of transformations that maps ABC to A'B'C'.
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what is the value of (mn)(8)? m(8)=15 n(8)=10
A. 25
B. 150
C. 88
D. 1,510
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Answer: I believe the answer would be B
Step-by-step explanation: Multiply m(8) and n(8) together to get (mn)(8) and you would get 150.
Sam borrowed money from a credit union for 3 years and was charged simple interest at an annual rate of 2%. The total interest that he paid was $48. How much money did he borrow?
show step by step
Answers
Answer:
the amount of money sam has borrowed was 2400
he paid interest 2%
total intreast 48$
48÷3 = $16 a year
2%= $16
16×5= $80 for 10%
$80× 10= 800
800×3= 2400 total amount was borrowed
so 2400-2%= 2352
2400- 2352= 48
What’s the correct answer answer asap for brainlist
Answers
Answer:
I would say B, because the entire purpose and main theme of The Interlopers is to show the futility and uselessness of feuds and arguments, and that sometimes its best to just live and let live, just to keep it short.
What is the exact circumference of a
circle with diameter 48 millimeters?
Answers
The circumference of circle is 150.72 millimetres.
In simpler terms, circumference in the perimeter of circle. Thus, it will represent the measure of length of the line of circle. The relationship between circumference of circle and it's diameter will be represented as formula -
C = πd, where C is circumference and d is diameter.
Keep the value of diameter in the formula to find the exact circumference of the circle.
Circumference = 3.14×48
Performing multiplication to find the circumference of the circle
Circumference = 150.72 millimetres.
Therefore, the circumference of the circle with 48 millimetres diameter is 150.72 millimetres.
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Jon earns $13.75 per hour at his landscaping job. if the hours he works is represented by the variable h, then which equation below could be solved to find out how many hours he must work to earn $50.(1) h/13.75 = 50/1. (3) 50h = 13.75(2) h + 13.75 = 50. (4) 13.75h=50
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13.75h = 50 This is the answer
I think it euqation 4
Quadrilateral EFGH has the given measurements. EF = 8 units FG = 15 units GH = 8 units EH = 15 units FH = 17 units EG = 17 units What conclusion can be made from the given information? A. The quadrilateral is both a parallelogram and a rectangle. B. The quadrilateral is a parallelogram, but not a rectangle. C. The quadrilateral is a rectangle, but not a parallelogram. D. The quadrilateral is neither a parallelogram nor a rectangle.
Answers
A conclusion which can be made from the given information about quadrilateral EFGH is that: A. the quadrilateral is both a parallelogram and a rectangle.
What is a quadrilateral?A quadrilateral can be defined as a type of polygon that has four (4) sides, four (4) vertices, four (4) edges and four (4) angles.
The types of quadrilateral.In Geometry, there are different types of quadrilateral and these include the following:
ParallelogramSquaresRectangleRhombusTrapeziumIn this scenario, we can reasonably infer and logically deduce that quadrilateral EFGH is both a parallelogram and a rectangle because the length of its diagonal are equal and it has adjacent sides of equal of length.
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The quotient of b and 5 is -30.
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[tex]b \div 5 = - 30 \\ \frac{b}{5} = - 30[/tex]
I CONVERTED b÷5 SO THAT I CAN USE CROSS MULTIPLICATION
[tex]b = - 30(5) \\ b = - 150[/tex]
b is -150
HOPE THIS HELPS.
which graph best represents the equation y=x-1i’ll send the other photo :)
Answers
The given equation is
[tex]y=x-1[/tex]This equation belongs to an uptrend line because the variable x is positive. The line has to have a y-intercept at -1 because that's the constant term.
Following the characteristics of the equation, we can deduct that the correct graph is A.
Given the functions:
f(x) = 2x^2
g(x) = 3x - 2
h(x)= -4x+5
Evaluate: f(g(h(2)))
A 162
B. 242
C. -44
D -123
Answers
Answer:
242
Step-by-step explanation:
I used Desmos.
what is i^30?A. -iB. -1c. 1D.i
Answers
Answer
Explanation
Given:
[tex]i^{30}[/tex]we can rewrite the expression as follows:
[tex][/tex]Please help me find what X equals for question 3.
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The value of the missing variable x in the given triangle is; x = 15
What is the missing angle?
We know that sum of angles on a straight line is 180 degrees. That means the angle on the horizontal line of (9x + 16)° inside the triangle will be;
180 - (9x + 16)°
Now, we know that sum of angles in a triangle is 180 degrees. Thus;
180 - (9x + 16)° + 5x - 14 = 90
180 - 4x - 30 = 90
150 - 4x = 90
150 - 90 = 4x
60 = 4x
x = 60/4
x = 15
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Solve by substitution.2x + y = -9(3x + 5y = 4([?],[])
Answers
The given system is
[tex]\mleft\{\begin{aligned}2x+y=-9 \\ 3x+5y=4\end{aligned}\mright.[/tex]Let's isolate y in the first equation.
[tex]\begin{gathered} 2x+y=-9 \\ y=-9-2x \end{gathered}[/tex]Then, we replace the equation we've got in the second equation of the system.
[tex]3x+5(-9-2x)=4[/tex]Now, we solve for x.
[tex]\begin{gathered} 3x-45-10x=4 \\ -7x=4+45 \\ -7x=49 \\ x=\frac{49}{-7} \\ x=-7 \end{gathered}[/tex]We use this value to find y.
[tex]\begin{gathered} y=-9-2(-7) \\ y=-9+14 \\ y=5 \end{gathered}[/tex]Therefore, the solutions are x = -7 and y = 5.Given: BC = EC, CA = CD Prove: BA= DE
PLEASE HELP ASP!
Answers
Answer:
hope it helps
Step-by-step explanation:
Are all of these choices correct?
Answers
Answer:
Yep! Got it alllllll right.
translation: 6 units right and 6 units up71-5,-5), 41-5,-4), 7(-3,-5)A) T(1, 1), u(1, 2), v(3, 1)B) T1-5,1), U(-5,2), V(-3,1)C) T(-4,-4), U(-4,-3), V(-2,-4)D) T(2,-2), U12, -1), V(4, -2)
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step1
translation in x direction is left or right
translation in y direction is down or up
so
translation 6 units righ is the same that
x+6
translation 6 units up is the same that
y+6
For (-5,-5) we have -----> (-5+6,-5+6)-------> (1,1)
For (-5,-4) we have -----> (-5+6,-4+6) ----> (1,2)
For (-3,-5) we have ----> (-3+6,-5+6) -----> (3,1)
the solution is the option A
help it’ll be greatly appreciated:(
Answers
The complete table of the algae area is:
Day 1 2 3 4 5 6
Area 1/2 1 2 4 8 16
What is meant by an equation?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.
The initial area exists given as;
A(1) = 1/2 square meters
When it doubles every day, we have:
A(n) = 2 × A(n - 1) where A(1) = 2
So, we contain the following values
A(2) = 1/2 × 2 = 1
A(3) = 1 × 2 = 2
A(4) = 2 × 2 = 4
A(5) = 4 × 2 = 8
A(6) = 8 × 2 = 16
Therefore, the complete table is:
Day 1 2 3 4 5 6
Area 1/2 1 2 4 8 16
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Given a pair of points on each line, use the slope formula to determine whether AB and CD are parallel. perpendicular, or neither. AB: A(-2,-5) and B(4.7) CD: C(0, 2) and D(8,-2)
Answers
Answer:
Perpendicular
Explanation:
Given the coordinates
A(-2,-5), B(4.7) C(0, 2) and D(8,-2), we are to determine whether AB is parallel, perpendicular to CD or neither
To do this, we need to find he slope of AB and CD first. Using the formula for calculating slope as shown;
m = y2-y1/x2-x1
For AB:
slope = 7-(-5)/4-(-2)
Slope = 7+5/4+2
Slope of AB = 12/6
Slope of AB = 2
For CD
Slope of CD = -2-2/8-0
Slope of CD = -4/8
Slope of CD = -1/2
Note that for AB to be perpendicular to CD, the product of their slope must be -1.
Given
mAB = 2
mCD = -1/2
Taking their product
mAB * mCD = 2 * -1/2
mAB * mCD = -1
Since the product is -1, hence AB is PERPENDICULAR to CD
The scatterplot shows the number of cars and trucks sold by 10 differentemployees at a car and truck dealership during a month.How many employees sold more trucks than cars?
Answers
Let's us tabulate the data that is on the scatterplot.
As you can see in the fourth column, "Sold More Trucks than Cars?", some rows say "Yes" and some say "No".
Let's take a look at Employee No.1. Employee No.1 sold 4 trucks and sold 2 cars. Therefore, employee no.1 was able to sell more trucks than cars and thus, the answer to the fourth column is "Yes."
The same is done for all the employees.
In the end, there are 6 employees who were able to sell more trucks than cars.
The builder of a parking garage wants to build a ramp at an angle of 16 degrees covers a horizontal span of 55 feet. How long will the actual ramp be? Round yoursolution to four decimal places.
Answers
Let's draw the ramp to better understand the problem:
For us to be able to determine how long the ramp will be, we will be using the Cosine Function.
We get,
[tex]\text{ Cosine }\Theta\text{ = }\frac{\text{ b}}{\text{c}}[/tex][tex]Cosine(16^{\circ})\text{ = }\frac{55}{x}[/tex][tex]x\text{ = }\frac{55}{Cosine(16^{\circ})\text{ }}[/tex][tex]\text{ x = }\frac{55}{0.9612616959}[/tex][tex]x=57.216468974668940618504468175387[/tex][tex]\text{ x }\approx\text{ 57.2165 feet}[/tex]Therefore, the answer is 57.2165 feet.