The derivative and domain of the function f(x) = x²-2x³ is :[tex](-\infty, \infty)[/tex].
What is derivative of the function (differentiation)?The slope of a function's graph or, more precisely, the slope of the tangent line at a point can be used to interpret a function's derivative.
Its computation actually stems from the slope formula for a straight line, with the exception that curves require the employment of a limiting procedure.
Calculation for the derivative:
Step 1: Use the definition of the derivative. Remember that f(x+h) means plug (x+h) into everywhere there is an "x" in f(x).
[tex]\begin{aligned}f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\&=\lim _{h \rightarrow 0} \frac{(x+h)^{2}-2(x+h)^{3}-\left(x^{2}-2 x^{3}\right)}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(x^{2}+2 h x+h^{2}\right)-2(x+h)(x+h)^{2}-x^{2}+2 x^{3}}{h} \quad \text { cancel } x^{2}\end{aligned}[/tex]
[tex]\begin{aligned}&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2(x+h)\left(x^{2}+2 h x+h^{2}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{\left(2 h x+h^{2}\right)-2\left(x^{3}+2 h x^{2}+h^{2} x+h x^{2}+2 h^{2} x+h^{3}\right)+2 x^{3}}{h} \\&=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-2 x^{3}-4 h x^{2}-2 h^{2} x-2 h x^{2}-4 h^{2} x-2 h^{3}+2 x^{3}}{h}\end{aligned}[/tex]
[tex]=\lim _{h \rightarrow 0} \frac{2 h x+h^{2}-6 h x^{2}-6 h^{2} x-2 h^{3}}{h}[/tex]
Step 2: Cancel out a factor of h from each term in the numerator with the h in the denominator. Then direct substitute h=0.
[tex]\begin{aligned}&=\lim _{h \rightarrow 0}\left(2 x+h-6 x^{2}-6 h x-2 h^{2}\right) \\&=2 x+0-6 x^{2}-6(0) x-2(0)^{2} \\&=2 x-6 x^{2}\end{aligned}[/tex]
f and f' are polynomials, so their domains are all real numbers.
Therefore, for f(x) = x²-2x³ domain of f and f' :[tex](-\infty, \infty)[/tex].
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The complete question is -
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative f(x) = x²-2x³.
Can someone help me please
Answer:
[tex]the \: answer \: is \: choice \: b \: \frac{ - 2 \sin(x) }{1 + \cos(x) } [/tex]
will give you an explanation if you want tag me on comment.
Miguel is making pudding. The recipe yields
6 servings and calls for 2
cups of milk. How
many cups of milk will Miguel need for 18
servings of pudding?
Using the given recipe, he will need 6 cups of milk to make 18 servings.
How many cups of milk will Miguel need for 18 servings of pudding?We know that the recipe for 6 servings, calls for 2 cups of milk.
Miguel wants to do 18 servings, Then he needs to use the recipe:
18 = x*6
Solving the linear equation for x:
x = 18/6 = 3
So he needs to use his recipe 3 times.
And we know that for 6 servings he needs 2 cups of milk, then for 18 he needs:
3*(2 cups) = 6 cups of milk.
Using the given recipe, he will need 6 cups of milk to make 18 servings.
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IS ANYONE ABLE TO HELP ME PLEASE ASAP
Answer:
36
Step-by-step explanation:
To answer this question, we need to first understand the notation (f · x)(-2). This is a composite function, and another way to write this is f(g(-2)). This means that we have to find g(-2) and apply the function f to whatever this value is. Let's start by finding g(-2):
[tex]g(x)=x-2\\g(-2)=-2-2\\g(-2)=-4[/tex]
Now that we know g(-2) is -4, we can apply this to f:
[tex]f(x)=x^2-4x+4\\f(-4)=(-4)^2-4(-4)+4\\f(-4)=16-(-16)+4\\f(-4)=16+16+4\\f(-4)=36[/tex]
Therefore, our answer is 36
A university surveyed its students on their opinions of campus housing. the following two-way table displays data for the sample of 200200200 students who responded to the survey. how many students in this sample had a negative opinion of campus housing? students gender positive opinion negative opinion neutral opinion male 404040 363636 141414 female 424242 565656 1212
Of students in the sample, 90/200 fraction of the students were male.
What is a fraction?In mathematics, a fraction is used to represent a piece or part of a whole. It symbolizes the equal pieces of the whole. A fraction is made up of two parts: the numerator and the denominator. The top number is known as the numerator, and the bottom number is known as the denominator.To find the fraction of male students:
Number of male students = 90
Number of female students = 110
So, total students = 90 + 110 = 200
Fraction of students who were male = male students / total students
Fraction of students who were male = 90/200
So, 90/200 fraction of the students were male.
Therefore, 90/200 fraction of the students were male.
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The question you are looking for is given here:
A university surveyed its students on their opinions of campus housing. The following two-way table displays data for the sample of
200 students responded to the survey.
Help please!!! Due in 10 mins
Answer:
2.0
Step-by-step explanation:
How much work is performed lifting in order to lift a 50.0 kg block 7.00 meters? Use g=9.81 m/s^2
The work done in lifting the block is 3430 J
Calculating workFrom the question, we are to determine the work done in lifting the block
Using the formula,
Work done = mgh
Where m is the mass
g is the acceleration due to gravity
and h is the height
From the question,
m = 50.0 kg
g = 9.81 m/s²
h = 7.00 m
Putting the parameters into the formula,
Work done = 50.0 ×9.8 × 7.00
Work done = 3430 J
Hence, the work done in lifting the block is 3430 J
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If 15 actuators have failed today, what is the probability that a) at least 10 are repairable? b) from 3 to 8 are repairable? c) exactly 5 are repairable?
The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.
According to the statement
we have given that If 15 actuators have failed and we have to find the probabilities on some conditions.
we know that the formula of probabilities is
probability = possible outcomes / total outcomes
So,
at least 10 are repairable = 1 - (10 are not repairable)at least 10 are repairable = 1 - 10/15
at least 10 are repairable = (15 - 10)/15
at least 10 are repairable = (5)/15
at least 10 are repairable = 1/3
from 3 to 8 are repairable = 3/15 *8/15from 3 to 8 are repairable = 1/5 *8/15
exactly 5 are repairable = 5/15exactly 5 are repairable = 1/3
These are the probabilities of the given conditions.
So, The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.
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Use the drop-down menus to complete each equation so the statement about its solution is true. CLEAR CHECK No Solutions 3x 9 4x x
The complete equation so that the statement is true is 3x+9+4x+x= 8x + 1
How to complete the equation?The equation is given as:
3x+9+4x+x=
Represent the blank with y
So, we have:
3x + 9 + 4x + x = y
Evaluate the like terms
8x + 9 = y
For the equation to have no solution the value of y and 8x + 9 must not be equal.
Take for instance
y = 8x + 1
So, we have:
8x + 9 = 8x + 1
Subtract 8x from both sides
9 = 1
The above equation is false because 1 and 9 are not equal
Hence, the complete equation so that the statement is true is 3x+9+4x+x= 8x + 1
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Complete question
Use the drop-down menus to complete each equation so the statement about its solution is true.
No Solutions
3x+9+4x+x=
The volume of a rectangular box is 1.0 ft3. what is the length if the width is 1.0 ft and the height is 1.5 ft?
The length of the rectangular box is 0.67 feet
How to determine the length of the box?The given parameters are:
Length = ??
Width = 1.0 ft
Height = 1.5 ft
Volume = 1.0 ft^3
The volume is calculated as:
Volume = Length * Width * Height
Substitute the known values in the above equation
1 = Length * 1 * 1.5
Evaluate the product
1 = Length * 1.5
Divide both sides by 1.5
Length = 0.67
Hence, the length of the rectangular box is 0.67 feet
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the diffrence of 3-7p^2+11(4+n)
The expression 3 - 7p² + 11(4 + n) to find an example of each kind of expression is;
The first value is a number, 3 = constantThe second value, -7p² = factorThe third value, 11(4 + n) = binomial and monomial valuesExpression3 - 7p² + 11(4 + n)
The expression has 3 parts
3-7p²11(4 + n)3 - 7p² + 11(4 + n)
= 3 - 7p² + 44 + 11n
= 47 + 7p² + 11n
Complete question:
Use the expression 3 — 7p2 + 11(4 + n ) to find an example of each kind of expression
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Expand (1 + root 2) (3 - root 2)
Give your answer in the form a+b root 2 where a and b are integers.
Answer:
1+2√2
a = 1 b= 2
See the attached page, I've shown the calculation over there.
James has $0.50 more than his brother John. The sum of their money is $2.50. how much money does John have
Answer:
Step-by-step explanation:
John has $1.25
James has $1.75
i think
Answer:
$1.00
Step-by-step explanation:
James has $0.50 more than John.
When they put their money together, they have a combined total of $2.50.
[tex]2.50-0.50=2.00[/tex]
[tex]2.00\div2=1.00[/tex]
[tex]1.00+1.50=2.50[/tex]
[tex]=2.50[/tex]
John has one dollar.
Hope this helps.
Below is the market for funnel cakes at a small community fair.
a) what is the equilibrium price and quantity of the funnel cakes?
b) What is the total consumer spending on funnel cakes at the equilibrium point?
c) Calculate the producer surplus and consumer surplus at equilibrium. Show your work.
d) What would be the result if the suppliers charged $6? Explain.
e) Cheese pretzels are substitute good for funnel cakes at the same fair. Draw a correctly labeled graph of the market for cheese pretzels. Shade the area of consumer surplus.
f) A large number of new guests enter the fair. Illustrate the effect of this change on the same graph from part E.
g) does the consumer surplus increase, decrease, or stay the same as a result of the change from part F? Explain.
h) A new cheese pretzel stand opens at the same fair. On a new graph, illustrate the market for cheese pretzels at this fair before and after the new stand opened.
i) if the large number of new guests entering the fair happened at the same time that the new cheese pretzel stand open how would the equilibrium price and quantity for cheese pretzels change? Explain.
From the graph given, the equilibrium price and quantities are given respectively as;
Price = $5Quantity = 6 units of funnel cakes.What is the total consumer spending on funnel cakes at the equilibrium point?The total consumer spending(TCS) is achieved by finding the product of the price and the quantities at the equilibrium point.
Hence,
TCS = 5 * 6
= 60
A) The producer surplus is the area of the triangle shaded yellow in the attached image.
Recall that he area of triangle is:
1/2 b * h
b = 4
h = 6
Hence Area of the RA-Triangle (Producer Surplus) =
1/2 * 4 * 6
= 12
B) The Consumer surplus is the area of the triangle shaded Green in the attached image.
Recall that the area of triangle is:
1/2 b * h
b = 4
h = 6
Hence Area of the RA-Triangle (Consumer Surplus) =
1/2 * 4 * 6
= 12
What would be the result if the suppliers charged $6?
If the suppliers charged $6, then the quantity demanded would drop to 4 units of funnel cakes. It is demonstrated graphically. Recall that the law of demand and supply indicates that when prices goes up, demand comes down and vice versa.
Graph II illustrates what would happen if a large number of guests entered the fair. It would result to a right-ward shift in the demand curve.
Does the consumer surplus increase, decrease, or stay the same as a result of the change from part F?Yes, there is an increase in the consumer surplus. Now there are more consumers hence the total amount they are willing to pay as an aggregate sum has increased from what was.
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The functions f(x) and g(x) are shown on the graph.
f(x) = log(x)
g(x)
6
Using fix), what is the equation that represents gix)?
A) g(x) = log5(x) - 3
B) g(x) = log5(x) + 3
C) g(x) = log5(x-3)
D) g(x) = log5(x+3)
Using translation concepts, the equation of g(x) is:
D) [tex]\log_5{x + 3}[/tex]
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
Researching this problem on the internet, g(x) is a shift left of 3 units of [tex]f(x) = \log_5{x}[/tex], hence:
[tex]g(x) = f(x + 3) = \log_5{x + 3}[/tex], which means that option D is correct.
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the population of happy land is 15.8 happys per acre exactly 553 happys live in happy land. how many acres is happyland
Answer:
35 acres.
Step-by-step explanation:
Proportions:
15. 8 ⇒ 1 acre
553 ⇒ P acres
P = 553*1/15.8
P = 35 acres
Given f(x) = x² + 3x -3 and g(z) = 2x, find f(g(x)).
Solution
First, we need to decide which function will go inside which. Since
our function reads "f of g of z", we know that we are going to
substitute the function for g(x) into every a spot in the f(x)
function.
f(g(x))=(x
)²+3(x
)-3
Next, perform normal order of operations for each term. (Hint:
(2x)² = (2x)(2x))
f(g(x)) = 4
²+3
2-3
Let's look at how this example would be different if it was
performed in the opposite order. Given f(x) = x² + 3x - 3 and
g(x)=2x, find g(f(x)).
In this case, we'll need to plug the whole function for f(x) into the
input (2) spot in the g(x) function.
g(f(x)) = 4
(x² + 3x - 3)
Then we'll distribute to each term.
g(f(x)) = 4(x²)+2(3x)-3 3
Simplify to get the final answer.
The composition of the functions f(x) and g(x) gives the equation:
[tex]f(g(x)) = 4x^2 + 6x - 3[/tex]
How to get the composition of functions?Here we want to get the composition of the two functions:
[tex]f(x) = x^2 + 3x - 3\\\\g(x) = 2x[/tex]
f(x) is a quadratic and g(x) is a linear equation.
Now we want to get the composition:
[tex]f(g(x))[/tex]
This means that we need to evaluate function f(x) in g(x), so we can replace all the "x" in the function f(x) by the notation "g(x)"
[tex]f(g(x)) = g(x)^2 + 3*g(x) - 3[/tex]
Now we replace all the "g(x)" by the actual function g(x) = 2x, we wll get:
[tex]f(g(x)) = g(x)^2 + 3*g(x) - 3 = (2x)^2 + 3*(2x) - 3[/tex]
finally, we can simplify this to get the composition, which is a quadratic function just like f(x).
[tex]f(g(x)) = 4x^2 + 6x - 3[/tex]
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David has kept track of his family’s grocery bills for the past 10 weeks, as shown in the table.
Week 1 2 3 4 5 6 7 8 9 10
Bill ($) 92 106 129 115 100 84 110 156 98 87
Would you choose to use a histogram, a circle graph, or a line graph to display the data? Explain your choice. Then make a display.
The preferred choice would be a line graph (drawn in the figure attached) for displaying the data.
Why a line graph is the preferred option here?
For displaying the data of David's family's grocery bills for the past 10 weeks, a line graph is the most preferable choice since it would display the data's trend for us. This is similar to how we would notice times when there were huge costs.Another reason for choosing a line graph is that it is very easy to comprehend. It is simple to observe how the data are related and how they have changed.What is a line graph?
An individual data point is connected by a line in a line graph, sometimes referred to as a line plot or a line chart. A line graph shows numerical values over a predetermined period of time.
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Please someone help me with question a) and b). I’ve been having so much trouble with these types of questions. The booklet says that the answers are a) 134degrees and b) 14.3km. I just don’t get how to get there. Thank you!
The bearing to take is 134 degrees and the shortest distance is 14.3 km
The bearing it would takeThe attached diagram represents a labelled version of the ship movement
The angle at Q is
Q = 180 - 155 + 90 + 120 - 90
This gives
Q = 145
The distance PR is calculated as:
PR^2 =PQ^2 + QR^2 - 2 * PQ * QR * cos(Q)
This gives
PR^2 = 9^2 + 6^2 - 2 * 9 * 6 * cos(145)
Evaluate
PR^2 = 205.46
Take the square root
PR = 14.3 km
The angle at P is then calculated as:
QR/sin(P) = PR/sin(Q)
This gives
6/sin(P) = 14.3/sin(145)
Evaluate the quotient
6/sin(P) = 24.93
Make sin(P) the subject
sin(P) = 6/24.93
Divide
sin(P) = 0.2407
Take the arc sin of both sides
P = 14
The bearing is then calculated as:
Bearing = P + 120
This gives
Bearing = 14 + 120
Evaluate
Bearing = 134
Hence, the bearing to take is 134 degrees
The shortest distanceIn (a), we have
PR = 14.3 km
This represents the shortest distance
Hence, the shortest distance is 14.3 km
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If the ratio of areas of two similar solids is 25:36, what is the linear ratio.
Answer: Hi here go your answer is
25 divides by 36 is 0.69444 or 0.69
Step-by-step explanation:
Find the surface area of the composite figure. Round your answer to the nearest tenth if necessary.
If the length,breadth, height of cuboid are 16 m, 5 m,4 m , slant height of pyramid is 10m, height be 6 m then the surface area of composite figure be 444 [tex]m^{2}[/tex].
Given that the length,breadth,height of cuboid are 16m,5 m, 4m and slant height be 10 m and height be 6 m.
Surface area is basically sum of area of all the sides of a figure.
Surface area of composite figure = area of cuboid less area of common side + area of 2 slant sides of pyramid+area of two triangles.
=2(lb+bh+hl)-(lb)+2(slant height*breadth of cuboid)+2[1/2* length of cuboid* height of triangle)
=2(16*5+5*4+4*16)-16*5+2(10*5)+2[1/2*16*6]
=2(80+20+64)-80+2*50+2(48)
=2(164)-80+100+96
=328-80+196
=248+196
=444[tex]m^{2}[/tex]
Hence the surface area of composite figure having a cuboid and pyramid on top is 444[tex]m^{2}[/tex].
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When a large cake is cut into piece that each weighs 3 ounces, it yields 120 pieces. How many pieces would the cake yield if it were cut into 2- ounce pieces instead
when the cake is cut into [tex]2[/tex] ounce pieces instead of [tex]3[/tex] pieces it yields [tex]180[/tex] pieces.
How to find large cake pieces ?
Given in the question when cake cuts into piece that each weight is [tex]3[/tex] ounce it yields [tex]120[/tex]
So original large cake weight is
[tex]w=120*3\\\\w=360[/tex]
So we can find cake pieces when cake cuts into weight [tex]2[/tex] each piece
[tex]pieces=\frac{360}{2}\\=180[/tex]
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A softball pitcher has a 0.507 probability of throwing a strike for each pitch. If the softball pitcher throws 15 pitches, what is the probability that more than 8 of them are strikes
8/25 is the probability that more than 8 of them are strikes.
Lets simplify the problem,
We are given that a softball pitcher has a 0.507 probability of throwing a strike for each pitch. Also, the softball pitcher throws 15 pitches.
The above situation can be represented through Binomial distribution;
P(X = r) = 0,1,2,3,,.....
where, n = number of trials (samples) taken = 15 pitches
r = number of success = more than 8
p = probability of success which in our question is probability of
Throwing a strike for each pitch = 0.507
LET X = Number of strikes
So, it means X ~ n = 15, p = 0.507
So, Probability that more than 8 of them are strikes = P(X > 8)
P(X > 8) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)
After putting the values in it we get,
P = 0.323
P= 8/25 approx
8/25 is the probability that more than 8 of them are strikes.
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Help me:-
Distribute to create an equivalent expression with the fewest symbols possible.
4(5+y) =
Answer:
20 + 4y
Step-by-step explanation:
The 4 multiplies the 5 and the y, i.e., it distributes over the terms of the expression in ( ).
Complete the following tasks for this equation: b/7.8= -2.15.
1. give an estimate for the solution. be sure to explain how you arrived at the estimate, either by showing an equation or in words.
2. solve the equation. show or explain your work.
3. compared to your estimate, is your answer reasonable? explain your answer in complete sentences.
Answer:
Step-by-step explanation:
Factors
A term in multiplication for an entire variety by which a bigger complete quantity may be dividedA divisor is an integer that evenly divides a range without leaving the restFactorization is the decomposition of an item right into a product of other objectsInteger factorization, the manner of breaking down a composite quantity into smaller non-trivial divisorsA coefficient, a multiplicative thing in an expression, usually more than a fewThe act of forming a component institution or quotient ring in summary algebraA von Neumann algebra, with a trivial centerFactor (graph principle), a spanning subgraphAny finite contiguous sub-sequence of a word in a group concept(b)/(7.8)=-2.15
Multiply each term in the equation by 7.8.
(b)/(7.8)*7.8=-2.15*7.8
Cancel the common factor of (1)/(7.8).
1b=-2.15*7.8
Multiply -2.15 by 7.8 to get -16.77.
1b=-16.77
Divide each term in the equation by 1.
(1b)/(1)=-(16.77)/(1)
Cancel the common factor of 1.
b=-(16.77)/(1)
Divide -16.77 by 1 to get -16.77.
b=-16.77
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Please help me fast
What is the equation of this graph
Answer-4,5
Step-by-step explanation:
What is the domain of the function graphed below?
X<7
x≤7
-2
all real numbers
Answer:
x < 7
Step-by-step explanation:
The domain encompasses all the "x" values (or inputs) of a function.
The arrow on the left line symbolizes that the function continues into the negative region for infinity. As such, this eliminates option C.
Closed dots are points whose "x" values are included in the function. Open-dots are on points whose "x" values are not included in the function. Although the open-dot on the left line at x = -1 generally means it is not included in the function, the closed dot on the right line overrides this. Therefore, x = -1 is in the domain. The value x = 7 is not within the domain of the function due to the open-dot. This eliminates option B and option D.
With this said, the domain of the function is x < 7.
Ava bought 7.73 pounds of fruit for a class party. the class ate 3.59 pounds of the fruit. how much fruit is left?
Ava bought [tex]7.73[/tex] pounds of fruit for a class party , class ate [tex]3.59[/tex] pounds fruits then [tex]4.14[/tex] pounds fruit left after party.
How can we find how much fruit is left after party ?
Ava bought [tex]=7.73[/tex] pounds fruit
ate by class [tex]=3.59[/tex] pounds fruit
fruit left after party [tex]= total fruit -fruit ate by class[/tex]
[tex]= 7.73-3.59\\=4.14[/tex]
So [tex]4.14[/tex] pound fruit is left .
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What is the simplified form of startroot 100 x superscript 36 baseline endroot ?
The simplified form of the expression is [tex]10 x^{18}[/tex].
What are indices in maths?
An index, or a power, is the small floating number that goes next to a number or letter.The plural of index is indices. Indices show how many times a number or letter has been multiplied by itself.The given expression is [tex]\sqrt{100x^{36} }[/tex]
Applying the rule, [tex]\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}[/tex]
Now, [tex]\sqrt{100} \sqrt{x^{36} }[/tex]
Simplifying, we get;
[tex]10\sqrt{x^{36} }[/tex]
Applying the exponent rule, [tex]a^{bc} = (a^{b} )^{c}[/tex]
We get
[tex]10 \sqrt{(x^{18} })^{2}[/tex]
[tex]10 x^{18}[/tex]
Therefore, the simplified form of the expression is [tex]10 x^{18}[/tex].
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Answer:
10 x 18
Step-by-step explanation:
d) Arrange 8/9,7/18,10/16 in ascending order of magnitude
8/9 = 0.888888....
7/18 = 0.388888888889
10/16 = 0.625
Ascending order - means going up, smallest to largest. (Imagine climbing a ladder, where you start of low, then, go higher up the ladder.)
Thus, answer is 7/18, 10/16, 8/9
Hope this helps!
can someone show me how to do this
The force of a 5.0 kg bowling ball with an acceleration of 3m/s 2 is 15 N.
True or False
Answer:
True
Step-by-step explanation:
We got the formula for that:
F=ma
now if you replace the given data, it's gonna be
F=5 kg×3m/s²====> F=15kg.m/s²