If you're just starting calculus, perhaps you're asking about using the definition of the derivative to differentiate [tex]x^4[/tex].
We have
[tex]\dfrac{d}{dx} x^4 = \displaystyle \lim_{h\to0} \frac{(x+h)^4 - x^4}h[/tex]
Expand the numerator using the binomial theorem, then simplify and compute the limit.
[tex]\dfrac{d}{dx} x^4 = \displaystyle \lim_{h\to0} \frac{(x^4+4hx^3 + 6h^2x^2 + 4h^3x + h^4) - x^4}h \\\\ ~~~~~~~~ = \lim_{h\to0} \frac{4hx^3 + 6h^2x^2 + 4h^3x + h^4}h \\\\ ~~~~~~~~ = \lim_{h\to0} (4x^3 + 6hx^2 + 4h^2x + h^3) = \boxed{4x^3}[/tex]
In general, the derivative of a power function [tex]f(x) = x^n[/tex] is [tex]\frac{df}{dx} = nx^{n-1}[/tex]. (This is the aptly-named "power rule" for differentiation.)
the Royal fruit company produces two types of fruit drinks the first type is 45% pure fruit juice and the second type is 95% pure fruit juice. the company is attempting to produce a fruit drink that contains 55% pure fruit juice how many pints of each of the two existing types of drink must be used to make 180 pints of a mixture that is 55% pure fruit juice
Answer:
first type: 144 pints
second type: 36 pints
Step-by-step explanation:
So let's try to interpret the given information. So the first thing to know is how to calculate x%, to calculate this, you simply multiply the number by x/100, which is what x is in decimal form.
So let's just start off by assigning variables to each brand, with the variable representing the amount of pints. Let's say "x" is the amount of pints of the first brand, and "y" is the amount of pints of the second brand.
The next thing to do is make equations using the given information. So we want a fruit drink that is 55% pure fruit juice, and it's 180 pints. So to find what 55% is, we multiply by 55/100 or 0.55. This means that 55% is 0.55(180) = 99. So using the given information this means that 99 pints is pure fruit juice in the mixture. This comes from mixing the first two brands, so that means if we take the pure fruit juice from the first brand and add it to the second brand we get 99, and remember we are given the information that 45% of the first brand is pure fruit juice and the second is 95%, this means that 0.45x is pure fruit juice in the first brand and 0.95y is pure fruit juice in the second brand. This sets up the following equation: [tex]99 = 0.45x + 0.95y[/tex]
The second equation to set up is using the total amount of pints. x represents how much is from the first brand, and y represents how much is from the second brand. Since the mixture consists of these two brands, and the mixture is 180 pints we can form the equation: [tex]180 = x+y[/tex]
The last step is to solve the systems of equations using the two equations we formed. This can be done via substitution. We simply need to solve for either x or y, in the total amount of pints equation and then substitute it into the pure fruit juice equation. In this example I'll solve for x
Original Equation:
[tex]180 = x+y[/tex]
Subtract y from both sides
[tex]180-y=x[/tex]
So now that we solved for y in terms of x we can substitute this into the pure fruit juice equation, so that we're only dealing with one variable.
Original Equation:
[tex]99 = 0.45x + 0.95y[/tex]
Substitute 180-y as x
[tex]99 = 0.45(180-y) + 0.95y[/tex]
Distribute the 0.45
[tex]99=81-0.45y+0.95y[/tex]
Add like terms
[tex]99=81+0.5y[/tex]
subtract 81 from both sides
[tex]18=0.5y[/tex]
Divide both sides by 0.5 (same thing as multiply by 2)
[tex]36=y[/tex]
This means we have to use 36 pints of the second brand. We can substitute this into either equation to solve for x, but it's easier to substitute into the total pints equation since there are no coefficients (technically a 1 in front, but it's kind of be ignored for now), so it's pretty straightforward to solve for x
Original Equation:
[tex]180=x+y[/tex]
Substitute 36 as y
[tex]180=x+36[/tex]
Subtract 36 from both sides
[tex]144=x[/tex]
This means 144 pints from the first brand and 36 from the second
Sarah has 15 naira and bobo has 39 naira .how much must Sarah give to bobo so that bibo shall have 5 times as much as Sarah
Please help! will give 14 points and brainliest!
Answer:
EF = 22 , AB = 1
Step-by-step explanation:
the midsegment of a trapezoid is equal to half the sum of the parallel bases
EF = [tex]\frac{1}{2}[/tex] (AB + DC ) ← substitute values
x + 6 = [tex]\frac{1}{2}[/tex] (2x - 9 + x + 5) ← multiply both sides by 2 to clear the fraction
2x + 12 = 3x - 4 ( subtract 3x from both sides )
- x + 12 = - 4 ( subtract 12 from both sides )
- x = - 16 ( multiply both sides by - 1 )
x = 16
Then
EF = x + 6 = 16 + 6 = 22
Similarly
EF = [tex]\frac{1}{2}[/tex] (AB + DC ) , that is
x + 3 = [tex]\frac{1}{2}[/tex] (4x - 3 + 2x + 5 ) ← multiply both sides by 2 to clear the fraction
2x + 6 = 6x + 2 ( subtract 6x from both sides )
- 4x + 6 = 2 ( subtract 6 from both sides )
- 4x = - 4 ( divide both sides by - 4 )
x = 1
Then
AB = 4x - 3 = 4(1) - 3 = 4 - 3 = 1
Which is equivalent to (3/125)*?
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Question reads:}[/tex]
[tex]\large\textbf{Which is equivalent to }\rm{\bf \dfrac{3}{125}}\large\textbf{ ?}[/tex]
[tex]\huge\textbf{Solving for your fraction:}[/tex]
[tex]\mathbf{\dfrac{3}{125}}[/tex]
[tex]\mathbf{= \dfrac{3\times2}{125\times2}}[/tex]
[tex]\mathbf{\approx \dfrac{3 + 3}{125 + 12{5}}}[/tex]
[tex]\mathbf{= \dfrac{6}{250}}[/tex]
[tex]\huge\textbf{Therefore, your answer should be:}[/tex]
[tex]\huge\boxed{\frak{\dfrac{6}{250}}}\huge\checkmark[/tex]
[tex]\large\text{By the way, they are a lot more fractions that are equivalent to}\\\large\text{the given equation (}\rm{\dfrac{3}{125}}\large\text{) but we only labeled one that is the very}\\\large\text{first fraction that is equal to it. }[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]From the diagram below, if AC =40, CD=10, and DB=8, then what would be the length of AB?
The length of AB would be 38 units
How to determine the length AB?The attached image represents the missing piece in the question
Considering the triangle BCD;
Calculate the length BC using the following Pythagoras theorem
BC^2 = CD^2 + BD^2
This gives
BC^2 = 10^2 + 8^2
Evaluate
BC^2 = 164
Considering the triangle CBA;
Calculate the length AB using the following Pythagoras theorem
AB^2 = AC^2 - BC^2
This gives
AB^2 = 40^2 - 164
Evaluate
AB^2 = 1436
Take the square root
AB = 38 (approximated)
Hence, the length of AB would be 38 units
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A store sells both cold and hot beverages. Cold beverages, c, cost $1.50, while hot beverages, h, cost $2.00. On Saturday, drink receipts totaled $360, and 4 times as many cold beverages were sold as hot beverages.
Which system of linear equations represents the beverage sales on Saturday?
Answer:
There are two equations and two variables, so this is a system of linear equations.
c + h = 360
4c = h
Step-by-step explanation:
Answer:
The linear equations would be c = 4h and 1.5c + 2h = 360.
Step-by-step explanation:
First we have to create a table for our data.
c = total number of cold beverages sold.
h = total number of hot beverages sold.
The total cost of both beverages sold -
1.5c + 2h = 360
The amount of cold beverages -
c = 4h
Now we just need to substitute the second equation to the first one.
1.5 * (4h) + 2h = 360
= 6h + 2h = 360
= 8h = 360
= h = 45
Now we need to find the value of c.
c = 4* 45 = 180
Therefore 180 cold beverages were sold and 45 hot beverages were sold.
But the linear equations would be c = 4h and 1.5c + 2h = 360.
Hope this helped! :D
A bank employee is filling an empty cash machine with bundles of $100.00,$500.00 and $1000.00 bills each bundle has 100bills in it and the machine hold 10 bundles of each type. What amount of money is required to fill the machines?
Answer:
65fghxtt
Step-by-step explanation:
Find the area of the shaded region. (drawing is not to scale) GEOMETRY!!! HELP PLEASE
Answer:
150
Step-by-step explanation:
The area of the trapezoid is (½)(14+10)(2) = 24.
The area of the rectangle is (14)(9) = 126.
Adding these together, we get 126 + 24 = 150.
I need the volume of that figure
Answer:
708π in³ = 2224.25 in.³
Step-by-step explanation:
The composite solid is made up of a cone, a cylinder, and a half sphere.
Recall the volume formulas:
cone: πr²h/3
cylinder: πr²h
sphere: 4πr³/3
half sphere: (1/2) × 4πr³/3
We need to find teh heoght of the cone. The slant height is 10 in. The radius is 6 in. We can use the Pythaogorean theorem to find the height of the cone.
h = √(10² - 6²) = 8
total volume = π(6 in.)²(8 in.)/3 + π(6 in.)²(13 in.) + (1/2) × 4π(6 in.)³/3
total volume = π(96 in.³ + 468 in.³ + 144 in.³)
total volume = 708π in³ = 2224.25 in.³
A plane takes off from an airport on a bearing of 270° and travels at a speed of 320 mph. after sometime and flight, space the plane encounters a 35 mph wind blowing directly north. Find the speed of the airplane after it encounters the wind.
A.322mph B.330mph C.285mph D.355mph
The speed of the plane after it encounters the wind is C.285mph
How to calculate the speed of the plane when it encounters the wind?Since the plane takes off from an airport on a bearing of 270° and travels at a speed of 320 mph it's velocity is v = (320cos270°)i + (320sin270°)j
= (320 × 0)i + (320 × -1)j
= 0i - 320j
= - 320j mph
Also, the plane encounters a 35 mph wind blowing directly north. The velocity of the wind is v' = 35j mph
So, the velocity of the plane after it encounters the wind is the resultant velocity, V = v + v'
= -320j mph + 35j mph
= -285j mph
So, the speed of the plane after it encounters the wind is the magnitude of V = |-285j| mph
= 285 mph
So, the speed of the plane after it encounters the wind is C.285mph
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Look at this triangle.
A
Work out length BC.
13 cm
8 cm
B
С
In the given triangle, the length of BC is 10.25 cm
Solving a right triangle
From the question, we are to determine the length BC
The triangle is a right triangle, then we can write that
/AB/² = /AC/² + /BC/² (Pythagorean theorem)
From the given information,
/AB/ = 13 cm
/AC/ = 8 cm
Putting the parameters into the equation, we get
13² = 8² + /BC/²
169 = 64 + /BC/²
/BC/² = 169 - 64
/BC/² = 105
/BC/ = √105
/BC/ = 10.25 cm
Hence, the length of BC is 10.25 cm
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HELP THIS IS MY LAST QUESTION
Answer:
0.71
Step-by-step explanation:
- Please see the attached picture for full solution!:)
------------ HappY LearninG <3 -----------
So this is not a school related question, I am a trader looking to see my APY if I take an initial investment amount and make 5% on it every day of the year. I've tried the compound interest formula but it doesn't work, I only get about a 64.8% increase which I know is false. So here is what I'm looking for: I need to know the official APY of making 5% everyday on a 1,000 investment for 1 year and then the final amount your left with.
Answer:
um tbh me dont know
Step-by-step explanation:
Ted has 45 apples he give his one friend 20 apples how many apples do ted have left
What is the volume of the sphere shown below with a radius of 9? OA. 108 cu. units B. 972* cu units OC. 2916 cu units D. 1458, cu units
The volume of the sphere shown below with a radius of 9 is 3052.08 cubic units
Volume of a sphereThe formula for calculating the volume of a sphere is expressed as;
V = 4/3πr³
where
r is the radius of the sphere
Substitute
V = 4/3π(9)³
V = 4/3(3.14)*729
V = 3052.08 cubic units
Hence the volume of the sphere shown below with a radius of 9 is 3052.08 cubic units
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Graph a system of equations to solve Log(-5.6x+1.3)=-1-x
From least to greatest the solutions are:
By seeing where the curves intercept, we conclude that the solutions are:
(0.22, -1.22) and (-2.12, 1.12)
How to solve the system of equations?
Here we just need to graph the equations:
y = log(-5.6*x + 1.3)
y = -1 - x
And see in which points the curves intercept.
The graph can be seen below:
There we can see that the intersections (and solutions) are the points:
(0.22, -1.22) and (-2.12, 1.12)
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Can some one help !!???
Answer:
Yes
Step-by-step explanation:
please help solve inequality question c
The solution to the given inequality is; x ≥ -9/4 or x ≥ 1
How to solve Inequality problems?
We want to solve the inequality;
(4x² + 5x - 9)/(x² - x - 6) ≥ 0
Let us factorize both numerator and denominator to get;
4x² + 5x - 9 = (4x + 9)(x - 1)
Similarly, x² - x - 6 when factorized gives;
x² - x - 6 = (x + 2)(x - 3)
Thus, we now have;
[(4x + 9)(x - 1)]/[(x + 2)(x - 3)] ≥ 0
Multiply both sides by [(x + 2)(x - 3)] to get;
(4x + 9)(x - 1) ≥ 0
Thus;
(4x + 9) ≥ 0 or (x - 1) ≥ 0
Thus;
x ≥ -9/4 or x ≥ 1
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Listen
If 2/3 is the fraction representing the ratio of corresponding sides of two similar
triangles, the ratio of the areas of the triangles would be:
2/3
3/2
1.41/1.73
4/9
Question 3 (1 point)
6
help me please, anwser me fast please
Answer:
Yes, it is a function
Step-by-step explanation:
A function has only one output for each input. So a function can have the same output for different input, but one input can only have one output. In this case none of the inputs output more than one output, so it is a function.
I set off on a 15 miles cycle ride at a speed of 10mile per hour. how long does the ride take me
The ride takes 1.5 hours for a speed of 10 miles per hour.
What is speed?The rate at which an object travels a certain distance can be described using the speed formula. The distance that a body travels in a certain amount of time is a common way to measure speed.
A body's speed can be defined as how quickly it is going. The equation for a given body's speed can be written as,
Speed=Distance/Time
According to the question,
Speed=10 miles per hour
Distance=15 miles
Time=[tex]\frac{15}{10}[/tex]=1.5 hours
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Last years freshman class at big state university totaled 5,305 students of those 1258 received a merit scholarship to help offset tuition costs their freshman year.the amount received was n(3456, 478) if the full cost was 4250 what percentage of students receive a merit scholarship did not receive enough to cover full tuition
Using the normal distribution, it is found that 95.15% of students receive a merit scholarship did not receive enough to cover full tuition.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.The mean and the standard deviation for the amounts are given as follows:
[tex]\mu = 3456, \sigma = 478[/tex]
The proportion is the p-value of Z when X = 4250, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{4250 - 3456}{478}[/tex]
Z = 1.66
Z = 1.66 has a p-value of 0.9515.
Hence 95.15% of students receive a merit scholarship did not receive enough to cover full tuition.
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Help me with this problems
Answer:
its easy
Step-by-step explanation:
you should change all into degrees
HELPPPP PLEASEEE ASAPP!! The population of a small farming community is declining at a rate of
7% per year. The decline can be expressed by the exponential
equation P=C (1-0.07), where P is the population after t years and
C is the current population. If the population was 8500 in 2004, when
will the population be less than 6000?
Answer:
365/8500*7=0.30058823529411-0.07/100*7=0.9951
PLEASE HELP Which of the following expressions could not be used to determine the exact value of cos 60°?
The trigonometric expression cos 60° is equivalent to the expressions cos² 30° - sin² 30°, 2 · cos² 30° - 1 and 1 - 2 · sin² 30°. Then, not equivalent to 2 · sin 30° · cos 30°. (Correct choice: A)
How to find an expression not equivalent to a trigonometric equation
In this question we have a trigonometric equation and we are supposed to find an expression not equivalent to the former. Herein we have that the cosine of a double angle is equivalent to the following expressions:
cos 2θ = cos² θ - sin² θ (1)
cos 2θ = 2 · cos² θ - 1 (2)
cos 2θ = 1 - 2 · sin² θ (3)
The trigonometric expression cos 60° is equivalent to the expressions cos² 30° - sin² 30°, 2 · cos² 30° - 1 and 1 - 2 · sin² 30°. Then, not equivalent to 2 · sin 30° · cos 30°. (Correct choice: A)
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A trader sold 100 boxes of a $8.00 per box, 800 boxes at $6.00 per box and 600 boxes at $4.00 per box .find the average selling price per box
The average selling price per box is $5.33
What is the average selling price?The core value of a set of data is expressed as the average of a list of variables. It is defined mathematically as the ratio of the total number of data points to the number of units in the list. The average of a specific set of numerical data is also known as the mean in statistics.
Average = Sum of Values/Number of Values
The total selling price of all boxes=(8*100)+(800*6)+(600*4)
=$8000
Total number of boxes=100+800+600
=1500
Average selling price per box=8000/1500=$5.33
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In 2012, the population of a city was 6.93 million. The exponential growth rate was 2.18% per year.
a) Find the exponential growth function.
b) Estimate the population of the city in 2018.
e) When will the population of the city be 10 million?
d) Find the doubling time.
a) The growth function has the form:
[tex]P(x) = P*(1 + r)^x[/tex],where:
P(x) - population after x years, P- initial population, r - rate of change, x - number of years since 2012.Considering the values we get the function:
[tex]P(x) = 6.93*(1 + 2.18/100 ) ^x=6.93*1.0218^x[/tex]b) Number of years between 2012 and 2018 is:
x = 2018 - 2012 = 6Find the population, using the above equation:
[tex]P(6) = 6.93*1.0218^6 = 7.89[/tex] million (rounded)c) Find the value of x when P(x) = 10 million
[tex]10 = 6.93*1.0218^x[/tex][tex]1.0218^x = 10/6.93[/tex][tex]1.0218^x=1.443[/tex][tex]log1.0218^x=log1.443[/tex][tex]x = log1.443/log1.0218[/tex][tex]x=17[/tex] yearsd) Find the doubling time:
[tex]2*6.93 =6.93*1.0218^x[/tex][tex]1.0218^x=2[/tex][tex]x=log2/log1.0218[/tex][tex]x=32.14[/tex] yearsAnswer:
[tex]\textsf{a)} \quad f(x)=6.93(1.0218)^x[/tex]
b) 7.89 million (2 d.p.)
c) 2019
d) 32.14 years (2 d.p.) after 2012
Step-by-step explanation:
Exponential Function
[tex]y=ab^x[/tex]
where:
a is the initial value (y-intercept)b is the base (growth/decay factor) in decimal formx is the independent variabley is the dependent variableIf b > 1 then it is an increasing function
If 0 < b < 1 then it is a decreasing function
Part (a)
If the exponential growth rate was 2.18% per year, then each year there would be 102.18% of the previous year's population.
⇒ b = 1.0218
Given:
a = 6.93 millionb = 1.0218x = time (in years after 2012)y = population (in millions)Substitute the given values into the formula to create an exponential growth function:
[tex]f(x)=6.93(1.0218)^x[/tex]
Part (b)
To estimate the population of the city in 2018, determine the value of x:[tex]x = 2018 - 2012 = 6[/tex]
Substitute the found value of x into the found exponential function:
[tex]\implies f(6)=6.93(1.0218)^6=7.89 \: \sf million\:(2\:d.p.)[/tex]
Part (c)
To determine when the population of the city will be 10 million, set the function to 10 and solve for x:
[tex]\implies 6.93(1.0218)^x=10[/tex]
[tex]\implies (1.0218)^x=\dfrac{10}{6.93}[/tex]
[tex]\implies \ln (1.0218)^x= \ln \left(\dfrac{10}{6.93}\right)[/tex]
[tex]\implies x\ln (1.0218)= \ln \left(\dfrac{10}{6.93}\right)[/tex]
[tex]\implies x= \dfrac{\ln \left(\frac{10}{6.93}\right)}{\ln (1.0218)}[/tex]
[tex]\implies x=17.00496413...[/tex]
To find the year, add the found value of x to 2012:
[tex]\implies \sf 2012 + 17.00496413... = 2019[/tex]
Part (d)
To find the doubling time, set the function to double the initial population and solve for x:
[tex]\implies 6.93(1.0218)^x=6.93 \cdot 2[/tex]
[tex]\implies (1.0218)^x=2[/tex]
[tex]\implies \ln (1.0218)^x= \ln (2)[/tex]
[tex]\implies x \ln (1.0218)= \ln (2)[/tex]
[tex]\implies x = \dfrac{\ln (2)}{\ln (1.0218)}[/tex]
[tex]\implies x=32.14107014...[/tex]
Therefore, the doubling time is 32.14 years (2 d.p.) after 2012.
Student Name:
Lesson Number: 22
4th Proportion
Directions:
Complete the following constructions. Leave all construction arcs on your paper.
a
Math Assignment
Level: 1007
b
1 Given a, b, and c, the length of three segments. Find the fourth proportional.
Find the fourth proprtional to r, s, and t.
The fourth proportionals are bc/a and st/r
How to determine the fourth proportional?Segments a, b and c
Let the fourth segment be x.
So, we have the following equivalent ratio
a : b = c : x
Express as fraction
a/b = c/x
Make x the subject
x = bc/a
Segments r, s and t
Let the fourth segment be x.
So, we have the following equivalent ratio
r : s = t : x
Express as fraction
r/s = t/x
Make x the subject
x = st/r
Hence, the fourth proportionals are bc/a and st/r
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The functions fand g are defined as follows.
g(x) = -2x³-5
f(x) = -2x-3
Find f(5) and g (-2).
Simplify your answers as much as possible.
ƒ (5) = 0
8 (-2) =
믐
X S
2.
?
Answer:
f(x) = -2x - 3
g(x) = -2x³ - 5
f(5) = -2(5) - 3
f(5) = -10 - 3
f(5) = -13
g(-2) = -2(-2)³ - 5
g(-2) =-2(-8) - 5
g(-2) = 16 - 5
g(-2) = 11
Create an inequality:
The sun of the numbers is not more than 12
Answer:
x + y ≤ 12
Step-by-step explanation:
Inequality:
Not more than 12 means less than or equal to 12
x + y ≤ 12