Answer:
[tex]\displaystyle y' = 2(2x - 5)(x^5 - 5)(12x^5 - 25x^4 - 10)[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightDistributive Property
Algebra I
Terms/CoefficientsFactoringCalculus
Derivatives
Derivative Notation
Derivative of a constant is 0
Basic Power Rule:
f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
y = (2x - 5)²(5 - x⁵)²
Step 2: Differentiate
Derivative Rule [Product Rule]: [tex]\displaystyle y' = \frac{d}{dx}[(2x - 5)^2](5 - x^5)^2 + (2x - 5)^2\frac{d}{dx}[(5 - x^5)^2][/tex]Chain Rule [Basic Power Rule]: [tex]\displaystyle y' = [2(2x - 5)^{2-1} \cdot \frac{d}{dx}[2x]](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5)^{2-1} \cdot \frac{d}{dx}[-x^5]][/tex]Simplify: [tex]\displaystyle y' = [2(2x - 5) \cdot \frac{d}{dx}[2x]](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot \frac{d}{dx}[-x^5]][/tex]Basic Power Rule: [tex]\displaystyle y' = [2(2x - 5) \cdot 1(2x^{1 - 1})](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot -5x^{5 - 1}][/tex]Simplify: [tex]\displaystyle y' = [2(2x - 5) \cdot 2](5 - x^5)^2 + (2x - 5)^2[2(5 - x^5) \cdot -5x^4][/tex]Multiply: [tex]\displaystyle y' = 4(2x - 5)(5 - x^5)^2 - 10x^4(2x - 5)^2(5 - x^5)[/tex]Factor: [tex]\displaystyle y' = 2(2x - 5)(5 - x^5)[2(5 - x^5) - 5x^4(2x - 5)][/tex][Distributive Property] Distribute 2: [tex]\displaystyle y' = 2(2x - 5)(5 - x^5)[10 - 2x^5 - 5x^4(2x - 5)][/tex][Distributive Property] Distribute 5x⁴: [tex]\displaystyle y' = 2(2x - 5)(5 - x^5)[10 - 2x^5 - 10x^5 + 25x^4][/tex][Addition] Combine like terms (x⁵): [tex]\displaystyle y' = 2(2x - 5)(5 - x^5)(10 - 12x^5 + 25x^4)[/tex]Rewrite: [tex]\displaystyle y' = 2(2x - 5)(x^5 - 5)(12x^5 - 25x^4 - 10)[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
There are 4 teams. Each team plays each other team once. How many games are played?
A 3
B 4
C. 6
D. 12
E 16
Answer:
E) 16
Step-by-step explanation:
4 * 4 = 16
Hope this helps
The solution is Option C.
The number of games played by 4 teams if they play each other only once is given by combinations and is 6 games
What are Combinations?
The number of ways of selecting r objects from n unlike objects is:
ⁿCₓ = n! / ( ( n - x )! x! )
where
n = total number of objects
x = number of choosing objects from the set
Given data ,
Let the total number of teams be n = 4 teams
Each team plays the other team only once , so x = 2
The number of games played by 4 teams if they play each other only once is given by Combination
ⁿCₓ = n! / ( ( n - x )! x! )
Substituting the values in the equation , we get
⁴C₂ = ( 4! ) / 2! x 2!
On simplifying the equation , we get
⁴C₂ = ( 4 x 3 ) / 2 x 1
⁴C₂ = 2 x 3
⁴C₂ = 6 games
Therefore , the value of ⁴C₂ =6
Hence , the number of games is 6 games
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find the value of x. if necessary, round to the nearest tenth
a. 13.7 in
b. 11.7 in
c. 6.9 in
d. 9.7 in
The sum of and is -15/32 and 7/32 is
Answer:
-8/32 or -1/4(if simplified)
Step-by-step explanation:
-15+7 = -8
-8/32 or -1/4
To test for the significance of a regression model involving 14 independent variables and 255 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are:_________
a. 14 and 255.
b. 255 and 14.
c. 13 and 240.
d. 14 and 240.
Answer:
d). 14 and 240
Step-by-step explanation:
According to the Question,
Given that, The regression model involving 14 independent variables and 255 observationsWe need to find out the numerator and denominator freedom degrees ,
k = 14 regression variable
= n - k - 1
= 255 - 14 - 1
= 240 residuals
df=14,240
Now The degree of freedom for F-critical isF_{k,n-k-1}=F_{14,240}
So, The df of the numerator is 14 and df of the denominator is 240
There are currently 441 dairy cows at Dancing Dairy Farm. Due to some considerations, the number of dairy cows is decreasing at the rate of 13 cows per year. Currently, each cow produces an average of 1157 gallons of milk per year, and production of milk is increasing at the rate of 39 gallons per cow per year. Use the product rule to determine the rate at which milk production at Dancing Dairy Farm is currently changing.
Answer:
the production of milk at the Dancing Dairy Farm is increasing by 2158 gallons/year
Step-by-step explanation:
Given the data in the question;
Let us represent the number of cows at the farm with x and
each cow produces y gallons of milk
Total mil production will be T.
so
T = x × y
now, differentiating with respect to t
dT/dt = d( xy )/dt
dT/dt = xdy/dt + ydx/dt
given that;
x = 441
y = 1157
dx/dt = -13
dy/dt = 39
so we substitute
dT/dt = ( 441 )( 39 ) + ( 1157 )( -13 )
dT/dt = 17199 - 15041
dt/dT = 2158
Therefore, the production of milk at the Dancing Dairy Farm is increasing by 2158 gallons/year
HELP ME PLEASEEEEEEEEEEEEEE
Answer:
a
Step-by-step explanation:
You have savings of 30,000. You invest in a bond mutual fund that pays 3% simple interest. What are your total savings after 20 years?
Answer:
54,183.34
Step-by-step explanation:
a = 30,000(1.03)²⁰
a = 54,183.34
 Marsha has a bag that contains 4 green marbles, 8 yellow marbles , and 20 red marbles . If she chooses one marble from the bag, what is the probability that the marble is not red?
PLEASE HELP IF YOURE GOOD AT GEOMETRY!!
Answer:
C. 3/8
HOPE THIS HELPS :)
Answer:
c. 3/8
Step-by-step explanation:
first you need the denomerator by adding all marbles together which equals 32. now for the munerator you need the sum of the green and yellow marbles. this equals 12. so your fraction is 12/32. next we simplify. we can divide both numbers by 4. getting us a fraction of 3/8.
Marcus likes to go for a run each morning before school. He recorded the number of minutes he spends running and the distance he covers. The scatter plot represents the data he collected
Answer:
D. 0.85
Step-by-step explanation:
The data points in the scatter plot are closer to each other along the line of best fit, this means that there is a strong positive association between minutes and distance and therefore, the correlation coefficient would be relatively closer to 1.
The correlation is positive since both variables tend to increase together in the same direction.
Therefore, the best estimate of the correlation coefficient out of the given options would be 0.85
parabola
Given that tanθ= [tex]-\frac{9}{4}[/tex] and [tex]\frac{\pi }{2\\}[/tex]<θ<π , find the exact values of the trigonometric functions.
9514 1404 393
Answer:
sin(θ) = (9√97)/97cos(θ) = (-4√97)/97csc(θ) = (√97)/9sec(θ) = (-√97)/4cot(θ) = -4/9Step-by-step explanation:
The angle is in the 2nd quadrant, where the sine is positive and the cosine is negative.
tan^2(θ) +1 = sec^2(θ) = (-9/4)^2 +1 = 97/16 ⇒ sec = -(1/4)√97
cot(θ) = 1/tan(θ) = -4/9
csc^2(θ) = cot^2(θ) +1 = (-4/9)^2 +1 = 97/81 ⇒ csc = (1/9)√97
sin(θ) = 1/csc(θ) = (9√97)/97
cos(θ) = 1/sec(θ) = (-4√97)/97
Can I get an awnser to both of these questions? Thank you!
Answer:
y+1=7(x+4) for first one
y-4=5(x-0) for second one
Step-by-step explanation:
point slope form for both of those questions.
the formula for point slope form is
y-y1(y coordinate) = slope(x+x2(x coordinate))
y+1=7(x+4)
Second question:
-1,-1 to 0,4
change in x +1, change in y +5.
5/1 = 5
slope is 5
y-4=5(x-0)
How many sticks would be in pattern number n
Answer:
hey. pls post a pic of the pattern or type in the number of sticks in each term so we could help tou determine the nth term
A market has two check out lines. Let X be the number of customers at the express checkout line at a particular time of day. Let Y denote the number of customers in the super-express line at the same time. The joint probability mass function of (X, Y) is given below. What is the probability of having only one customer at the express checkout line when there are more than two customers in the super-express line at the same time
I need help with this math problem
Answer: A) (x + 12) and (x + 9)
Step-by-step explanation:
You need to get x^2 + 21x + 108.
A) (x + 12) (x + 9) = x^2 + 12x + 9x + 108 = x^2 + 21x + 108
B) (x - 12) (x - 9) = x^2 - 9x - 12x + 108 = x^2 -21x + 108
C) (x - 27) (x - 4) = x^2 - 4x - 27x + 108 = x^2 - 31x + 108
D) (x + 27) (x + 4) = x^2 + 27x + 4x + 31 = x^2 + 31x + 31
So, the answer is A).
The rate of inflation measures the percentage increase in the price of consumer goods. The rate of inflation in the year 2014 was 2% per year. To get a sense of what this rate would mean in the long run, let's suppose that it persists through 2034.
What would be the cost in 2034 of an item that costs $100 in 2014? (Round your answer to the nearest cent.)
$
Answer:
148.59
Step-by-step explanation:
Inflation is kind of similar to interest and with that bieng said we can (kind of ) use the same formula
the time between 2034 and 2014 is 2034-2014= 20
100(1.02)²⁰=148.5947396
which we can round to 148.59
148.5 is the cost in 2034 of an item that costs $100 in 2014
What is Simple Interest?Simple interest is a quick and easy method of calculating the interest charge on a loan
[tex]A=P(1+r)^{t}[/tex]
where A is the Actual amount
P is Initial amount
t is time period
r is rate of interest
Given,
The rate of inflation measures the percentage increase in the price of consumer goods
The rate of inflation in the year 2014 was 2% per year.
2% is converted to decimal by dividing with 100
2/100=0.02
The time gap between 2014 and 2034 is
2034-2014=20 years
The initial amount is 100
A=100(1+0.02)²⁰
A=100(1.02)²⁰
A=100×1.485
A=148.5
Hence 148.5 is the cost in 2034 of an item that costs $100 in 2014.
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This holiday season, Ms. Bastarache organized to give each 9th grader a $10 Dunkin Donuts gift card. Because she bought a lot of cards, Ms. Bastarache tipped the Dunkin Donuts employee $20.
This function represents the relationship between the number of gift cards, c, and T(c), the total cost of the cards.
T(c) = 10c + 20
Be sure to answer BOTH questions below.
1. Determine the total cost, in dollars, for Ms. Bastarache to give out 93 gift cards.
2. Last year, Ms. Bastarache spent $820, how many cards did she purchase?
need help asap plzzzzzz
Answer:
1. $950
2. 80
Step-by-step explanation:
What is the missing number in the table?
Answer:
1296
Step-by-step explanation:
the y is multiplying by six each time 6*6 = 36
36*6 = 216
216*6 = 1296
2196 = 7776
A restaurant charges large parties an amount that depends on the number of people
that are eating. The restaurant charges $650 for 25 people and $1,850 for 80 people.
What is the restaurant charging per person (unit rate)?
Answer:
$26 per person or $23.125 per person for the bigger group
Step-by-step explanation:
How much would you pay for: 240 textbooks at $15.40 each, 100 pens at $1.25 each and 30
dozen erasers at 95c each?
Answer:
384950c or $3849.50
Step-by-step explanation:
240x1540=369600
100x125=12500
30x95=2850
369600+12500+2850=384950
A landscaper buys 1 gallon of plant fertilizer. He uses 1/5 of the fertilizer, and then divides the rest into 3 smaller bottles. How much does he put in each bottle?
Answer:
4/15 gallon per bottle
Step-by-step explanation:
First, we find the remaining fraction of fertilizer after using 1/5. Using our fraction calculator, we see:
1 - 1/5 = 4/5
To find the amount of fertilizer per bottle, we then divide 4/5 by 3 and we get:
4/15 gallon per bottle
John made this model to show 4/7 * 13/9
Using John's model, what is 4/7 * 13/9? (has to be in fractions)
Answer:
52/63
Step-by-step explanation:
Given the expression
4/7 * 13/9
Taking the product
= (4*13)/(7*9)
= 52/63
Hence the product of both fractions is 52/63
Which pairs in the form x,y are solutions to the equation 7x-5y=28?
Answer:
x=4 y=0
Step-by-step explanation:
7(4)-5(0)=28
28-0=28
28=28
At a real estate agency, an agent sold a house for $367,000 The commission rate is
6.5% for the real estate agency and the commission rate for the agent is 25 % of the amount the real estate agency gets. How much did the agency make on the house? How much did the agent earn in commission?
The agency made $___ on the house.
Answer:
The agency makes 23855
The agent makes 5963.75
Step-by-step explanation:
First find the commission for the agency
367000 * 6.5%
367000*.065
23855
Now the agent gets 25% of this amount
23855 *25%
23855 *.25
5963.75
Answer:
Step-by-step explanation:
The agency made $24,440.00
The Agent made $6,110.00
The resale value V, in thousands of dollars, of a boat is a function of the number of years t since the start of 2011, and the formula is
V = 12.5 − 1.3t.
(a) Calculate V(3).
(b) In what year will the resale value be 7.3 thousand dollars?
(c) Solve for t in the formula above to obtain a formula expressing t as a function of V.
(d) In what year will the resale value be 3.4 thousand dollars?
Answer:
a) V(3) = 8.6.
b) The resale value will be 7.3 thousand dollars at the start of 2015.
c) [tex]t(V) = \frac{12.5 - V}{1.3}[/tex]
d) 2017
Step-by-step explanation:
We are given the following function:
[tex]V(t) = 12.5 - 1.3t[/tex]
(a) Calculate V(3).
This is V when t = 3. So
[tex]V(3) = 12.5 - 1.3(3) = 8.6[/tex]
So
V(3) = 8.6.
(b) In what year will the resale value be 7.3 thousand dollars?
t years after the start of 2011, and t is found when [tex]V(t) = 7.3[/tex]. So
[tex]V(t) = 12.5 - 1.3t[/tex]
[tex]7.3 = 12.5 - 1.3t[/tex]
[tex]1.3t = 5.2[/tex]
[tex]t = \frac{5.2}{1.3}[/tex]
[tex]t = 4[/tex]
2011 + 4 = 2015
The resale value will be 7.3 thousand dollars at the start of 2015.
(c) Solve for t in the formula above to obtain a formula expressing t as a function of V.
[tex]V(t) = 12.5 - 1.3t[/tex]
[tex]1.3t(V) = 12.5 - V[/tex]
[tex]t(V) = \frac{12.5 - V}{1.3}[/tex]
(d) In what year will the resale value be 3.4 thousand dollars?
t years after 2011, and t is found t when [tex]V = 3.4[/tex]. So
[tex]V(t) = 12.5 - 1.3t[/tex]
[tex]3.9 = 12.5 - 1.3t[/tex]
[tex]1.3t = 8.6[/tex]
[tex]t = \frac{8.6}{1.3}[/tex]
[tex]t = 6.62[/tex]
2011 + 6.62 = 2017
So the year is 2017.
Jed bought a generator that will run for 2 hours on a liter of gas. The gas tank on the generator is a rectangular prism with dimensions 20 cm by 15 cm by 10 cm, as shown. If Jed fills the tank with gas, how long will the generator run? Show how you arrived at your answer. [1000 cm 3= 1 liter] Explain your answer.
Answer:
The generator will run for 6 hours.
Step-by-step explanation:
First, find the volume of the gas tank that is on the generator. I am told that the gas tank is a rectangular prism that has the dimensions of 20 cm, 15 cm, and 10cm.
Area of any prism = height * width * length
Area of prism = 20 cm * 15 cm * 10 cm = 3,000 cm cubed
I am told the important info that 1,000 cm cubed = 1 liter
Therefore, 3,000 cm cubed = 3 liters
I am also told that the generator will run for 2 hours on 1 liter of gas
Lets make a ratio table:
Hours the generator will run for : liters of gas
2 : 1
6 : 3
the generator will run for 6 hours.
The mean height of women in a country (ages 20-29) is 64 4 inches A random sample of 50 women in this age group is selected What is the probability that the mean height for the sample is greater than 65 inches? Assume o = 2.91 The probability that the mean height for the sample is greater than 65 inches is
Answer:
0.0721 = 7.21% probability that the mean height for the sample is greater than 65 inches.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean of 64.4 inches, standard deviation of 2.91
This means that [tex]\mu = 64.4, \sigma = 2.91[/tex]
Sample of 50 women
This means that [tex]n = 50, s = \frac{2.91}{\sqrt{50}}[/tex]
What is the probability that the mean height for the sample is greater than 65 inches?
This is 1 subtracted by the p-value of Z when X = 65. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{65 - 64.4}{\frac{2.91}{\sqrt{50}}}[/tex]
[tex]Z = 1.46[/tex]
[tex]Z = 1.46[/tex] has a p-value of 0.9279
1 - 0.9279 = 0.0721
0.0721 = 7.21% probability that the mean height for the sample is greater than 65 inches.
What are the zeros of the function y = 2x2 + 5x + 2?
A. x =
- 3x =
x = -2
B. X =
1
2
.X = 2
C. X =
3.x=-2
O D. x=-1.x = 2
Answer: not sure
but I need points
Step-by-step explanation:
JUST KIDDING It’s A I did this in school before and got it correct :)
anyone help me, let's prove
Answer:
In my opinion the limit is equal to 1 not 0, sorry.
Step-by-step explanation: 6 25 13 43
lim n ⇒∞ ((2n - 1)/2n)
lim n ⇒∞ (2n/2n) - 1)/2n) 2n/2n = 1 1/∞ = 0
= 1 - 0
= 1
when I graphed the function I also got 1
giải phương trình y′′ − 7y′ + 6y = sin x.
Start with the underlying homogeneous equation:
[tex]y''-7y'+6y=0[/tex]
which has characteristic equation
[tex]r^2-7r+6=(r-6)(r-1)=0[/tex]
with roots at r = 6 and r = 1. So the characteristic solution is
[tex]y_c=C_1e^{6x}+C_2e^x[/tex]
Now for the particular solution, we can use the method of undetermined coefficients, with the following ansatz (the "guess" solution) and its derivatives,
[tex]y_p=a\cos x+b\sin x[/tex]
[tex]{y_p}'=-a\sin x+b\cos x[/tex]
[tex]{y_p}''=-a\cos x-b\sin x[/tex]
Substituting these into the original equation gives
[tex](-a\cos x-b\sin x)-7(-a\sin x+b\cos x)+6(a\cos x+b\sin x)=\sin x[/tex]
[tex](5a-7b)\cos x+(7a+5b)\sin x=\sin x[/tex]
[tex]\implies\begin{cases}5a-7b=0\\7a+5b=1\end{cases}\implies a=\dfrac7{74},b=\dfrac5{74}[/tex]
So the particular solution is
[tex]y_p=\dfrac7{74}\cos x+\dfrac5{74}\sin x[/tex]
and hence the general solution is
[tex]y=y_c+y+p=\boxed{C_1e^{6x}+C_2e^x+\dfrac7{74}\cos x+\dfrac5{74}\sin x}[/tex]
Simplify:......................................................
Answer:
...
Step-by-step explanation:...