Answers
Answer to part (A) is y = 42x+9
Answer to part (B) is 98
========================================================
Explanation:
Part (A)
Let's plug x = 0 into the 1st derivative of f(x)
[tex]f'(\text{x}) = \cos(\pi \text{x}) + \text{x}^5 + 6\\\\f'(0) = \cos(\pi *0) + (0)^5 + 6\\\\f'(0) = 1 + 0 + 6\\\\f'(0) = 7\\\\[/tex]
We'll use that later in the steps below, which show computing the derivative value of h(x) at x = 0.
[tex]h(\text{x}) = ( f(\text{x}) )^2\\\\h'(\text{x}) = 2( f(\text{x}) )*f'(\text{x}) \ \ \text{ ... chain rule}\\\\h'(0) = 2( f(0) )*f'(0)\\\\h'(0) = 2( 3 )*7\\\\h'(0) = 42\\\\[/tex]
This is the slope of the tangent line to h(x) at x = 0.
Now plug x = 0 into the h(x) function itself, without any derivatives applied.
[tex]h(\text{x}) = ( f(\text{x}) )^2\\\\h(0) = ( f(0) )^2\\\\h(0) = ( 3 )^2\\\\h(0) = 9\\\\[/tex]
This is the y intercept of the line, i.e. the b value.
We found that
m = 42 = slope of the tangentb = 9 = y intercept of the tangent lineWe go from y = mx+b to y = 42x+9 as the equation of the tangent line.
===================================================
Part (B)
In the previous part, we already calculated the first derivative. Differentiate that with respect to x to get the second derivative.
[tex]h'(\text{x}) = 2( f(\text{x}) )*f'(\text{x})\\\\h''(\text{x}) = \frac{d}{dx}\left[h'(x)\right]\\\\h''(\text{x}) = \frac{d}{dx}\left[2( f(\text{x}) )*f'(\text{x})\right]\\\\h''(\text{x}) = \frac{d}{dx}\left[2( f(\text{x}) )\right]*f'(\text{x})+2*f(\text{x})*\frac{d}{dx}\left[f'(\text{x})\right] \ \ \text{ ... product rule}\\\\h''(\text{x}) = 2f'(\text{x})*f'(\text{x})+2*f(\text{x})*f''(\text{x})\\\\h''(\text{x}) = 2\left(f'(\text{x})\right)^2+2*f(\text{x})*f''(\text{x})\\\\[/tex]
The second derivative is useful to determine where the function is concave up or concave down. And also to determine points of inflection.
The h''(x) function involves f''(x), so we'll need to find the second derivative of the f(x) function.
[tex]f'(\text{x}) = \cos(\pi \text{x}) + \text{x}^5 + 6\\\\f''(\text{x}) = \frac{d}{dx}\left[\cos(\pi \text{x}) + \text{x}^5 + 6\right]\\\\f''(\text{x}) = -\pi\sin(\pi \text{x}) + 5\text{x}^4\\\\[/tex]
Then plug in x = 0
[tex]f''(\text{x}) = -\pi\sin(\pi \text{x}) + 5\text{x}^4\\\\f''(0) = -\pi\sin(\pi *0) + 5(0)^4\\\\f''(0) = 0\\\\[/tex]
We have enough info to find h''(0) finally.
[tex]h''(\text{x}) = 2\left(f'(\text{x})\right)^2+2*f(\text{x})*f''(\text{x})\\\\h''(0) = 2\left(f'(0)\right)^2+2*f(0)*f''(0)\\\\h''(0) = 2\left(7\right)^2+2*3*0\\\\h''(0) = 98\\\\[/tex]
Side notes:
Refer back to the previous section when we found f'(0) = 7.The h''(0) is positive. It tells us that h(x) is concave up when x = 0.Related Questions
What is the intermediate step in the form (x+a)^2=b as a result of completing the square for the following equation?
x^2+32x+116=12x
Answers
Answer:
x=246 or x=-266
Step-by-step explanation:
First you bring 12x to the other side and simplify to give
x²+20x+116=0
Then solve by completing the square:
(x+10)²-100+116=0
(x+10)²+16=0
(x+10)²=-16
x+10=±256
x=±256-10
x=246 or x=-266
Hope this helps
How many grams of H2SO4 are needed to prepare 5.0 L of a 2 M H2SO4 solution?
Answers
980g of sulphuric acid (H₂SO₄) is needed to prepare 5.0L of a 2M solution.
What is Sulfuric acid?Sulfuric acid, also known as sulphuric acid or oil of vitriol in the past, is a mineral acid made up of the elements hydrogen, oxygen, and sulfur with the chemical formula H₂SO₄. It is a viscous liquid that is miscible with water. It has no color or smell.Sulfuric acid can irritate the nose and throat, burn the eyes, burn holes in the stomach if eaten, cause severe skin burns, and make breathing difficult if inhaled.So, the grams of H₂SO₄ needed will be:
Calculate to prepare 5.0L of a 2M solution.98g/mol × 5mol × 2mol /L = 980g H₂SO₄Therefore, 980g of sulphuric acid (H₂SO₄) is needed to prepare 5.0L of a 2M solution.
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Fractions and Decimals
Write each fraction as a decimal.
4. //
2/5
Answers
2/5 fraction as a decimal is 0.4.
Any number that is expressed as a fraction has two components, the numerator and the denominator. Typically, we divide the numerator by the denominator to convert a fractional number to decimal form. The formula p/q, where q=0, is used to denote fractions. While the decimal numbers, such as 7.575, are created by joining the entire number and fractional parts with a decimal point.
There are two methods to convert fraction to decimal which are given below:
Long division methodBy converting the denominator of the fraction into a power of 10For 2/5
We will divide them as mentioned above,
2/5 = 0.4
Hence, 2/5 is equal to 0.4 in decimal form.
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Malik created a model of a rectangular flag.
His model is 10 inches long and the actual flag is 25 inches long.
How many times wider is the actual flag than Malik's model?
Answers
Answer:
2.5x
Step-by-step explanation:
Given information for width:
Model: 10 in.
Real: 25 in.
Solve: How much times wider is real than model?
To find this, lets divide 25in. by 10in to find the scale factor.
25/10 = 2.5
2.5
We can verify this by multiplying 10 by 2.5
10 x 2.5 = 25
(a) The amount of pure acid in 40 oz of 20% acid solution is oz.
(Type an integer or a decimal.)
Answers
neverminding the jumbled wording
how much is 20% of 40?
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{20\% of 40}}{\left( \cfrac{20}{100} \right)40}\implies 8[/tex]
Si un bolso costaba$40.000 y tiene un descuento de un 10%¿cual es el precio final del bolso?
Answers
El precio final, esto es, el precio con descuento, del bolso es igual a $ 36.000.
¿Cómo calcular el precio final del bolso?
En este problema debemos determinar el precio final de un bolso (C') adquirido por un cliente con descuento, igual al precio original (C) menos el descuento, descrito en la siguiente fórmula:
C' = C · (1 + r / 100)
Donde r es la tasa de descuento. Nótese que existe descuento para 0 < r < 100.
Si sabemos que C = 40.000 y r = - 10, entonces el precio final del bolso es:
C' = 40.000 · (1 - 10 / 100)
C' = 36.000
El precio final del bolso es igual a $ 36.000.
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Caroline invested $900 in an account paying an interest rate of 8\tfrac{7}{8}8 8 7 % compounded continuously. Jaxon invested $900 in an account paying an interest rate of 8\tfrac{5}{8}8 8 5 % compounded daily. After 8 years, how much more money would Caroline have in her account than Jaxon, ?
Answers
Caroline would have $36 more money in her account than Jaxon.
From the question, we have
Caroline amount,
A = P* e^rt
= 900*e^(71*8)/800
= 900*e^(71/100)
=$1830.59
Jaxon amount,
A = P* e^rt
= 900*e^(69*8)/800
= 900*e^(69/100)
=$1794.34
Difference = $1830.59 - $1794.34
= $36.25
= $36 ( to the nearest dollar)
Subtraction:
Subtraction represents the operation of removing objects from a collection. The minus sign signifies subtraction −. For example, there are nine oranges arranged as a stack (as shown in the above figure), out of which four oranges are transferred to a basket, then there will be 9 – 4 oranges left in the stack, i.e. five oranges. Therefore, the difference between 9 and 4 is 5, i.e., 9 − 4 = 5. Subtraction is not only applied to natural numbers but also can be incorporated for different types of numbers.
The letter "-" stands for subtraction. Minuend, subtrahend, and difference are the three numerical components that make up the subtraction operation. A minuend is the first number in a subtraction process and is the number from which we subtract another integer in a subtraction phrase.
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please answer this question. I get my cat taken away if I dont
Answers
Answer: -3
Step-by-step explanation: Used calculator
Help Pls Help I need it in 10 mins!!!!!!!!!!!!!!PLS
The point (6,2) is reflected over the -axis and then the image is translated 4 units right and 3 units down. What is the absolute difference between the coordinates of the final image?
Answers
Answer:
When we reflect over the x-axis the answer will be (6, -2)
Then when we go 4 units right it will turn into (10,-2)
And lastly, when we go 3 units down, it will be (10,-5)
The Answer is (10,-2)
(Brainiest is optional)
Zach loves to compete in triathlons. At a race a few months ago, he finished in 280 minutes. Yesterday, he completed another triathlon and finished in 294 minutes. What was the percent of increase in Zach's finishing time?
Answers
The percent of increase in Zach's finishing time is 5%.
How to calculate the percentage?From the information, at a race a few months ago, he finished in 280 minutes and yesterday, he completed another triathlon and finished in 294 minutes.
The increase in time will be:
= New time - Old time
= 294 - 280
= 14 minutes
The percentage increase will be:
= Increase in time / Original time × 100
= 14 / 280 × 100
= 1/20 × 100
= 5%
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If the triangle equals 12cm in perimeter, what's the equation for the triangle with 5cm,3cm, and a missing number labeled as P
Answers
Hope it helps
Thank you
I need help with the properties of exponents. my work is past due and I'm failing my classes!
Answers
Answer:
answers found in pdf
Step-by-step explanation:
mathaway will help check your answers that what I did.
the mean age for all foothill college students for a recent fall term was 33.2. the population standard deviation has been pretty consistent at 15. suppose that twenty-five winter students were randomly selected. the mean age for the sample was 30.4. we are interested in the true mean age for winter foothill college students. let x
Answers
Lower Limit = 24.52
Upper Limit = 36.28
Error Bound = 5.88
Lower Limit: - The Lower Limit for every class is the smallest value in that class. For example: 5 is a lower lower limit for the set S= { 5,8,42,34}
Upper Limit: - A value that is greater than or equal to every element of a set of data. For example: - { 3,5,11,20,22} is a upper bound of 22.
Given that,
Population Mean,μ = 33.2
sample Mean, χ (x bar) = 30.4
sample Size, n = 25
α (Alpha) = 0.05
Population standard deviation,σ = 15
95% confidence interval:
μ ± z critical σ / √n
Putting the values, we get,
z critical at α0.05 = ± 1.96
30.4 ±1.96( 15/√25) = 30.4 ± 5.88 = (24.52,36.28)
a) Lower Limit = 24.52
b) Upper Limit = 36.28
c) Error Bound =
Margin of Error = z critical × σ /√n
= 1.96 × 15/√25
= 1.96 × 15/5
= 5.88
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50 Points pls help!
Question: Fred drew the footprint of a stage he was planning to build for his band on a coordinate plane. He decided he wanted to make it smaller because he wanted to make sure it fit at every venue.
Please help me solve letter A and B :)
Answers
Answer:
A) Picture added,B) 26 units.=========================
Part AThe smaller figure is added, considering the coordinates of the image follow the rule:
(x, y) → (0.5x, 0.5y)Smaller stage is in blue. See attached
Part BSince each side is half the original, the perimeter of the image is half the larger one:
P = 52/2 = 26 unitsLESSON 8 SESSION 2
Name
Practice Showing That the
Slope of a Line Is Constant
> Study the Example showing that the slope of a
line is constant. Then solve problems 1-5.
Example
Cece draws a graph representing her
family's car trip to the beach. She wants
to show her father that their gas
mileage in miles per gallon was the
same for the entire trip. Cece chose
three points on the line and drew
vertical line segments from the points to
the x-axis. How can she use these
segments to prove that the gas mileage
was constant for the trip?
Distance (mi)
250
150
100
30
0
(S. 155)
(1,31)
8 10
What was the gas mileage for the trip in the Example?
6
Gasoline Used (gal)
The vertical line segments create three
similar right triangles. Because the
triangles are similar, the quotient of the length of the vertical side to the length
of the horizontal side is the same for each triangle. These quotients are slopes
between (0, 0) and the points Cece chose. Because Cece could have chosen any
points on the line to make the similar triangles, the slope is constant for the
whole line. So, the gas mileage is constant for the whole trip
For the graph in the Example, find the slope between the origin and each of the
points Cece chose.
Vocabul
slope
for any two
a line, the
shange
change
of
Answers
The slope of origin (0, 0) with (1, 31); (5, 155), and (9, 279) is 31
The slope of a line:The slope of a line will indicate the direction and steepness. It is also defined as the change in the y-coordinate for a corresponding change in the x-coordinate.
Slope of a line with given points (x₁, y₁) and (x₂, y₂) is
Slope, m = (y₂ -y₁) /(x₂-x₁)Here we have
Points (1, 31) (5, 155) and (9, 279)
We need to find slopes between two points
As we know the formula for slope with given points is
Slope, m = (y₂ -y₁) /(x₂-x₁)
Here coordinates of the origin are (0, 0)
The slope between (0, 0) and (1, 31)
=> m = (31 - 0) /(1 - 0) = 31/1 = 31
The slope between (0, 0) and (5, 155)
=> m = (155 - 0) /(5- 0) = 155/5 = 31
The slope between (0, 0) and (9, 279)
=> m = (279 - 0) /(9 - 0) = 279/9 = 31
Therefore,
The slope of origin (0, 0) with (1, 31); (5, 155), and (9, 279) is 31
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Mitzi looked up prices of thirteen used Chevrolet HHR retro" tracks in the classified ads and found these prices: 8,500, 8,500, 8,500, 9,900, $10,800, 10,800, 11,000, 12,500, $12,500, $13,000, 13,000, $14,500, and $23,000
Answers
The price range of the Chevrolet HHR retro will be $14,500.
What is the range of the data?The range of the data is defined as the difference between the maximum value of data and the minimum value of the data.
The given data as:
$8,500, $8,500, $8,500, $9,900, $10,800, 10,800, 11,000, 12,500, $12,500, $13,000, 13,000, $14,500, and $23,000
According to the given information, the required solution would be as:
The maximum value of data = $23,000
The minimum value of data = $8,500
The range of the data = The maximum value of data - The minimum value of data
Substitute the values in the above formula,
The range of the data = 23,000 - 8,500
Apply the subtraction operation to get the answer
The range of the data = 14,500
Therefore, the price range of the Chevrolet HHR retro will be $14,500.
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The question seems to be incomplete the correct question would be:
Mitzi looked up prices of thirteen used Chevrolet HHR retro" tracks in the classified ads and found these prices: 8,500, 8,500, 8,500, 9,900, $10,800, 10,800, 11,000, 12,500, $12,500, $13,000, 13,000, $14,500, and $23,000. Find the range of the data.
4. Find the measure of
TIL
110
40 50 60
140 130 120 110 100 sp 80 70 60
0 120 130
35 degrees
135 degrees
145 degrees
O95 degrees
Answers
Answer:
145 degrees
Step-by-step explanation:
The side through L is at a 35 degree angle, and the side through N is at a 180 degree angle. Taking the difference, we get 145 degrees.
Which theorem or postulate could be used to prove the congruence of the triangles?
Answers
The reason is that the picture shows 2 angles and a side with the side sandwitched between the angles. You read it as “angle, side, angle”, so the answer is ASA
Angelica purchases $25.75 worth of cosmetics online. There is a 2% processing fee. What is the total cost of the cosmetics? Round to the nearest cent
26.27
Step by step
2/100 = 0.02
0.02 x 25.75 = 0.51
0.51 + 25.75 = 26.26 rounding is 26.27
Answers
The total cost of the cosmetics including 2% processing fee rounded up to the nearest cent is $26.27
The cost of the cosmetics is $25.75
The 2% processing fee is calculated as follows:
2/100 × $25.75 = $ 0.515
The total cost including processing fee is $25.75 + $ 0.515 = $26.265 rounded up to $26.27
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What is the slope-intercept form of the equation of the line that passes through the points (7,-5) and (3,- 9)?
y=x-4
y = x + 12
y=x-12
y = x + 5
Answers
Answer:
y = x - 12
Step-by-step explanation:
slope = (-9 - (-5))/(3 - 7) = -4/(-4) = 1
y = mx + b
y = x + b
-5 = 7 + b
b = -12
y = x - 12
What is the sum of (3x + 4) + (9x – 10)?
Answers
Answer:
6(2x - 1)
Step-by-step explanation:
Just look up the answer on the calculator
Put the following equation of a line into slope-intercept form, simplifying all fractions. 4x−3y= −21
Answers
Answer:
y=4/3x +7
Step-by-step explanation:
4x - 3y = -21
Move 4x to the other side
-3y = -21 -4x
Next divide the other side by -3
y= 7 +4/3x
Move numbers around
y = 4/3x + 7
Evelyn deposited $90 in an account
earning 10% interest compounded
annually.
To the nearest cent, how much interest
will she earn in 1 year?
Use the formula B = p(1 + r)², where B is
the balance (final amount), p is the
principal (starting amount), r is the
interest rate expressed as a decimal, and t
is the time in years.
Answers
The compound interest earned by Evelyn in one year is $9.
Evelyn has a bank account. She deposited $90 in her account. She earns 10% interest on her deposited money. The interest earned is compounded annually. We need to calculate the interest earned by her in one year, to the nearest cent.
We need to calculate the compound interest. The formula is given below.
B = p(1 + r)^t
The variables 'B', 'p', 't', and 'r' represent the final amount, the initial amount, the time duration, and the rate of interest, respectively.
B = $90(1 + 0.1)^1
B = $90*1.1
B = $99
The final amount in her bank account after one year is $99. The interest earned is the difference between the final and the initial amount. Let the variable 'I' represent the interest earned.
I = B - p
I = $99 - $90
I = $9
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Which option shows the equation for the line in the graph?
Responses
y=−5x−5y is equal to negative 5 x minus 5
y=3x−10y is equal to 3 x minus 10
y=−3x−10y is equal to negative 3 x minus 10
y=−10x−3
Answers
The option that shows the equation for the line in the graph is
y=−3x−10y is equal to negative 3 x minus 10How to obtain the equation of the line in the graphA graph is a pictorial representation of data. The graph shows a linear function which will be written in the slope intercept form form as
y = mx + c
where
m = slope
c = intercept
x = input variables
y = output variables
calculation of slope
The points used for the slope is gotten from the graph as B(-5. 5) (-2, -4)
m = (y₂ - y₁) / (x₂ - x₁)
substituting the values
= ( -4 - 5 ) / ( -2 - -5 )
= -9 / -2 + 5
= -9 / 3
= -3
y intercept, c
this is a point where the graph cuts the y axis
c = -10
hence the equation is written using y = mx + c and substituting the values
y = -3x - 10
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how is random error different from systematic error? provide an example that illustrates this difference.
Answers
Systematic errors are consistently in the same direction
In contrast , random errors produce different values in random directions
Random error causes a slight difference of one measurement to from the next. It is quite unpredictable in it's variation during an experiment.
Systematic error always affects measurements by the same proportion, provided that a reading is taken the same way each time. It is predictable
An example that illustrates this difference are
Systematic errors are consistently in the same direction
for example they are always 50g , 1% or 99 mm too large or too small
In contrast , random errors produce different values in random directions
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What’s the answer someone please help
Answers
Answer:
Step-by-step explanation:
Ok:
1. This is a percent problem!
45/100*15=
675/100=
6.75
2. Subtract:
15-6.75=
8.25$
Answer: 8.25$
Lamar will run at most 37 miles this week. So far, he has run 22 miles. What are the possible numbers of additional miles he will run?
Answers
The possible numbers of additional miles that Lamar can run are all the solutions of the inequality:
0 ≤ x ≤ 15
What are the possible numbers of additional miles?We know that Lamar will run at most 37 miles this week, and he already did run 22 miles, then the (maximum) distance left this week is:
D = 37mi - 22mi = 15mi
So he can run at most 15 miles, this means that he can run any distance between 0 miles and 15 miles by the end of the week.
So, if we define x as the distance that he will run on the week (in miles), then we can write the inequality:
0 ≤ x ≤ 15
so the possible numbers of additional miles he will run on this week are all the values of x that are a solution for the above inequality.
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helpppppppppppppppppppppppppp asapppppppppppppppppppppppppppppppp
Answers
ANSWER THIS QUESTION FOR BRAINLISTEST
Answers
The transformation rule used to transform Triangle PQR to P'Q'R' is; (x, y) → (⁵/₂x, ⁵/₂y)
How to Interpret Mathematical Transformations?We are given from the question that the triangle P'Q'R' was transformed from the triangle PQR.
We see from the image that the coordinates of the original triangle PQR are;
P(-1, 1)
Q(2, 0)
R(-1, -2)
We also see that the coordinates of the transformed triangle P'Q'R' are;
P(-5/2, 5/2)
Q(5, 0)
R(-5/2, -5)
Thus, we can conclude that the transformation rule used here is;
(x, y) → (⁵/₂x, ⁵/₂y)
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helppppppppppppppppppppppppppppppppppp pls can you show me how to write out each problem and solve it
Answers
Answer:
1) 1.96
2) 6
3) 4/3
Step-by-step explanation:
think of exponents as repeated multiplication. the exponent is how many times you're going to multiply the number itself by.
1) [tex]1.4^{2}[/tex] is really just 1.4 * 1.4, which you can solve by long multiplication or a calculator to get 1.96
2) for this one, you can rewrite the expression as a fraction. you'll also want to use PEMDAS, starting with the parentheses
[tex]\frac{300}{ 2(0.5+4.5)^{2}}\\\\\frac{300}{2*(5)^2}\\\\\frac{300}{2*25} \\\\\frac{300}{50} = 6[/tex]
3) [tex](\frac{1}{2})^{3} = \frac{1}{2} *\frac{1}{2} *\frac{1}{2} = \frac{1}{8} \\[/tex]
you can rewrite the division as a fraction again, only this time. you can change the subtraction to the reciprocal of the denominator
[tex]\frac{1}{3} / (2*\frac{1}{8} )= \frac{1}{3} / \frac{1}{4} = \frac{1}{3} * 4 = \frac{4}{3}[/tex]
hope this helped!