Answer:
The answers agree.
[tex]f^{\prime}(x) = - 20x^{3} + 15x^{2} - 2x + 1[/tex]
Step-by-step explanation:
The product rule is:
[tex]f(x) = g(x)h(x)[/tex]
[tex]f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)[/tex]
In this problem, we have that:
[tex]f(x) = (1 + 5x^{2})(x - x^{2})[/tex]
So
[tex]g(x) = 1 + 5x^{2}[/tex]
[tex]g^{\prime}(x) = 10x[/tex]
[tex]h(x) = x - x^{2}[/tex]
[tex]h^{\prime}(x) = 1 - 2x[/tex].
[tex]f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)[/tex]
[tex]f^{\prime}(x) = (10x)*(x - x^{2}) + (1 + 5x^{2})(1-2x)[/tex]
[tex]f^{\prime}(x) = 10x^{2} - 10x^{3} + 1 - 2x + 5x^{2} - 10x^{3}[/tex]
[tex]f^{\prime}(x) = - 20x^{3} + 15x^{2} - 2x + 1[/tex]
Multiplication first:
[tex]f(x) = (1 + 5x^{2})(x - x^{2})[/tex]
[tex]f(x) = x - x^{2} + 5x^{3} - 5x^{4}[/tex]
[tex]f(x) = -5x^{4} + 5x^{3} - x^{2} + x[/tex]
[tex]f^{\prime}(x) = -20x^{3} + 15x^{2} - 2x + 1[/tex]
Comparing the results, we can see that the derivatives obtained using the two methods are indeed the same. The answer is yes.
What's the function about?The derivative of f(x) = (1 + 5x2)(x − x2) can be found using the product rule and by performing the multiplication first.
Using the product rule:
The product rule states that the derivative of f(x)g(x) is f′(x)g(x)+f(x)g′(x).
In this case, f(x) = 1 + 5x2 and g(x) = x − x2. So the derivative of f(x)g(x) is:
f′(x)g(x)+f(x)g′(x)
=(10x+10x3)(x−x2)+(1+5x2)(2x−2x3)
=10x2−10x3+10x3−10x4+2x−2x3+1+10x2−10x3
=−10x4+12x2+2x
Performing the multiplication first:
We can first multiply out the expression (1 + 5x2)(x − x2) to get:
x2−5x4+x3−5x3+x−x2
=−5x4+12x2+x
Then, we can take the derivative of this expression to get the same derivative as we found using the product rule:
f′(x)=−10x4+12x2+2x
Therefore, the derivative of f(x) = (1 + 5x2)(x − x2) is -10x4+12x2+2x, regardless of whether we use the product rule or perform the multiplication first.
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The interior angles formed by the sides of a hexagon have measures that sum to 720 degrees. What is the measure of angle F?
-2(bx - 5) = 16
What’s does b and x equal?
Z = 6.8 e j 2.57 + 5.5 e j 2.64 put z in the form z = r ∠ θ , where r > 0 , θ is in degrees, and − 180 ∘ < θ ≤ 180 ∘ .
Use the normal distribution of fish lengths for which the mean is 1111 inches and the standard deviation is 44 inches. assume the variable x is normally distributed. left parenthesis a right parenthesis(a) what percent of the fish are longer than 1212 inches? left parenthesis b right parenthesis(b) if 300300 fish are randomly selected, about how many would you expect to be shorter than 77 inches?
How is the graph of y=^3√0.5x related to its parent function,y=^3√x ? A.It is horizontally stretched by a factor of 0.5 B. It is horizontally compressed by a factor of 0.5 C. It is translated left by 0.5 units D. It is translated right by 0.5 units
If 555 is divided by 2, what is the remainder? A) 1 B) 2 C) 3 D) 5
Step-by-step explanation:
If 555 is divided by 2 then the quotient is 277 and remainder is 1.
HELP ASAP How do I find the answer to this
PLEASE HELP ME ILL GIVE U A BRAINLIST a man who is 6 feet tall cats a shadow that is 15 feet long.A tree casts a similar shadow is 22 FT how tall is the tree?
20 points Easy question
what is 5/20 simplify
The circumference of a circle is 28 pi inches. What is the length of the radius of this circle?
Answer:
The length of the radius of this circle is 14 inches.
Step-by-step explanation:
Circumference of the circle = [tex]28\pi[/tex]
Let's take the length of the radius of the circle as [tex]a[/tex]
Circumference of a circle = [tex]2\pi r[/tex]
Where,
r is the radius of the circle.
Therefore,
[tex]2\pi a=28\pi[/tex]
[tex]2a=28[/tex]
[tex]a=14[/tex]
Therefore, the length of the radius of this circle is 14 inches.
A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 199 milligrams with s = 16.5 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs.
To construct a 95% confidence interval, use the formula: Confidence Interval = sample mean ± (critical value * (sample standard deviation / sqrt(sample size))). The confidence interval for the true mean cholesterol content is approximately 190.37 to 207.63 milligrams.
Explanation:To construct a 95% confidence interval for the true mean cholesterol content of all chicken eggs, we can use the formula:
Confidence Interval = sample mean ± (critical value * (sample standard deviation / sqrt(sample size)))
Given that the sample mean is 199 milligrams, the sample standard deviation is 16.5 milligrams, and the sample size is 12, we need to determine the critical value for a 95% confidence level. Since we have a small sample size, we need to use the t-distribution.
Using a t-table or a statistical calculator, we find that the critical value for a 95% confidence level with a sample size of 12 is approximately 2.179.
Substituting the values into the formula, we get:
Confidence Interval = 199 ± (2.179 * (16.5 / sqrt(12)))
Simplifying and calculating, the confidence interval is approximately 190.37 to 207.63 milligrams.
To construct a 95% confidence interval for the mean cholesterol content of chicken eggs, we use the sample data provided, along with the t-distribution for a sample size less than 30, and apply the standard formula for confidence intervals.
Explanation:The question asks to construct a 95% confidence interval for the true mean cholesterol content of chicken eggs based on the given sample data. To calculate this, we first need to use the sample mean (μ), which is 199 milligrams, and the standard deviation (s), which is 16.5 milligrams. Since the sample size is less than 30, we will use the t-distribution to find the critical value. With n=12, the degrees of freedom (df) are 11.
First, look up the critical t-value for df=11 at a 95% confidence level. Let's assume it is approximately 2.201. Next, calculate the standard error (SE) of the mean by dividing the sample standard deviation by the square root of the sample size. That's SE = 16.5 / sqrt(12).
Now, we can construct the confidence interval using the formula: mean ± (t-value * SE). Plugging in the numbers, we get 199 ± (2.201 * SE).
d. A 95 percent confidence interval means that if we took many samples and built a confidence interval from each of them, we'd expect about 95 percent of those intervals to contain the true mean cholesterol level of all such eggs. Because of sample variability, the interval provides a range of values within which we are 95% confident the true mean lies.
What is the total surface area of the cone? 90π cm2 65π cm2 120π cm2 115π cm2
The total surface area of a cone can be found using the formula A = πr(r + l), where A is the total surface area, r is the radius of the base, and l is the slant height. In this case, the correct answer is 115π cm².
Explanation:The total surface area of a cone can be found using the formula A = πr(r + l), where A is the total surface area, r is the radius of the base, and l is the slant height.
In this case, the correct answer is 115π cm². Let's calculate it:
Let's assume the radius of the cone is r.And let's assume the slant height of the cone is s.The total surface area of the cone is A = πr(r + s).We can substitute the given values into the formula: A = π(4)(4 + 7) = 115π cm².Therefore, the correct answer is 115π cm².
corey has 42 feet of fencing around his garden. The garden is rectangular in shape, and its length is equal to twice the width minus 3 feet. Define the variables, and write a system of equations to find the length and width of the garden
The variables as length are L and width is W and the system of equations is L = 2W - 3 and 42 = 2(L + W) and the length and width are 13 feet and 8 feet respectively.
What is a rectangle?A rectangle is a geometrical figure in which opposite sides are equal.
The angle between any two consecutive sides will be 90 degrees.
Area of rectangle = length × width.
As per the given total fencing = 42 feet
Let's say the length is L and the width is W.
As per the given,
L = 2W - 3
Perimeter = 2(length + width)
42 = 2(L + W)
21 = 2W - 3 + W
21 + 3 = 3W
W = 8 feet
And, L = 2(8) - 3 = 13 feet
Hence "The variables as lengths are L and width is W and the system of equations is L = 2W - 3 and 42 = 2(L + W) and the length and width are 13 feet and 8 feet respectively".
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Easy 20 points!
Which expression finds the length of side y in this right triangle?
The Last one is your answer
Answer:
The last one i did this before
Step-by-step explanation:
The front wall of your shed is in the shape of a trapezoid. The bottom measures 22 feet, the top measures 20 feet, and the height is 10 feet. Calculate the area of the front wall of your shed.
Answer:
the answer on edge is D. 66^2
Step-by-step explanation:
The mean output of a certain type of amplifier is 117 watts with a variance of 100. if 42 amplifiers are sampled, what is the probability that the mean of the sample would differ from the population mean by less than 3.3 watts? round your answer to four decimal places.
Solve the equation by factoring. z2 − 6z − 27 = 0
To solve the quadratic equation z² -6z - 27 = 0, factor it to get (z - 9)(z + 3) = 0. Using the Zero Product Property, we find that the solutions are z = 9 and z = -3.
Solving Quadratic Equations by Factoring
To solve the quadratic equation z ²
- 6z - 27 = 0 by factoring, we must find two numbers that add up to -6 and
multiply to -27. After identifying these numbers, we can factor the quadratic expression
into a product of two binomials and then apply the Zero Product Property to find the
values of z.
The two numbers that meet these criteria are -9 and 3, as (-9)*(3) = -27 and (-9)+(3) =
-6. Therefore, we can express the quadratic equation as:
(z - 9)(z + 3) = 0
By the Zero Product Property, we find the solutions to the equation by setting each
binomial equal to zero:
z - 9 = 0
z = 9
z + 3 = 0
z = -3
Thus, the solutions to the quadratic equation are z = 9 and z = -3. These are the z-values that satisfy the original equation.
5:36, 2:9, 3:18 , 1:3 which ratio is the largest?
What is 7/10 expressed as a decimal? Enter your answer in the box.
the length of johns patio is 5 feet less than 3 times the length of howards patio .johns patio has a length of 40 feet . what is the length of howards patio
Type the correct answer in each blank. Use numerals instead of words. If necessary, use / for the fraction bar(s). John hosts an art workshop on the weekends. He has an average of 14 students in each session and charges a fee of $12 per session. He estimates that for every $2 increases in the fee, the average number of students reduces by 1. Complete the equation that models this scenario, where c(x) is the revenue generated and x is the number of $2 fee increases.
c(x) =_x^2 +_x+_
Answer:
c(x) = -2x^2+16x+168
Step-by-step explanation:
just did it on edmentum
Which expression is equivalent to 10x^6y^12/-5x^-2y^-6 ? Assume . –50x8y18 –2x8y18 –2x12y72 5x8y18
Answer:
Option B is correct, i.e. -2 x^8 y^18.
Step-by-step explanation:
To find the equivalent expression to (10)(x^6)(y^12) / (-5)(x^-2)(y^-6)
Step 1: simplify numerical terms
(10) / (-5) = -2
Step 2: simplify exponents of x-terms
(x^6) / (x^-2) = (x^6) * (x^2) = x^(6+2) = x^8
Step 3: simplify exponents of y-terms
(y^12) / (y^-6) = (y^12) * (y^6) = y^(12+6) = y^18
It means (10)(x^6)(y^12) / (-5)(x^-2)(y^-6) = -2 x^8 y^18
Hence, option B is correct, i.e. -2 x^8 y^18.
ralph is a electrician he charges an initial fee of 24$ , plus 24 per hour. If ralph earned 144$ on the job , how long did the job take.
10,456 to the nearest tenth
6 + 12 ÷ 3 – 2 × 2 =
Haseem was given the following enlargement. What is the perimeter, in centimeters, of the enlarged triangle?
A line contains points M(1, 3) and N(5, 0). What is the slope of MN?
The slope of the line that points M and N lies on is [tex]-\frac{3}{4}[/tex]
Given:
[tex]M(1, 3)\\N(5, 0)[/tex]
Formula for slope of a line is:
[tex]slope (m) = \frac{y_2 - y_1}{x_2 - x_1}[/tex]
Let,
[tex]M(1, 3) = (x_1, y_1)\\N(5, 0) = (x_2, y_2)[/tex]
Substitute
[tex]Slope(m) = \frac{0-3}{5-1} \\= \frac{-3}{4} \\= -\frac{3}{4}[/tex]
Therefore, the slope of the line that contains M(1, 3) and N(5, 0) is: [tex]-\frac{3}{4}[/tex]
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Can someone help me with this question, please?
Hey can you please help me posted picture of question
Eight students were absent the following number of days in a year: 4,8,0,1,7,2,6,and 3. Decide if the range or interquartile range better describes the data set, and explain your reasoning.