The tangent line to the curve at (0, 2) is y = 2x + 2, and the normal line is y = -1/2x + 2.
To find the equations of the tangent line and normal line to the curve at the given point (0, 2) for the function y = [tex]x^{4} +2e^{x}[/tex], we first need to determine the derivative of the function at that point.
1. Differentiate the function y =[tex]x^{4} +2e^{x}[/tex] with respect to x:
dy/dx = [tex]4x^{3} +2e^{x}[/tex]
2. Evaluate the derivative at the given point (0, 2):
dy/dx |(0, 2) = [tex]4(0)^{3}[/tex] + 2[tex]e^{0}[/tex] = 2
3. The slope of the tangent line at the point (0, 2) is 2.
4. Use the point-slope form to find the equation of the tangent line:
y - 2 = 2(x - 0)
y = 2x + 2
5. The slope of the normal line is the negative reciprocal of 2, which is -1/2.
6. Use the point-slope form to find the equation of the normal line:
y - 2 = (-1/2)(x - 0)
Simplify:
y = -1/2x + 2.
Question- Find the equation of the tangent line and normal line to the curve at a given point y=[tex]x^{4} +2e^{x}[/tex], (0, 2).
78·112=7278·112=72 . how can you tell that her answer is wrong?
Simplify the function f(x)=1/3(81)^3x/4 . Then determine the key aspects of the function. The initial value is . The simplified base is . The domain is . The range is .
1.1 Factoring: x3-81
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0+b3 =
a3+b3
Check : 81 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
1.2 Find roots (zeroes) of : F(x) = x3-81
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is -81.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9 ,27 ,81
Let us test ....
Polynomial Roots Calculator found no rational roots
2.1 Solve : x3-81 = 0
Add 81 to both sides of the equation :
x3 = 81
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:
x = ∛ 81
Can ∛ 81 be simplified ?
Yes! The prime factorization of 81 is
3•3•3•3
To be able to remove something from under the radical, there have to be 3 instances of it (because we are taking a cube i.e. cube root).
∛ 81 = ∛ 3•3•3•3 =
3 • ∛ 3
The equation has one real solution
This solution is x = 3 • ∛3 = 4.3267
Answer:
The initial value is
✔ 1/3
The simplified base is
✔ 27
The domain is
✔ all real numbers
The range is
✔ y > 0
Step-by-step explanation:
In the formula l=prt, what does the p stand for
A house costs $600 more than 12 times the lot on which it was built. If the lot cost x dollars, what did the house cost?
Which expression represents this?
$600 + 12x
$600x + 12
$720x
The expression that represents the situation is:
$ 600+12x
Step-by-step explanation:cost of lot is: $ x
This means that 12 times the lot on which it was built is: $ 12x
Also, it is given that:
A house costs $600 more than 12 times the lot on which it was built.
This means that the total cost of the house is:
600 plus the total cost of the lot on which it was built.
i.e. The expression that represents the total cost of the house is:
$ 600+12x
Find two numbers if their difference is 16 and their ratio is 5:7.
To determine two numbers where their difference is 16, and their ratio is 5:7, we let the smaller number be 5x and the larger one be 7x, solve for x with the equation 7x - 5x = 16, and find that the numbers are 40 and 56.
Explanation:To find two numbers with a difference of 16 and a ratio of 5:7, first let's denote the smaller number as 5x and the larger number as 7x. The difference between the two numbers is 16, so we can write the equation 7x - 5x = 16.
Solving for x gives us 2x = 16 or x = 8. Thus, the two numbers we are looking for are 5x = 5 × 8 = 40 and 7x = 7 × 8 = 56.
Check: The difference between 56 and 40 is indeed 16, and the ratio of 40 to 56 simplifies to 5:7, confirming our solution.
There are 3 brothers Adam Ben and Callum Callum is 3/8 of adams age Ben is 4/5 of Adams age given that Callum is 2 years younger than Ben find Adams age
Jenna hikes 1/2mile every 1/4 hour. how far does she hike in 1 hour?
Jenna hikes 2 miles in 1 hour, as she covers 1/2 mile in each 1/4 hour and there are four 1/4 hours in a full hour.
Jenna hikes 1/2 mile every 1/4 hour. To determine how far she hikes in 1 hour, we need to calculate the distance she can cover in four such intervals since 1 hour is composed of four 1/4 hours.
So, the calculation would be:
1/2 mile in 1/4 hour
1/2 mile x 4 (since there are 4 quarters in a full hour) = 2 miles
Therefore, Jenna hikes 2 miles in 1 hour.
What does the standard error of the estimate tell us?
The random variable x is the number of shots it takes before you make the first free throw in basketball. assuming the probability of success (making a free throw) is constant from trial to trial, what type of distribution does x follow?
Which answer represents a speed, not a velocity?
A. 35m/s east
B. -35 m/s down
C. 35 m/s
D. 35 m/s south
The expression that represents the speed will be 35 m/s. Then the correct option is C.
What is a vector?The quantity which has magnitude, and direction, and follows the law of vector addition is called a vector.
The velocity is the rate of change of the displacement. And it is a vector quantity.
The distance covered by the particle or the body in an hour is called speed. It is a scalar quantity. It is the ratio of distance to time.
In options A, B, and D, then speed depend on the direction which means they are velocity.
The expression that represents the speed will be 35 m/s. Then the correct option is C.
More about the vector link is given below.
https://brainly.com/question/13188123
#SPJ2
What is the solution to the proportion? Round the answer to the nearest tenth if necessary.
52/x = 9.5/3.25
A: x=14.6
B: x=17.8
C: x=157.8
D:x=169
Yeah it's B:17.8 I agree with you
which action illustrates rational behavior
8 1/2% of 4000
I hope u understood that.
select the proper order from least to greatest 2/3 , 7/6 , 1/8 , 9/10
a smoothie recipe calls for 2 1/4 cups of frozen mango. if the recipe serves two, how many cups of frozen mango is needed to serve one?
Answer: The required number of cups of frozen mango is 1.125.
Step-by-step explanation: Given that a smoothie recipe calls for [tex]2\dfrac{1}{4}[/tex] cups of frozen mango and the recipe serves two.
We are to find the number of cups of frozen mango that is needed to serve one.
We will be using the UNITARY method to solve the problem.
We have
The number of cups of frozen mango that needs to serve two is
[tex]2\dfrac{1}{4}=\dfrac{9}{4}=2.25.[/tex]
Therefore, the number of frozen mango that needs to serve one is given by
[tex]\dfrac{2.25}{2}=1.125.[/tex]
Thus, the required number of cups of frozen mango is 1.125.
How to find mean and standard deviation for binomial distribution?
To calculate mean and standard deviation for a binomial distribution, use the formulas μ = np and σ = √npq. Ensure both np and nq are greater than five to use the normal approximation.
Explanation:To find the mean (μ) and standard deviation (σ) for a binomial distribution, you use the formulas μ = np and σ = √npq respectively. Here, 'n' represents the number of trials, 'p' is the probability of success on a single trial, and 'q' is the probability of failure on a single trial, which is calculated as q=1-p.
If you want to determine whether you can use the normal approximation to the binomial, it is essential to ensure that both np and nq are greater than five. This is because the normal approximation is most accurate when the binomial distribution is symmetrical, which typically occurs when both np and nq exceed five.
Your bowl contains 8 red balls, and 9 blue balls, and you draw 4 balls. in how many ways can the selection be made so that at least one of each color is drawn?
In 2156 ways the selection be made so that at least one of each color is drawn?
The question is asking for the number of ways to select balls from a bowl containing 8 red balls and 9 blue balls if 4 balls are drawn and at least one of each color is drawn.
To solve this, we must consider the different combinations of red and blue balls that meet the requirement.
Calculate the total possible outcomes: 17C4 = 2380
Calculate the outcomes where all balls of one color are drawn: 9C4 + 8C4 = 224
Subtract the outcomes where only one color is drawn from the total: 2380 - 224 = 2156 ways
Angles opposite to one another at the intersection of two lines are called _ _ _ _ _ _ _ _ _ _ _ _ _ _ angles.
Answer:
Step-by-step explanation: vertical angel
What is math compass used for ?
What is the median of this set of data: 10, 11, 12, 12, 15, 19, 20, 21, and 22? Express your answer as an integer.
Question 7 options:
12
15
16
19
The volume of a cube depends on the length of its sides. This can be written in function notation as v(s). What is the best interpretation of v(2) = 8?
Los angles is about 385 miles from san francisco. how far apart should the city's be on a map with a scale of 1in=1515mil
A sampling distribution is constructed based on a sample size of 9. if the sampling distribution has a mean of 500 and a standard error of 15, what is the standard deviation of the original comparison population
The standard deviation of the original comparison population is 45.
Explanation:The standard deviation of the original comparison population can be calculated using the formula:
Standard deviation of the sampling distribution = standard error * sqrt(sample size)
Given that the standard error of the sampling distribution is 15 and the sample size is 9, we can substitute these values into the formula:
The standard deviation of the original comparison population can be determined by using the relationship between the standard deviation, standard error, and sample size in a sampling distribution. The formula is:
[tex]\[ \text{Standard deviation of the original comparison population} = \text{Standard error} \times \sqrt{\text{Sample size}} \][/tex]
Given a standard error of 15 and a sample size of 9, substitute these values into the formula:
\[ \text{Standard deviation of the original comparison population} =[tex]15 \times \sqrt{9} = 15 \times 3 = 45 \][/tex]
Therefore, the standard deviation of the original comparison population is 45.
Standard deviation of the original comparison population = 15 * sqrt(9) = 15 * 3 = 45
Therefore, the standard deviation of the original comparison population is 45.
The table gives the numbers of teachers and students in grades 6 to 8 at Earhart Middle School. Grade Teachers Students 6 1 25 7 2 50 8 3 75 Which teacher-to-student ratio maintains the proportional relationship described in the table?
Answer:
Teachers 4 and Students 100, Teachers 5 and Students 125
Step-by-step explanation:
Going by the chart, by each teacher, the students are increasing by 25. The only ones to follow this pattern is 4:100 and 5:25
what's the equation of this line.
12 packs of juice cost $3.00 after a 40% discount. Which equation would allow you to find the original price of a 12-pack
Ralphie has 23 more toys than his little brother Joey. If they were to get all of their toys out and lay them on the floor, there would be 384 toys on the floor.
please help! Algebra 1 math.
Find the ordered pairs for the x- and y-intercepts of the equation 3x − 2y = 18 and select the appropriate option below.
A. The x-intercept is (0, 6); the y-intercept is (–9, 0).
B.The x-intercept is (6, 0); the y-intercept is (0, –9).
C. The x-intercept is (3, 0); the y-intercept is (0, –2).
D.The x-intercept is (0, 3); the y-intercept is (–2, 0).
Farmer Fred has 200 sheep on his farm.He sells 40 of them.What percentage of his sheep does he have left?
If you have 256 data points, how many classes (bins) would sturges' rule suggest
Final answer:
Sturges' rule suggests using 9 classes or bins for 256 data points.
Explanation:
Sturges' rule is a method for determining the number of classes or bins required for a histogram. According to Sturges' rule, the formula to calculate the number of classes is:
K = 1 + log2(n)
Where K is the number of classes and n is the number of data points. In this case, since you have 256 data points, the number of classes suggested by Sturges' rule would be:
K = 1 + log2(256)
K = 1 + log2(28)
K = 1 + 8
K = 9
Therefore, Sturges' rule would suggest using 9 classes or bins for the histogram.