Find the exact value of the following limit: the limit as x goes to 0 of the quotient of the quantity e raised to the 3 times x power minus 3 times x minus 1 and x squared.
Type answer as a decimal.
The value of the limit is 4.5 .
What is limit?An output value that a function approaches for the specified input values is referred to as a limit.
Calculus and mathematical analysis depend on limits, which are also used to determine integrals, derivatives, and continuity.
The given limit is,
F(x) = [tex]lim_{x- > 0} (e^{3x}-3x-1)/x^2[/tex]
Differentiate with respect to x,
F'(x) = [tex]lim_{x- > 0} d/{dx}(e^{3x}-3x-1)/x^2[/tex]
= [tex]lim_{x- > 0} d/{dx}(e^{3x}-3x-1)/d/dx(x^2)[/tex]
= [tex]lim_{x- > 0}\ 3e^{3x}-3/2x[/tex]
Differentiate again with respect to x,
F''(x) = [tex]lim_{x- > 0}\ d/dx(3e^{3x}-3/2x)[/tex]
= [tex]lim_{x- > 0} (9e^{3x)}/2}[/tex]
= [tex]9 .e^0/2[/tex]
= [tex]9/2[/tex]
= [tex]4.5[/tex]
The limit of the function is 4.5 .
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How much water can be held by a cylindrical tank with a radius of 12 feet and a height of 30 feet?
Line LJ is a diameter of circle K. If tangents to circle K are constructed through points L and J, what relationship would exist between the two tangents? Explain.
What is the prime factorization of -96? tell me how do u get the answer
Answer: -1 x 2^5 x 3
Step-by-step explanation: To find the prime factorization of -96, we need to first factor out -1, which gives us 1 x (-1) x 2 x 2 x 2 x 2 x 2 x 3. Then, we can rewrite -1 as -1^1, and combine the 2's to get 2^5. So the prime factorization of -96 is -1^1 x 2^5 x 3.
Remember, negative numbers can also have prime factorizations.
URGENT!!!!!!!!!!11!! Plz HElP
A school typically sells 500 yearbooks each year for $50 each. the economics class does a project and discovers that they can sell 100 more yearbooks for every $5 decrease in price.the revenue for the yearbook sales is equal to the number of yaerbooks sold times the price of the yearbook. let x represent the number of $5 decreases in price. if the expression that represents the revenue is written in the form r(x)=(500+ax)(50-bx). find the values of a and b.
5⁄6 · n = 10 (solve for n)
An open box is formed by cutting squares with side lengths of 3 inches from each corner of a square piece of paper. what is a side length of the original paper if the box has a volume of 675 cubic inches?
What are the explicit equation and domain for a geometric sequence with a first term of 3 and a second term of −9?
Answer: [tex]\ a_{n}=3(-3)^{n-1}[/tex]
Step-by-step explanation:
Given: A geometric sequence with its first term [tex]a_1=a=3[/tex]
and second term [tex]a_2=-9[/tex]
We know that the common ratio in of a geometric sequence=[tex]\frac{a_{n}}{a_{n-1}}[/tex]
Thus, common ratio [tex]r=\frac{-9}{3}=-3[/tex]
We know that the explicit rule for geometric sequence is written as
[tex]a_{n}=ar^{n-1}\\\Rightarrow\ a_{n}=3(-3)^{n-1}..[\text{by substituting the values of 'a' and 'r' in it }][/tex]
Thus, the explicit rule for the given geometric sequence is [tex]\ a_{n}=3(-3)^{n-1}[/tex] for every n ,a natural number.
What is the logarithmic function modeled by the following table
x
8
16
32
f(x)
3
4
5
A. F(x)= log_x2
B. F(x)= log_2 x
C. F(x)= 2 log_10 x
D. F(x)= x log_10 2
Log_2 8 = 3 (because 2^3 = 8), log_2 16 = 4 and log_2 32 = 5.
What are Logarithms?A base must be raised to a certain exponent or power, or logarithm, in order to produce a specific number. If bx = n, then x is the logarithm of n to the base b, which is expressed mathematically as x = logb n.
For instance, 23 = 8; hence, 3 is the base-2 logarithm of 8 or 3 = log2 8. In the same way, 2 = log10 100 since 102 = 100. The latter type of logarithms, those with base 10, are known as common or Briggsian logarithms and are denoted by the letter log n.
Logarithms, which were developed in the 17th century to expedite calculations, significantly decreased the amount of time needed to multiply integers with numerous digits.
Therefore, Log_2 8 = 3 (because 2^3 = 8), log_2 16 = 4 and log_2 32 = 5.
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house blend coffee is 50% columbian beans and special blend coffee is 80% columbian beans. how much of each should be used to produce 100kg of a blend that is 68% columbian beans
Let us say that,
x = mass of 50% Columbian beans required
y = mass of 80% Columbian beans required
To solve this problem, we set up two mass balance equations:
Overall mass balance: 100 = x + y
x = 100 – y ---> 1
Coffee mass balance: 0.68 (100) = 0.50 x + 0.80 y
68 = 0.50 x + 0.80 y ---> 2
Combining equations 1 and 2:
68 = 0.50 (100 – y) + 0.80 y
68 = 50 – 0.50 y + 0.80 y
18 = 0.30 y
y = 60 kg
Calculating for x using equation 1:
x = 100 – y
x = 100 – 60
x = 40 kg
Answer:
40 kg of 50% Columbian beans required
60 kg of 80% Columbian beans required
One number is 6 less than another. their sum is 12 find the larger number
y=x-6
x+y=12
x+x-6 =12
2x-6=12
2x=18
x=18/2 =9
x=9
y =9-6 =3
9+3 =12
Larger number is 9
A comic-strip writer churns out a different number of comic strips each day. For 16 days, the writer logged the number of comic strips written each day (sorted low to high): {1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7}. What type of skew can be observed in this distribution?
positive skew
negative skew
zero skew
skew cannot be observed
a culture started with 5000 bacteria. after 2 hours, it grew to 6500 bacteria. Predict how many bacteria will be present after 18 hrs
What are the vertical asymptotes of the function f(x) = the quantity of 2 x plus 8, all over x squared plus 5 x plus 6? x = −3 and x = −2 x = −3 and x = 2 x = 1 and x = −2 x = 1 and x = 2?
Find the area of a square with apothem 9 in. Round to the nearest whole number.
281 in2
305 in2
458 in2
324 in2
Answer:
The area of the square is 324 square inches.
Step-by-step explanation:
The apothem of the square is 9 inches.
The side of the square is twice the length of the apothem.
Hence, the side of the square is given by
[tex]a=2\times 9=18\text{ in}[/tex]
The area of a square is the given by
[tex]A=a^2\\A=18^2\\A=324\text{ in}^2[/tex]
Therefore, the area of the square is 324 square inches.
PLEASE CAN SOMEONE HELP!!!????
Use elimination to solve for x and y:
−2x−y=9
2x−9y=1
My Choices are....
a. (−4,−1)
b. (−1,−4)
c. (5,1)
d. (−1,−7)
The values of x and y after solving both the equations by the elimination method are -4 and -1. So option A is correct
What is an Equation ?An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.
For example, 3x+2y=0.
Types of equation
1. Linear Equation
2. Quadratic Equation
3. Cubic Equation
Given that,
Two linear equations
−2x − y = 9
2x − 9y = 1
by using elimination method,
−2x − y = 9
2x − 9y = 1
0 - 10y = 10
y = -1
2x − 9× -1 = 1
2x = -8
x = -4
Hence, the values are, -4 and -1
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Accuracy is a measure of how close an answer is to the actual or expected value
A ship travels 10 miles from Point A to Point B, makes a turn of 112, and travels 16 miles to Point C. If the ship travels directly from Point C back to Point A, how many miles will it travel on the last leg of the trip (from Point C to Point A)? Round your answer to the nearest tenth of a mile.
If you guess an answer on two multiple choice questions with the options a, b, or c, what is the probability of you guessing the answer to both questions correctly?
Sophie has a hard rubber ball whose circumference measures 13 inches. she wants to store it in a box. what is the number of cubic inches in the volume of the smallest cube- shaped box with integer dimensions that she can use?
The stack on the left is made up of 15 pennies, and the stack on the right is also made up of 15 pennies. If the volume of the stack of pennies on the left is 360 mm3, what is the volume of the stack of pennies on the right in cubic millimeters? Use 3.14 for pi. (Hint: only enter numerals in the answer blank)
Answer:
360 is the answer. Just type in 360 in the answer blank.
Step-by-step explanation:
Explain what needs to happen to the inequality sign when dividing or multiplying by a negative number. a. nothing happens c. change the inequality sign to an equals sign b. flip the inequality sign d. the inequality needs to be graphed on a number line
Answer:
(B) flip the inequality sign.
Step-by-step explanation:
If we consider an inequality such that [tex]-x\leq7[/tex], then if we multiply the inequality with a negative number such as [tex]-1[/tex], then the inequality becomes [tex]x\geq-7[/tex].
Also, if we divide the above inequality [tex]-x\leq7[/tex], by a negative number that is [tex]-1[/tex], then the inequality becomes [tex]x\geq-7[/tex].
Therefore, if we multiply or divide an inequality by a negative number, then it flips the inequality sign.
Hence, option (B) is correct.
Factoring help. Need step by step guidance ....
35n ^ 2 + 22n + 3
Indicate the equation of the given line in standard form. The line that contains the point Q( 1, -2) and is parallel to the line whose equation is y - 4 = 2/3 (x - 3)
From a group of six people, two individuals are to be selected at random. how many possible selections are there
Final answer:
To find the number of possible selections when choosing two individuals from a group of six people at random, use the combinations formula C(6, 2) = 6! / (2!(6-2)!) which equals 15 possible selections.
Explanation:
From a group of six people, selecting two individuals at random involves calculating the number of combinations. The formula for combinations is C(n, k) = n! / (k!(n-k)!), where 'n' represents the total number of items and 'k' represents the number of items to choose.
In this scenario, with n being 6 (the total number of people) and k being 2 (the number of people to select), the calculation would be:
C(6, 2) = 6! / (2!(6-2)!) = (6 * 5) / (2 * 1) = 15
Therefore, there are 15 possible selections when choosing two individuals from a group of six people at random.
Flying against the wind, a jet travels 4400mi in 8 hours. Flying with the wind, the same jet travels 5820mi in 6 hours. What is the rate of the jet in still air and what is the rate of the wind?
In how many different ways can a president vice-president and secretary be elected from a class of 15 students
Why does a bag of chips puff up in an airplane?
What are the domain, range, and asymptote of h(x) = (1.4)x + 5?
A model for a company's revenue is R=-15p^2+300p+12,000, where p is the price in dollars of the company's product. What prize will maximize revenue?