Final answer:
The notation f'(x) represents the derivative of the function f with respect to x, signifying the rate of change of the function or the slope of the tangent to its curve at a specific point.
Explanation:
The notation f prime of x, often written as f'(x), represents the derivative of the function f with respect to x. This concept is a fundamental part of calculus, which deals with the rate at which quantities change. The derivative f'(x) is the instant rate of change of the function f(x) at a particular value of x, or the slope of the tangent line to the curve f(x) at that point.
For example, if f(x) = x2, then the derivative f'(x) would equal 2x, as the rate of change of x2 with respect to x is 2x. This means that for every unit increase in x, the value of x2 increases by an amount corresponding to 2x.
The derivative can also be understood in the context of physics, where it might represent things like velocity, which is the derivative of position with respect to time. Regardless of the context, f'(x) is about understanding how a function changes at a specific point.
A rectangular athletic field is twice as long as it is wide. if the perimeter of the athletic field is 192 yards, what are its dimensions
Find two consecutive even integers such that five times the smaller integer is ten more than three times the larger integer.
The problem can be solved by setting up and solving an algebraic equation based on the given conditions. With 'x' representing the smaller integer, and 'x + 2' for the next consecutive even integer, 'x' can be found to identify these integers.
Explanation:To find the two consecutive even integers specified in the question, we should first assign variables to these numbers. Let's say the smaller even integer is 'x', then the next consecutive even integer will be 'x+2' (because even numbers increment by 2).
According to the problem, five times the smaller integer equals ten more than three times the larger integer. We can translate this into a mathematical equation: 5*x = 3*(x+2) + 10.
By simplifying this equation through multiplication and subtraction, we can make 'x' the subject to find the value of the smaller even integer. The larger integer is then x + 2. Using these formulas, we can identify the two consecutive integers that meet the criteria specified in the problem.
Learn more about Even Integers here:https://brainly.com/question/490943
#SPJ2
Choose what the expressions below best represent within the context of the word problem. The tens digit of a number is twice the ones digit. The sum of the digits in the number is 12. What is the number? x represents digit 2x represents digit
Answer:
I see people ask this everywhere, THEY ARE NOT ASKING YOU TO SOLVE IT. It's just a matter of whether what digit does x represent and what digit does 2x represent.
Step-by-step explanation:
The answer to your question is;
x represents the ones digit; and;
2x represents the tens digit.
Mr. Plum's math class of 25 students had an average of 85 on a test. Miss Scarlett's class of 22 students had an average of 87 on the same test. What is the average of the two classes combined?
A mean is an arithmetic average of a set of observations. The average of the two classes combined is 85.936.
What is Mean?A mean is an arithmetic average of a set of observations. it is given by the formula,
Mean = (Sum of observations)/Number of observations
Mr. Plum's math class of 25 students had an average of 85 on a test.
Average = Total marks /Total number of students
Total marks of Mr. Plum's class = Total number of students × Average
Total marks of Mr. Plum's class = 25 × 85
Total marks of Mr. Plum's class = 2,125
Miss Scarlett's class of 22 students had an average of 87 on a test.
Average = Total marks /Total number of students
Total marks of Miss Scarlett's class = Total number of students × Average
Total marks of Miss Scarlett's class = 22 × 87
Total marks of Miss Scarlett's class = 1,914
Further, the average of the two classes combined is,
Average = Total marks / Total students
= (2,125 + 1914)/(25+22)
= 4039/47
= 85.936
Hence, the average of the two classes combined is 85.936.
Learn more about Mean here:
https://brainly.com/question/16967035
#SPJ2
The half-life of a substance is how long it takes for half of the substance to decay or become harmless (for certain radioactive materials). The half-life of a substance is 8.6 days and there is an amount equal to 15 grams now. What is the expression for the amount A(t) that remains after t days, and what is the amount of the substance remaining (rounded to the nearest tenth) after 37 days? Hint: The exponential equation for half-life is A(t) = A0(0.5)t/H, where A(t) is the final amount remaining, A0 is the initial amount, t is time, and H is the half-life.
Final answer:
The amount of a substance remaining after a period of time can be calculated using the half-life formula A(t) = [tex]\frac{A0(0.5)t}{H}[/tex]. After 37 days, with a half-life of 8.6 days starting with 15 grams, approximately 2.2 grams will remain.
Explanation:
The half-life of a substance is the amount of time it takes for half of the substance to decay. For a substance with a given initial amount, the expression for the amount A(t) that remains after t days is A(t) = [tex]\frac{A0(0.5)t}{H}[/tex], where A0 is the initial amount, t is the time in days, and H is the half-life.
Given that the half-life H is 8.6 days and the introductory sum A0 is 15 grams, able to utilize this condition to discover the remaining sum after a certain number of days. To decide the sum of the substance remaining after 37 days, we substitute these values into the equation: A(37) = [tex]\frac{15(0.5)37}{8.6}[/tex]
Calculating this expression, we get that approximately 2.2 grams remain after 37 days, rounded to the nearest tenth.
Graph Jkl and its image after a reflection in the given line J(5,3), K(1,-2), l(-3,4); y-axis
When a shape is reflected, it must be reflected across a line.
See attachment for the graphs of JKL and the image of JKL
The coordinates of JKL are given as:
[tex]J = (5,3)[/tex]
[tex]K = (1,-2)[/tex]
[tex]L = (-3,4)[/tex]
The rule of reflection across the y-axis is:
[tex](x,y) \to (-x,y)[/tex]
So, we have:
[tex]J' =(-5,3)[/tex]
[tex]K' =(-1,-2)[/tex]
[tex]L' = (3,4)[/tex]
See attachment for the graphs of JKL and the image of JKL
Read more about reflections at:
https://brainly.com/question/938117
The first aircraft has 75 more seats than the second aircraft. The third aircraft has 49 fewer seats than the second aircraft. If their total number of seats is 401, find the number of seats for each aircraft.
1st = 75+x
2nd = x
3rd = x-49
401 = 75 +x+ x+ +x-49
401 = 75+3x-49
401 = 3x+26
375=3x
x=375/3 = 125
1st = 125+75 = 200
2nd = 125
3rd = 125-49 = 76
200 + 125 +76 = 401
Joe has $1,800 from his summer job to invest. If Joe wants to have $2,340 altogether and invests the money at 5% simple interest, in how many years will Joe have $2,340?
A sprinkler rotates through an angle of 115° spraying water outward for a distance of 12 ft. find the exact area watered, then round the result to the nearest tenth of a square foot.
Helppppppppppppp!!!!!
When the towels from a hotel were divided evenly among 5 laundry baskets, each basket contained 18 towels. Which equation, when solved, will show how many towels there were in all? t ÷ 5 = 18 5 ÷ t = 18 90 ÷ t = 18 18 – t = 5
How you will go about preparing yourself to confidently teach numeracy?
In preparation of yourself in having to teach numeracy, it is best to substantiate if learning the numeracy will have benefits or advantages, such as having to help other people or even empower them as an adult people learning and think of ways how will this could be important to them.
Factor the expression. k2 + kf – 2f2
Answer:
Factorized form of the given expression is (k + 2f)(k - f).
Step-by-step explanation:
The given expression is k² + kf - 2f² and we have to factorize the given expression.
k² + kf - 2f² = k² + 2kf - kf - 2f²
= k(k + 2f) - f(k + 2f)
= (k + 2f)(k - f)
So the factorized form of the expression is (k + 2f)(k - f).
Which term describes the set of all possible outcomes for a probability event?
Find the zeros with multiplicity for the function p(x) = (x^3 – 8)(x^5 – 4x^3)
Please show steps
Solve the following system of linear equations. 3x + 2y = 10 2x + 3y = 15/2 No solution y = (-3/2)x + 5 x = 3, y = -1/2 x = 3, y = 1/2
The solution of the equations is the point(3,1/2)
What is the solution to a linear equation?The solution of a linear equation is defined as the points, in which the lines represent the intersection of two linear equations. In other words, the solution set of the system of linear equations is the set of all possible values to the variables that satisfies the given linear equation.
Given here: 3x + 2y = 10 2x + 3y = 15/2 Solving the two equations we get
the solution as x=3 and y=1/2
Hence, the solution of the equations is the point(3,1/2)
Learn more about the solution to a linear equation here:
https://brainly.com/question/545403
#SPJ2
Final answer:
Using the elimination method, we solved the system of linear equations to find that the solution is x = 3 and y = 1/2.
Explanation:
The student is asking us to solve a system of linear equations. The equations given are:
3x + 2y = 10
2x + 3y = 15/2
To solve these equations, we can use either substitution or elimination method. Let's go ahead with the elimination method to find the values of x and y.
Multiply the first equation by 3 and the second equation by 2 to make the coefficients of x the same:
9x + 6y = 30 (multiplying the first equation by 3)
4x + 6y = 15 (multiplying the second equation by 2)
Now, subtract the second equation from the first equation to eliminate y:
9x + 6y - (4x + 6y) = 30 - 15
5x = 15
x = 3
Now plug x=3 into the first original equation:
3(3) + 2y = 10
9 + 2y = 10
2y = 1
y = 1/2
Therefore, the solution to the system of equations is x = 3 and y = 1/2.
Which of the following inequalities contains points only in the first and second quadrants?
y > 4x2 − 1y ≤ 2x2 + 1y > −2x2 + 3y > 2x2 − 3x + 2
If h(x) = 5 for the function h(x) = 2x + 1, what is the value of x?
Which ordered pair is a solution to the system of inequalities?
A. (0,4)
B. (-2, -3)
C. (1, 4)
D. (-3, 5)
How many license plates can be made using 2 digits then 5 letters if repeated digits and letters are not allowed?
Simplify
Y+ z -2y. If y=4 and z=2
Expand the following using either the Binomial Theorem or Pascal’s Triangle. You must show your work for credit. (x - 5)5
Answer:
The expansion is [tex](x-5)^5= x^5-25x^4+250x^3-1250x^2+3125 x-3125[/tex]
Step-by-step explanation:
Given : Expression [tex](x-5)^5[/tex]
To find : Expand the following using either the Binomial Theorem or Pascal’s Triangle?
Solution :
Applying binomial theorem,
Defined as [tex](p+q)^n=\sum_{r=0}^{n} ^nC_rp^{n-r} q^r[/tex]
Where, [tex]^nC_r=\frac{n!}{(n-r)!r!}[/tex]
Now, expanding [tex](x-5)^5[/tex] by binomial formula,
We have,
[tex](x-5)^5=\sum_{r=0}^{5} ^5C_rx^{5-r} (-5)^r[/tex]
Open the summation,
[tex]\Rightarrow ^5C_0 x^{5-0} (-5)^0 + ^5C_1 x^{5-1} (-5)^1 + ^5C_2 x^{5-2} (-5)^2 + ^5C_3 x^{5-3} (-5)^3 \\+ ^5C_4x^{5-4} (-5)^4 + ^5C_5x^{5-5} (-5)^5[/tex]
[tex]\Rightarrow (1)(x^5)(1)+(5)(x^4)(-5)+(10)(x^3)(25)+(10)(x^2)(-125)+(5)(x^1)(625)+(1)(1)(-3125)[/tex]
[tex]\Rightarrow x^5-25x^4+250x^3-1250x^2+3125 x-3125[/tex]
Therefore, The expansion is [tex](x-5)^5= x^5-25x^4+250x^3-1250x^2+3125 x-3125[/tex]
A person 6 ft tall casts a shadow 5ft long. at the same time, a nearby tree casts a shadow 31 ft long. find the height of the tree
Therefore, the tree's height is 37.2 feet.
We can use proportions to find the height of the tree based on the shadow it casts and compare it to the height and shadow of a known object (in this case, a person). Given that a person 6 ft tall casts a 5 ft long shadow, we can set up a proportion with the tree's 31 ft long shadow to find its height. The formula we use for this comparison is:
Person's Height / Person's Shadow = Tree's Height / Tree's Shadow
Plugging in the numbers:
6 ft / 5 ft = Tree's Height / 31 ft
Now, we solve for the Tree's Height:
6 ft * (31 ft / 5 ft) = Tree's Height
Tree's Height = 37.2 ft
Therefore, the height of the tree is 37.2 feet.
Amy is eight years older than twice her cousin Alicia’s age. The sum of their ages is less than 32. Let x represent Alicia's age. Which inequality represents Alicia’s possible age?
To find Alicia's possible age, we can write an inequality based on the given information, solve it, and express Alicia's age in terms of a variable.
Explanation:To find the inequality that represents Alicia's possible age, we can start by representing Amy's age in terms of Alicia's age. Let x represent Alicia's age. Amy is eight years older than twice Alicia's age, so Amy's age is 2x + 8. The sum of their ages is less than 32, so we can write the inequality x + (2x + 8) < 32 to represent their ages. Simplifying the inequality gives us 3x + 8 < 32. We can now solve for x by subtracting 8 from both sides of the inequality: 3x < 24. Divide both sides of the inequality by 3 to isolate x: x < 8. Therefore, Alicia's possible age is less than 8, and the inequality that represents her age is x < 8.
Learn more about Inequality for Alicia's age here:https://brainly.com/question/4935653
#SPJ12
Nine less than the product of ten and a number d is equal to eleven.
For a sample with m = 20 and s = 4, a score of x = 17 would be considered an extremely low score.
a. True
b. False
Point E is drawn on the graph so that the line EF is parallel to line CD
Answer:
-8
Step-by-step explanation:
edg 2020
if last years sales were $200,000 and this years increase was 250% how much are this years sales
Answer:
$700000
Step-by-step explanation:
We are given that last year sales=$200,000
This year sales increases=250%
We have to find the earn money from this year sales
Let x be the earn money in this year
According to question
[tex]\frac{250}{100}\times 200000+200000=x[/tex]
[tex]x=500000+200000=700000[/tex]
Hence, this year sales=$700000
Find the area of the triangle
area of triangle =1/2 * b *h
b=7
h=4.6
area = 1/2 *7*4.6 = 16.1 square mi
I need an answer ASAP!
An experiment is broken up into two parts. In the first part of the experiment a penny is tossed in the air. If the coin lands on heads, then the coin is flipped a second time. If the coin lands on tails, then a six-sided die is rolled. What is the probability of getting exactly one head?