The graph of f ′ (x), the derivative of f of x, is continuous for all x and consists of five line segments as shown below. Given f (0) = 7, find the absolute minimum value of f (x) over the interval [–3, 0].
0
2.5
4.5
11.5
Answers
[tex]f'(x)\ge0[/tex] for all [tex]x[/tex] in [-3, 0], so [tex]f(x)[/tex] is non-decreasing over this interval, and in particular we know right away that its minimum value must occur at [tex]x=-3[/tex].
From the plot, it's clear that on [-3, 0] we have [tex]f'(x)=-x[/tex]. So
[tex]f(x)=\displaystyle\int(-x)\,\mathrm dx=-\dfrac{x^2}2+C[/tex]
for some constant [tex]C[/tex]. Given that [tex]f(0)=7[/tex], we find that
[tex]7=-\dfrac{0^2}2+C\implies C=7[/tex]
so that on [-3, 0] we have
[tex]f(x)=-\dfrac{x^2}2+7[/tex]
and
[tex]f(-3)=\dfrac52=2.5[/tex]
Related Questions
For questions 4-5, how many solutions does the system have?
4. -4y+4=x
2x+8y=8
A. One Solution
B. Two Solutions
C. Infinitely Many Solutions
D. No Solution
5. 6x+2=y
3y-18x=12
A. One Solution
B. Two Solutions
C. Infinitely Many Solutions
D. No Solution
Answers
Answer:
[tex]\large\boxed{Q4.\ C.\ \text{Infinitely Many Solutions}}\\\boxed{Q5.\ D.\ \text{No Solution}}[/tex]
Step-by-step explanation:
[tex]Q4.\\\\\left\{\begin{array}{ccc}-4y+4=x&(1)\\2x+8y=8&(2)\end{array}\right\\\\\text{Substitute (1) to (2):}\\\\2(-4y+4)+8y=8\qquad\text{divide both sides by 2}\\-4y+4+4y=4\qquad\text{combine like terms}\\4=4\qquad\bold{TRUE}\\\\\text{Infinitely Many Solutions}[/tex]
[tex]Q5.\\\\\left\{\begin{array}{ccc}6x+2=y&(1)\\3y-18x=12&(2)\end{array}\right\\\\\text{Substitute (1) to (2):}\\\\3(6x+2)-18x=12\qquad\text{divide both sides by 3}\\6x+2-6x=4\qquad\text{combine like terms}\\2=4\qquad\bold{FALSE}\\\\\text{No Solution}[/tex]
Given the table below for selected values of f(x), use 6 left rectangles to estimate the value of
.
x
1
3
4
6
7
9
10
f(x)
4
8
6
10
10
12
16
Numerical Answers Expected!
Answer for Blank 1:
Answers
Divide the area into 6 portions and find the sum of each one to find the value of the final integral:
f(x)dx = 2(8+4)/2 + 1 (8+6)/2 + 2 (6+10)/2 + 1(10+10)/2 + 2 (10+12)/2 + 1 (12+16)/2
= 12 + 7 + 16 + 10 + 22 + 14 = 81 sq units
Answer:
Step-by-step explanation:
Given is a table of x, f(x) as below
x 1 3 4 6 7 9 10
f(x) 4 8 6 10 10 12 16
Mid pt 6 7 8 10 11 14
width of
interval 2 1 2 1 2 1
dA1 12 7 16 10 22 14
area = area of all rectangles= 81 sq units.
Find the value of x and the value of y.
Answers
This would be D cause these are 30,60, 90 angles. 30 (X) 60(X square root of 3 and 90 2x).
Answer:
C
Step-by-step explanation:
Using the sine ratio on the right triangle on the right side
sin30° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{y}{8}[/tex]
Multiply both sides by 8 × sin30° = y
⇒ y = 8 × [tex]\frac{1}{2}[/tex] = 4 → C
Graph the function. Describe its position relative to the graph of the indicated function
Answers
Answer:
C
Step-by-step explanation:
If the parent function is function [tex]y=f(x)[/tex] and [tex]a>0,[/tex] then
the graph of the function [tex]y=f(x-a)[/tex] is translated a units to the right graph of the parent function; the graph of the function [tex]y=f(x+a)[/tex] is translated a units to the left graph of the parent function;the graph of the function [tex]y=f(x)+a[/tex] is translated a units up graph of the parent function;the graph of the function [tex]y=f(x)-a[/tex] is translated a units down graph of the parent function.In your case, the grapgh of the function [tex]y=2^x+2[/tex] is translated 2 units up the graph of the function [tex]y=2^x.[/tex]
Answer:
To graph the function, we first need to know what means:[tex]f(x)=2^{x} +2[/tex] relative to [tex]f(x)=2^{x}[/tex]
This is called translation of functions, and it refers to the movement of a specific function, either upwards, downwards, leftwards or rightwards. To be able to translate a function, we have to use the next rules.
Moving upwards or downwards.If we sum a number to the whole function, we are gonna move that function upwards. For example, if we have [tex]f(x)=x^{2}[/tex] and sum 3 to it, we would have [tex]f(x)=x^{2} +3[/tex], which is the same function but moved 3 units upwards.If we subtract a number to the whole function, we are gonna move that function downwards. For example, if we have [tex]f(x)=x^{2}[/tex] and subtract 3 from it, we would have [tex]f(x)=x^{2} -3[/tex], which is the same function but moved 3 units downwards.In this case, the problem is saying that we have to move the function upwards by two units, because [tex]f(x)=2^{x} +2[/tex] is the same function as [tex]f(x)=2^{x}[/tex], but moved 2 units upwards.
In addition, the graph is attached, there you can see the difference between those two functions, you will se that they are the same, but one is moved two units upwards.
How would you solve mathematical problem?
Answers
Answer:
[tex]r=\dfrac{I}{Pt}[/tex]
Step-by-step explanation:
Solve this the way you do any "solve for" problem. Locate the variable of interest and identify what has been done to it. Undo that.
Here, the variable of interest is "r". It has been multiplied by Pt. We undo that multiplication by dividing by that product:
[tex]I=Prt\\\\\dfrac{I}{Pt}=\dfrac{Prt}{Pt}\\\\\dfrac{I}{Pt}=r \quad\text{simplify}[/tex]
Is there a way to calculate sin, cos, and tan without a calculator?
If so, could you please do it with this problem...
Tan(90°) = DE/4
Also, could you please explain it step by step.
Thanks!
Answers
Answer:
yes, and no
Step-by-step explanation:
There are a few trig functions that are "simple" and can be memorized. Usually, these are the ones for angles of 0°, 30°, 45°, 60°, 90°. See below for a table.
To solve an equation like this for DE, you multiply both sides of the equation by 4 (cancelling the denominator).
4·tan(90°) = DE
The tangent of 90° is "infinity." So, the length of DE will be infinite for an angle of 90°.
_____
Trig functions of multiple angles and half angles can be derived from those in the table using the appropriate formulas. The math can get messy fast.
Of course, SOH CAH TOA reminds you of the relationships between trig function values and right triangle side measures. You may not know the angle measure (and it may be irrelevant), but you can find the trig function value.
Answer:
See below.
Step-by-step explanation:
You can find them by constructing a right angled triangle with the given angle and measuring the sides giving sin = opposite / hypotenuse , cos = adjacent / hypotenuse and tan = opposite / adjacent ( SOH-CAH-TOA).
For tan 90 degrees you think of an 'imaginary' triangle where opposite side = hypotenuse and the adjacent side = 0 ( in fact a vertical line!) .Now tan x = opposite / adjacent = opposite / 0 which is indeterminate. There is no value for the tan of 90 degrees and so there is no solution to your problem.
You get paid $18 for 3 hours of work. How much money would make if you worked 7 hours?
Answers
Answer:
$42
Step-by-step explanation:
Divide $18 by 3 to get a unit rate of $6 per hour. Then, multiply by 7 hours to get $42.
He gets paid $6 each hour
Courney buys 2bags of apples.Each bag has 20 apples.How many apples does she buy?
Answers
A bag contains 1 blue, 2 green, and 3 red marbles, as shown. What is the probability of drawing a green marble out of the bag without looking?
Answers
ANSWER
[tex] P(G)= \frac{1}{3}[/tex]
EXPLANATION
The number of green marbles in the bag is
[tex]n(G) = 2[/tex]
The total number of marbles in the bag is
[tex]n(S)=1+2+3 = 6[/tex]
The probability of selecting a green marble from the bag without looking is
[tex]P(G)= \frac{n(G)}{n(S)} [/tex]
Substitute the values to get,
[tex]P(G)= \frac{2}{6} [/tex]
[tex]P(G)= \frac{1}{3} [/tex]
Answer:
The probability of drawing a green marble out of the bag without looking = 1/3
Step-by-step explanation:
It is given that, a bag contains 1 blue, 2 green, and 3 red marbles
Therefore total number of marble in the bag = 1 + 2 + 3 = 6
To find the probability
Total number of marble = 6
Number of green marble = 2
The probability of drawing a green marble = 2/6 = 1/3
Third-grade students at washington School read 530 books in march.They read 469 books in April.How many books did they read in the two months?
Answers
Answer:
999
530
+469
----------
999
Can anyone please help me!? Help!
Subject: Expanding and Simplifying Expressions
7th-grade Mathematics!
-------------------------------------------------------------------------------------
Question #1: Write an expression that is equivalent to 2/3 (4x + 9).
A) 8/3x + 6
B) 8/3x + 9
C) 2/3x + 6
D) 2/3x + 9
-------------------------------------------------------------------------------------
Question #2: Which expression is equivalent to 1/3p + 15?
A) 1/3(p + 45)
B) 1/3(p + 5)
C) 3(p + 45)
D) 3(p + 5)
Answers
Answer:
[tex]\large\boxed{Q\#1.\ A)\ \dfrac{8}{3}x+6}\\\boxed{Q\#2.\ A)\ \dfrac{1}{3}(p+45)}[/tex]
Step-by-step explanation:
[tex]Q\#1:\\\\\dfrac{2}{3}(4x+9)\qquad\text{use the distributive property}\\\\=\left(\dfrac{2}{3}\right)(4x)+\left(\dfrac{2}{3}\right)(9)=\dfrac{8}{3}x+(2)(3)=\dfrac{8}{3}x+6\\\\Q\#2:\\\\\dfrac{1}{3}p+15=\dfrac{1}{3}p+\dfrac{1}{3}\cdot(3)(15)=\dfrac{1}{3}p+\dfrac{1}{3}\cdot45=\dfrac{1}{3}(p+45)[/tex]
What is the value of the square root of 15 to the nearest tenth? Show or explain how you got your answer.
Answers
Answer:
3.9
Step-by-step explanation:
A calculator is useful for this.
Which is an equation of a direct proportion? y=−2x y=12x−2 y=−2x+3 y=4x
Answers
Answer: First option and last option.
Step-by-step explanation:
By definition, an equation of direct proportion has the following form:
[tex]y=kx[/tex]
Where k is the constant of proportionality.
The equation of the line [tex]y=mx+b[/tex], where m is the slope and b the y-intercept.
When b=0, means that the line passes through the origin. Therefore, it would be:
[tex]y=mx[/tex]
(which is similar to [tex]y=kx[/tex])
Then, the slope of the line would be the constant of proportionality.
By definition, the slope can be positive or negative.
Keeping on mind the information above, you can see that the equation that have that form are:
[tex]y=-2x\\\\y=4x[/tex]
Answer:
y=−2x
Step-by-step explanation:
I got it right on Imagine math.
Later 7 students joined group 2 (41 students)and one student left to join group 1. (44 students) Now 20 students play trombone and 7 more students play trumpet as play saxophone. How many students play each instrument?
Answers
48 students in group 2
43 students in group 1
So in total there are 91 students
7 students play the trump
And 20 play the trombone
Therefore it is known that 27 students play those two instruments in total
Because of this, Subtract 27 from 91 to get the num of students who doesn't play those instruments.
When solving equation which is the best first step to begin to simplify the equation -2(x+3)=-10
Answers
Final answer:
The best first step to simplify the equation -2(x+3)=-10 is to distribute the -2 to the terms inside the parentheses and simplify. Then, solve for x by isolating the variable.
Explanation:
The best first step to simplify the equation -2(x+3)=-10 is to distribute the -2 to the terms inside the parentheses. This can be done by multiplying -2 with each term inside the parentheses.
So, -2(x+3) becomes -2x-6. The equation now becomes -2x - 6 = -10.
Now, we can proceed with solving the equation by isolating the variable x.
We can begin by adding 6 to both sides of the equation to get -2x = -4.
Finally, we divide both sides of the equation by -2 to solve for x.
So, x = 2.
The correct answer is option c. [tex]\frac{-2}{\ 2}(x+3) = \frac{-10}{\ 2}[/tex]
When solving the equation -2(x+3)=-10, the best first step is to divide both sides by 2. This simplifies the equation by making it easier to solve.
Here are the steps:
Divide both sides by 2: [tex]\frac{-2}{\ 2}(x+3) = \frac{-10}{\ 2}[/tex]Equation simplifies to: -(x+3) = -5Open parantheses: -x-3 = -5Rearrange: x = 5-3 = 2By dividing first, you simplify the algebraic manipulation and make solving the equation straightforward.
The complete question is:
When solving equation which is the best first step to begin to simplify the equation -2(x+3)=-10
a. [tex](-2)(-2)(x+3) = -10(-2)[/tex]
b. [tex]\frac{-1}{\ 2}(-2)(x+3) = (-10)\frac{-1}{\ 2}[/tex]
c. [tex]\frac{-2}{\ 2}(x+3) = \frac{-10}{\ 2}[/tex]
d. [tex]\frac{-2}{-10}(x+3) = \frac{-10}{-10}[/tex]
Jill measured the length of her eraser. She wrote 5 on her paper without the unit which metric unit of measure should Jill include?
Answers
Answer:
Centimeter
Step-by-step explanation:
Erasers are pretty small so I would say it was about 5 cm.
lily worked 3 hours more than andrea. together they worked a total of 16 hours. andrea worked a hours. Which equation represents this situation?
A. a+3=16
B. a+6=16
C. a+(a-3)=16
D. a+(a+3)=16
Answers
Answer:
The times were Lily 9.5 hours and Andrea 6.5 hours.
Equation "D" is correct using those numbers.
Step-by-step explanation:
Marissa bought 2 1/2 of a gallon of orange juice to make punch how many qaurts of orange juice did marissa buy?
Answers
Answer:
10 quarts
Step-by-step explanation:
There are 4 quarts to a gallon, and she bought 2 1/2 gallons, or 2.5 gallons, so she bought 4(2.5) = 10 quarts
What is the discriminant of the polynomial below? 2x^2+3x-7
Answers
Answer:
Option C. [tex]65[/tex]
Step-by-step explanation:
we know that
The discriminant of a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to
[tex]D=b^{2}-4ac[/tex]
in this problem we have
[tex]2x^{2} +3x-7[/tex]
so
[tex]a=2\\b=3\\c=-7[/tex]
substitute
[tex]D=3^{2}-4(2)*(-7)[/tex]
[tex]D=9+56[/tex]
[tex]D=65[/tex]
The discriminant of the giving polynomial is 65. The correct option is C. 65
The discriminant of a quadratic functionFrom the question, we are to determine the discriminant of the given polynomial
The given polynomial is
2x²+3x-7
This is a quadratic function
The discriminant of a quadratic function is simply the value in the square root of the quadratic formula
That is,
Discriminant, D = b² - 4ac
In the given quadratic function, 2x²+3x-7
a = 2, b = 3 and c = -7
∴ D = 3² -4(2)(-7)
D = 9 +56
D = 65
Hence, the discriminant of the giving polynomial is 65. The correct option is C. 65
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For which equation would x = 5 be a solution?
4x - 7 = 27
3x + 6 = 21
12 - 2x = 13
7 + 6x = 27
Answers
Answer:
The second option.
3x + 6 = 21.
Step-by-step explanation:
3x + 6 = 21.
If x = 5, plug it in.
3 · 5 + 6 = 21
15 + 6 = 21
Hope this helps,
Davinia.
Answer:
Second Option
Step-by-step explanation:
Write the recursive formula and the explicit formula for the sequence {-15,-7,1,9, 17,...}.
Answers
ANSWER
Explicit formula:
[tex]a_n= 8n - 23[/tex]
Recursive
[tex]a_{n}= a_{n - 1} + 8,a_1=-15[/tex]
EXPLANATION
The first term of the sequence is
[tex]a_1=-15[/tex]
The common difference is
[tex]d = - 7 - - 15[/tex]
[tex]d = 8[/tex]
The nth term is given by the formula,
[tex]a_n=a_1 + d(n - 1)[/tex]
[tex]a_n= - 15+ 8(n - 1)[/tex]
[tex]a_n= - 15+ 8n -8[/tex]
The explicit formula is;
[tex]a_n= 8n - 23[/tex]
The recursive definition is
[tex]a_{n}= a_{n - 1} + 8,a_1=-15[/tex]
Explain how to solve 4x 2- 5x = 16 by completing the square. What are the solutions?
Answers
Good Luck:)
What is the coefficient of q in the sum of these two expressions
(2/3q-3/4(-1/6q-r)
Answers
The coefficient of q in the sum of the two expressions is 17/24.
We have,
The given expression is:
(2/3q - 3/4(-1/6q - r))
Let's simplify this expression step by step:
Step 1: Distribute -3/4 to (-1/6q - r):
(2/3q - 3/4 * -1/6q + 3/4 * r)
Step 2: Simplify the multiplication of fractions:
(2/3q + 1/8q + 3/4 * r)
Step 3: Combine like terms:
(17/24q + 3/4 * r)
Thus,
The coefficient of q in the sum of the two expressions is 17/24.
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The coefficient of q in the sum of the expressions is found by simplifying the terms individually and then combining them. After finding a common denominator and adding the terms, the coefficient of q is 19/24.
Explanation:To find the coefficient of q in the sum of the given expressions (2/3q - 3/4*(-1/6q - r)), we first need to simplify the expression step by step. Starting with the second term, we distribute the -3/4 across the parentheses:
-3/4 * (-1/6q) = 3/4 * 1/6 * q = 1/8q-3/4 * (-r) = 3/4 * rNow adding this to the first term:
(2/3q + 1/8q) - (3/4)rTo add the q terms, they need to have a common denominator. The common denominator for 3 and 8 is 24. We convert both fractions:
2/3q = (2*8)/(3*8)q = 16/24q1/8q = (1*3)/(8*3)q = 3/24qNow we can add them:
16/24q + 3/24q = 19/24qThe coefficient of q in the sum of the given expressions is 19/24.
A poll reported that 66% of adults were satisfied with the job the major airlines were doing. suppose 15 adults are selected at random and the number who are satisfied is recorded. complete parts (a) through (e) below.
Answers
The statistical technique applicable here is simple random sampling and binomial distribution. The formula for a binomial distribution is used to estimate the probabilities of a specific number of satisfactions among the randomly selected 15 adults. The average number and variability can also be calculated for this binomial distribution.
Explanation:This question is about statistics, particularly a technique called simple random sampling. In the given scenario, it's reported that 66% of adults are satisfied with the major airlines' job. We're asked to conjecture about a scenario where 15 adults are chosen at random and the number who are satisfied is observed.
First, this scenario is modeled with a binomial distribution. A binomial distribution is appropriate here because you're considering a certain number of 'trials' (15 adults), and for each trial, there are two possible outcomes - the adult is either satisfied or not satisfied with the airlines. The 'success' probability (satisfaction) remains constant (66%).
A. If we want to know the probability of exactly x adults satisfied, the formula for a binomial distribution is used: P(X = k) = C(n, k) * (p^k) * ((1 - p)^(n - k)), where C(n, k) is combinations of n things taken k at a time, n is number of trials (15), p is the probability of success (66% or 0.66), and k is the number of successes we want exact probability for.
B. If we want to know the expected value or the average number of adults satisfied, it's simply n*p, in this case 15*0.66 = 9.9, meaning, on average, about 10 adults out of 15 should be satisfied according to the poll report.
C. If we want to compute the variability, we can calculate the standard deviation. For a binomial distribution, the standard deviation, sigma, is square root of [ n*p*q ], where q = 1-p is the probability of 'failure', or an adult not satisfied.
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It was 8 degrees at nightfall. The temperature dropped 10 degrees by midnight. what was the temperature at midnight?
Answers
Answer: The temperature would be -2 degrees at midnight.
Step-by-step explanation:
If you take 8 and minus 10 then you get -2!
Answer:
-2
Step-by-step explanation:
hope it will help youu!!!
On a winter morning, the temperature before sunrise was 11 degrees. The temperature then rose by 1/2 degree each hour for 7 hours before dropping by 2 1/4 each hour for 3 hours. What was the temperature, in degrees Fahrenheit, after 10 hours/
Answers
Answer:
7.75 F
Step-by-step explanation:
you will do 7/2 *
convert 2.25 to imporper fraction
multiply the denomintor
Answer:
7.75
Step-by-step explanation:
Grades on a standardized test are known to have a mean of 1000 for students in the US. The test is administered to 453 randomly selected students from Queens College, and they obtained an average grade of 1013 and a standard deviation of 108. a. Construct a 95 % confidence interval for the true average test score for Queens College students. b. With 5% significance level, Is there a statistical evidence that Queens College students perform differently than other students in the US? The same 453 students selected earlier are now given a two-hour tutoring session and then asked to take the test a second time. The average change in their test scores is 9 points, and the standard deviation of the change is 60 points. Assume the changes are from a Normal distribution. c. You are asked by the school administration to study whether students have performed better in their second attempt. State in terms of , the relevant null and alternative hypothesis in conducting this study. d. Compute the t statistic for testing against ; obtain the p- value for the test. e. Do you reject at the 5% level? At the 1% level? f. Provide a short summary of your conclusions from this study. Comment on the practical versus statistical significance of this estimate.
Answers
Answer:
a: 1003 < µ < 1023
b: See attached photo for answers
c: see below
d and e: See second attached photo
f: See below
Step-by-step explanation:
a. to find a confidence interval, first find the error with the formula,
E = Z(σ/√n) where Z is the value of the z-score that relates to the confidence level, σ is the standard deviation of the population (or of the sample if the population standard deviation isn't given), and n is the sample size
For this problem, E = 1.96(108/√453) = 9.95 = 10
Now add and subtract that value form the sample average...
x - E < µ < x + E becomes 1013 - 10 < µ 1013 + 10
simplifies to 1003 < µ < 1023
b. See attached photo for hypothesis test results...
c. H0: µ = 1013
Ha: µ > 1013 (claim)
They want to test if it's greater than their first attempt, so this claim becomes the alternate hypothesis. We're testing against their first attempt, that's why µ = 1013, not 1000
d and e. See second attached photo for hypothesis test results...
f. There is enough evidence to support the claim that the students second attempt was an improvement from their first attempt. They happened to take a 2 hour tutoring course between tests. This doesn't prove that the tutoring was effective thought. It very well might have been effective, but also, the students taking the test again means they've already seen the test and know what to expect, which could have also boosted their scores.
If two angles form a linear pair and the m<1 =2x + 20 and m< 2 = 48x, find the measure of each angle
Answers
Answer:
[tex]m<1=26.4\°[/tex]
[tex]m<2=153.6\°[/tex]
Step-by-step explanation:
we know that
If m<1 and m<2 form a linear pair
then
[tex]m<1+m<2=180\°[/tex] -----> by supplementary angles
substitute the values and solve for x
[tex](2x+20)+48x=180\°[/tex]
[tex]50x=180\°-20\°[/tex]
[tex]50x=160\°[/tex]
[tex]x=3.2\°[/tex]
Find the measure of each angle
[tex]m<1=(2x+20)=2(3.2\°)+20\°=26.4\°[/tex]
[tex]m<2=48x=48(3.2\°)=153.6\°[/tex]
If 50 is increased by 40%, what is the new amount?
Answers
Answer:
70
Step-by-step explanation:
50+20 = 70
ANSWER
70
EXPLANATION
We want to increase 50 by 40%
Method 1
New amount =50+40% of 50
[tex] = 50 + \frac{40}{100} \times 50[/tex]
[tex] = 50 + \frac{40}{2} [/tex]
[tex] = 50 + 20[/tex]
[tex] = 70[/tex]
Method 2.
50 corresponds to 100%.
New amount corresponds to (100+40=140)%
This implies that, New amount
[tex] = 140\% \times 50[/tex]
[tex] = \frac{140}{100} \times 50[/tex]
[tex] = \frac{140}{2} [/tex]
[tex] = 70[/tex]
What is the value of x in the equation 3(2x + 4) = ?6? (4 points) ?3 1 12 19
Answers
Answer:
2
Step-by-step explanation:
3(2x + 4)=?
6x + 12
6x/6=x
12/6=2
x=2