The critical points for the given function f(x) are -√5 and √5.
so option d is the correct answer.
What is the critical point?A critical point is the part of the domain of a function where the derivative is either equal to zero or the function is not differentiable.
Differentiate the given function f(x)=x/(x²+5)
f'(x)=((x²+5)-x(2x))/(x²+5)²
Using the Quotient Rule for differentiation.
What is Quotient Rule?A method for finding the derivative or differentiation of a function that is given as a ratio or division of two differentiable functions in calculus is known as the quotient rule.
We get f'(x)=5-x²/(x²+5)²
So the derivative is zero at -√5 and √5 and non differentiable at -√5
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Answer to the question
Answer: [tex]m=\frac{y_2 -y_1}{x_2 -x_1}[/tex]
Step-by-step explanation:
[tex]y_2 =m(x_2 -x_1)+y_1\\\\y_2 -y_1=m(x_2 -x_1)\\\\m=\frac{y_2 -y_1}{x_2 -x_1}[/tex]
From the table below, determine whether the data shows an exponential function. Explain why or why not. x31-1-3y1234a.No; the domain values are at regular intervals and the range values have a common sum 1.b.No; the domain values are not at regular intervals.c.Yes; the domain values are at regular intervals and the range values have a common factor 2.d.Yes; the domain values are at regular intervals and the range values have a common sum 1.
Solution:
Given:
The table of values is given:
From the table,
We see the data is a linear function. This is because a linear function has domain values at regular intervals.
Also, the linear equation can be formed as shown below, indicating it is a linear function.
Considering two points, (3,1) and (1,2)
where,
[tex]\begin{gathered} x_1=3 \\ y_1=1 \\ x_2=1 \\ y_2=2 \\ \\ \text{Then,} \\ \text{slope, m is given by;} \\ m=\frac{y_2-y_1}{x_2-x_1} \\ \\ \text{Substituting the values into the formula above,} \\ m=\frac{2-1}{1-3} \\ m=\frac{1}{-2} \\ m=-\frac{1}{2} \end{gathered}[/tex]A linear equation is of the form;
[tex]\begin{gathered} y=mx+b \\ \text{where m is the slope} \\ b\text{ is the y-intercept} \\ \\ To\text{ get the value of the y-intercept, we use any given point} \\ U\sin g\text{ point (3,1)} \\ y=mx+b \\ 1=-\frac{1}{2}(3)+b \\ 1=-\frac{3}{2}+b \\ 1+\frac{3}{2}=b \\ 1+1.5=b \\ b=2.5 \\ \\ \\ \text{Thus, the linear equation is;} \\ y=-\frac{1}{2}x+2.5 \end{gathered}[/tex]From the above, has confirmed it is a linear function and not an exponential function, we can deduce that;
a) The function is not an exponential function.
b) The domain values (x-values) are at regular intervals
c) The range values (y-values) have a common difference of 1
Therefore, the correct answer is OPTION A
Bc is included between
Answer:
A. angle B and angle C
Step-by-step explanation:
Line segments are named after the two points where they begin and end.
Bella drove her car 81 km and used 9 liters of fuel. She wants to know how many kilometers (x) she can drive on 22 liters of fuel. She assumes her car will continue consuming fuel at the same rate.
Find the proportion between liters of fuel
that is find 22/9 = 2.444
thats how much liters have more to consume
now multiply 2.444 by 81 , the kilometers she has drived
It gives as result 2.444 x 81 = 198 kilometers
Consider the function f(x) = 6 - 7x ^ 2 on the interval [- 6, 7] Find the average or mean slope of the function on this interval , (7)-f(-6) 7-(-6) = boxed |
Answer:
• Mean Slope = -7
,• c=0.5
Explanation:
Given the function:
[tex]f\mleft(x\mright)=6-7x^2[/tex]Part A
We want to find the mean slope on the interval [-6, 7].
First, evaluate f(7) and f(-6):
[tex]\begin{gathered} f(7)=6-7(7^2)=6-7(49)=6-343=-337 \\ f(-6)=6-7(-6)^2=6-7(36)=6-252=-246 \end{gathered}[/tex]Next, substitute these values into the formula for the mean slope.
[tex]\begin{gathered} \text{ Mean Slope}=\frac{f(7)-f(-6)}{7-(-6)}=\frac{-337-(-246)}{7+6}=\frac{-337+246}{13} \\ =-\frac{91}{13} \\ =-7 \end{gathered}[/tex]The mean slope of the function over the interval [-6,7] is -7.
Part B
Given the function, f(x):
[tex]f\mleft(x\mright)=6-7x^2[/tex]Its derivative, f'(x) will be:
[tex]f^{\prime}(x)=-14x[/tex]Replace c for x:
[tex]f^{\prime}(c)=-14c[/tex]Equate f'(c) to the mean slope obtained in part a.
[tex]-14c=-7[/tex]Solve for c:
[tex]\begin{gathered} \frac{-14c}{-14}=\frac{-7}{-14} \\ c=0.5 \end{gathered}[/tex]The value of c that satisfies the mean value theorem is 0.5.
An integer is chosen at random from 1 to 50. find the probability that the chosen integer is not divisible by 2, 7 or 9a)13/50b)16/25c)9/25
There are a total of 50 numbers that are between 1 and 50. Halft of these numbers are even (divisible by 2 ) and half of then odd.
There are 25 integers that are not even and in total there are 50 integers; thereofre, the probablity of finding an even integer is
25/50 = 1/2
Tara makes and sells scarves for children and adults. She is able to sell the scarves for $18 per unit. Materials for the scarves cost $4 each. She has fixed cost per month of $280 and estimates that she can make and sell 80 scarves each month. How many scarves does Tara need to sell to break even?
Tara needs to sell 20 scarves to break even.
What is the break-even point?The break-even point is the sales level that the seller must attain to make the total revenue equal to the total costs (fixed and variable).
There is no profit or loss at the break-even point (either in units or dollar values).
Selling price per unit = $18
Product cost per unit = $4
Contribution margin per unit = $14
Fixed cost per month = $280
Estimated production and sales units per month = 80 scarves
Break-even sales units = fixed costs/contribution margin per unit
= 20 ($280/$14)
Check:
Total revenue at 20 units = $360 (20 x $18)
Total costs at 20 units = $360 ($280 + $4 x 20)
Thus, for Tara to break even, she needs to sell 20 scarves per month.
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solving right triangle find the missing side. round to the nearest tenth
Apply trigonometric functions:
Cos a = adjacent side / hypotenuse
Where:
a = angle = 59°
adjacent side = 34
Hypotenuse = x
Replacing:
Cos 59 = 34 / x
Solve for x:
x = 34 / cos 59
x = 66
a scuba diver descended 19 5/12 feet blow sea level. Then he descended another 3 3/5 feet. Which of the following is true about the scuba diver after both descents?
The position of the scuba diver is 23 1/60 feet.
How to calculate the fraction?From the information, the scuba diver descended 19 5/12 feet blow sea level and then he descended another 3 3/5 feet.
The position of the diver will be. the addition of the fraction for descending. This will be:
= 19 5/12 + 3 3/5
= 19 25/60 + 3 36/60
= 22 61/60
= 23 1/60
Note that your information is incomplete as the question was answered based on information given.
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Use a polar coordinate system to plot the point with the given polar coordinates. Then find another representation (r,θ) of this point in which:
Hello there. To solve this question, we have to remember some properties about polar coordinates.
Given a point (x, y) and we want to plot the graph for (r, theta) after making the transformation, the graph will be something like the following:
In this case, we want to graph the point (5, 3pi/4)
First, notice 3pi/4 = 75º, which is in the first quadrant.
Therefore the graph will indeed look like the one above:
Which is the option contained in the first answer.
what is the maximum amount ginger Logan can borrow today if it must be repaid in 23 months with simple interest at 6% and she knows that at the time she will be able to repay no more than $23,000?(round to the nearest dollar as needed)
Answer:
$20,628
Explanation:
The amount that Logan will repay can be calculated as:
[tex]A=P(1+rt)_{}[/tex]Where P is the amount that she will borrow, r is the annual rate and t is the time in years.
So, we can replace A by 23000, r by 6%, and t by 23/12 because a year has 12 months. Then:
[tex]23000=P(1+(0.06\cdot\frac{23}{12}))[/tex]Finally, to know the maximum amount that Logan can borrow, we need to solve the equation for P as:
[tex]\begin{gathered} 23000=P(1+0.115) \\ 23000=P\cdot1.115 \\ \frac{23000}{1.115}=P_{}_{} \\ 20628=P \end{gathered}[/tex]So, the answer is $20,628
W L is tangent to circle O at point W. If mW H A = 260 degrees , find m< AWL
Answer:
[tex]m\angle AWL\text{ = 130}\degree[/tex]Explanation:
Here, we want to get the measure of the angle marked AWL
The measure of this angle is simply half the measure of the big arc WHA
Mathematically, we have the measure of the angle as:
[tex]\begin{gathered} m\angle AWL\text{ = }\frac{1}{2}\text{ }\times\text{ mWHA} \\ \\ =\text{ }\frac{1}{2}\times\text{ 260 = 130}\degree \end{gathered}[/tex]The following are the standard equation of a circle with center at the origin and radius of 2, except: a. x^2-4=-y^2b. x^2+4=-y^2c. x^2+y^2=2^2d. x^2+y^2=4
The equation of a circle is defined as
[tex]\begin{gathered} x^2+y^2=r^2 \\ \text{where} \\ r\text{ is the radius} \end{gathered}[/tex]Given that the radius of the circle is 2, then the equation of the circle is
[tex]x^2+y^2=2^2\text{ (option C)}[/tex]Which can then be simplified to
[tex]x^2+y^2=4\text{ (option D)}[/tex]And we can rearrange the equation
[tex]x^2-4=-y^2\text{ (option A)}[/tex]Which means that it cannot be the equation
[tex]x^2+4=-y^2[/tex]90 minutes for 3 typed pages; 60 minutes for a a typed pages write a proportion for each phrase and solve it
90 minutes for 3 typed pages; 60 minutes for a a typed pages write a proportion for each phrase and solve it
we have that
90/3=60/a
solve for a
a=(60*3)/90
a=2 pagesIdentify the vertex, axis of symmetry, and if the graph has a maximum or minimum. Then write the function for the graph shown
Answer:
Step-by-step explanation:
n b
Suppose Set A contains 48 elements and Set B contains 16 elements. If the total number elements in either Set A or Set B is 54, how many elements do Sets A and B have in common?
Considering the Set A and B given, Applying inclusion - exclusion principle the number of elements common to both Sets is 10
What is inclusion - exclusion principle?This is a counting techniques that ensures that elements are not counted twice
It is achieved by the formula:
(A ∪ B) = n(A) + n(B) - n(A ∩ B)
Finding the elements Sets A and B have in commonThe information from the question include the following
Set A contains 48 elements
Set B contains 16 elements
The total number elements in either Set A or Set B is 54
Applying inclusion - exclusion principle gives the formula
Set A + Set B - ( Set A ∩ Set B ) = Set A ∪ Set B
substituting the values gives
48 + 16 - ( Set A ∩ Set B ) = 54
48 + 16 - 54 = ( Set A ∩ Set B )
10 = ( Set A ∩ Set B )
( Set A ∩ Set B ) = elements Sets A and B have in common
Therefore the number of the elements common to Sets A and B is 10
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A blueprint shows an apartment withan area of 15 square inches. Ifthe blueprint's scale is1 inch : 8 feet, what will the actualsquare footage of the apartment be?The actual area of the apartment willbe -square feet.
Given that;
The blueprint shows an apartment with an area of 15 square inches.
With scale
1 inch : 8 feet
Recall that;
Area of a square is;
[tex]\text{A}=l^2[/tex]Let l represent the length of the side on the blueprint;
The actual length will be;
[tex]\begin{gathered} 8\times l\text{ fe}et \\ 8l\text{ f}eet \end{gathered}[/tex]So, the actual Area will be;
[tex]\begin{gathered} A_f=(8l)^2 \\ A_f=64l^2 \\ A_f=64A\text{ square f}eet \end{gathered}[/tex]substituting the valuye of the blueprint Area;
find the height of the trapezoidA=51CM2b=10cmb=7cmH?
we must find b one of the parallel sides before proceeding to find h
from the diagram b = 7cm
[tex]\begin{gathered} \text{Area = }\frac{10\text{ +7}}{2}\times h \\ 51\text{ = }\frac{17}{2}\times h \end{gathered}[/tex][tex]\begin{gathered} 51\text{ x 2 = 17h} \\ h\text{ =}\frac{51\times2}{17} \\ h\text{ =6cm} \end{gathered}[/tex]Find the quotient of these complex numbers.(4 + 4i) (5 + 4i) =A.B.C.D.
Find the quotient given below:
[tex]\frac{4+4i}{5+4i}[/tex]When managing complex numbers, we must recall:
[tex]\begin{gathered} i^2=-1 \\ \text{ Or, equivalently:} \\ i=\sqrt{-1} \end{gathered}[/tex]Multiply and divide the expression by the conjugate of the denominator:
[tex]\frac{4+4i}{5+4i}\cdot\frac{5-4i}{5-4i}[/tex]Multiply the expressions in the numerator and in the denominator. We can apply the special product formula in the denominator:
[tex](a+b)(a-b)=a^2-b^2[/tex]Operating:
[tex]\frac{(4+4i)(5-4i)}{5^2-(4i)^2}[/tex]Operate and simplify:
[tex]\frac{20-16i+20i-16i^2}{25-16i^2}[/tex]Applying the property mentioned above:
[tex]\frac{20-16i+20i+16}{25+16}[/tex]Simplifying:
[tex]\frac{36+4i}{41}[/tex]
A Labrador Retriever puppy named Milo weighed 11 pounds and gained 2 pounds per week.
After how many weeks did Milo weigh 39 pounds? Weeks?
After 15 weeks Milo's weight is 39 pounds.
According to the question,
We have the following information:
Weight of Milo = 11 pounds
Milo gained weight at the rate of 2 pounds per week.
So, we have the following progression:
11, 13, 15, ....
Now, we will subtract the previous term from the next term to check whether it is an arithmetic progression or not.
15-13 = 2
13-11 = 2
So, it is an A.P.
We know that following formula is used to find the nth term:
an = a+(n-1)d where a is the first term, n is the number of term and d is the common difference
We have weight of Milo as 39 pounds.
11+(n-1)2 = 39
11+2n-2 = 29
2n+9 = 39
2n = 39-9
2n = 30
n = 30/2
n = 15
Hence, it will take 15 weeks to reach Milo's weight at 39 pounds.
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Nayeli bought Jamba juice smoothies for herself and Evelyn after school one day. The smoothies cost $4.95 each plus 8.5% tax. how much change did she receive from a $20 bill
Explanation
Step 1
remember
[tex]8.5\text{ \%}\Rightarrow\frac{8.5}{100}=0.085[/tex]then, to find the value of the tax, multiply 4.95 0 0.085
[tex]\text{tax}=4.95\cdot0.085=0.42075\text{ per smoothie}[/tex]so, the total cost is
total =2 smoothies +(taxes for 2 smoothies)
total=(2*4.95)+(2*0.42075)
total=9.9+0.8415
total=10.7415
so, Nayebi paid $10.7415
Theoretical Probability - Guided Practice#1 - All of the letters in the word Mississippi are written on separate pieces of paper and putin a hat. Find the probability in drawing the letter s from the hat.O 34.6%O 38.4%O 36.4%0 45.5%
The probability = outcome/total outcomes
The total of the outcomes is the total number of the letters of the given word, then
The total outcomes = 11
The outcome is the number of letter "s" in the word
The outcome = 4, then
The probability of "s" is
[tex]P(s)=\frac{4}{11}[/tex]To change it to percent multiply it by 100% and round it to the nearest 1 decimal place
[tex]\begin{gathered} P(s)=\frac{4}{11}\times100 \\ P(s)=36.4 \end{gathered}[/tex]The answer is 36.4%
Answer C
her player O oo Find the amount of simple interest that $400 would earn at 8% per year by the end of 3 years. O A. $96 OB. $11,200 O c. $3200 OD. D. $112 O E. $32
To answer this question, we need to use the next formula for simple interest:
[tex]FV=PV\cdot(1+in)[/tex]Where:
FV is the future value we need to find (in this case).
PV is the present value, that is, $400 (in this case).
i is the interest rate. In this case, we have 8% (0.08).
n is the number of periods (n = 3, in this case).
Then, we have:
[tex]FV=400\cdot(1+(0.08)\cdot3)\Rightarrow FV=496[/tex]That is, the FV is $496. Therefore, the simple interest is $(496-400 = 96).
Thus, the amount of simple interest that $400 would earn at 8% per year by the end of 3 years is $96 (option A).
In other words, the result can be obtained also if we have is $400 * (0.08)*3 = $96.
A person invested $3,700 in an account growing at a rate allowing the money to double every 6 years. How much money would be in the account after 14 years, to the nearest dollar?
Given :
The principal = 3,700
Assume a simple interest
The account growing at a rate allowing the money to double every 6 years.
So,
[tex]\begin{gathered} I=P\cdot r\cdot t \\ I=P \\ 3700=3700\cdot r\cdot6 \\ r=\frac{1}{6} \end{gathered}[/tex]How much money would be in the account after 14 years, to the nearest dollar?
So, we will substitute with r = 1/6, t = 14 years
So,
[tex]\begin{gathered} I=3700\cdot\frac{1}{6}\cdot14=8633.33 \\ \\ A=P+I=8633.33+3700=12333.33 \end{gathered}[/tex]Rounding to the nearest dollar
So, the answer will be $12,333
This year, Buffalo, New York had 45 inches of snow in January. Last year, Buffalo had 19 inches of snow in January. How much more snow did Buffalo receive this January? Show your work in the space below. Don't forget to label the units on your answer.
In January this year, 45 inches of snow. In January last year, it was 19 inches. The difference betwen the measures gives us how much more was received this year.
This difference
= 45 inches - 19 inches
= 26 inches
Solve the inequality and how do i graph ?
The most appropriate choice for linear inequation will be given by-
[tex]m > \frac{1}{2}[/tex] is the correct solution
What is linear inequation?
At first it is important to know about algebraic expressions.
Algebraic expressions consists of variables and numbers connected with addition, subtraction, multiplication and division.
Inequation shows the comparision between two algebraic expressions by connecting the two algebraic expressions by > , < , [tex]\geq, \leq[/tex]
A one degree inequation is known as linear inequation.
Here,
The given inequation is [tex]\frac{1}{2} - \frac{m}{4} < \frac{3}{8}[/tex]
Now,
[tex]\frac{1}{2} - \frac{m}{4} < \frac{3}{8}\\\\\frac{m}{4} > \frac{1}{2} - \frac{3}{8}\\\\\frac{m}{4} > \frac{4 - 3}{8}\\\\\frac{m}{4} > \frac{1}{8}\\\\m > \frac{1}{8} \times 4\\m > \frac{1}{2}[/tex]
The number line has been attached here.
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9) solve using substitution method and check your answer:4x - 3y + 2z = 16- 4y - Z = 7= 146x - y
Given the system of equations, solve the third equation for y, as shown below
[tex]\begin{gathered} 6x-y=14 \\ \Rightarrow y=6x-14 \end{gathered}[/tex]And, solve for z in the second equation,
[tex]\begin{gathered} -4y-z=7 \\ \Rightarrow z=-4y-7 \\ \Rightarrow z=-4(6x-14)-7=-24x+49 \end{gathered}[/tex]Thus, substitute the values of y and z in terms of x into the first equation, as shown below
[tex]\begin{gathered} \Rightarrow4x-3y+2z=4x-3(6x-14)+2(-24x+49)=4x-18x+42-48x+98 \\ \Rightarrow-62x+140=16 \\ \Rightarrow-62x=-124 \\ \Rightarrow x=2 \end{gathered}[/tex]Then, solving for y and z given x=2,
[tex]\begin{gathered} x=2 \\ \Rightarrow y=6*2-14=-2 \\ and \\ z=-24*2+49=-48+49=1 \end{gathered}[/tex]Therefore, the solution to the system of equations is x=2, y=-2, z=1To verify the solutions, substitute the values we found into the three equations of the system, as shown below
[tex]\begin{gathered} x=2,y=-2,z=1 \\ \Rightarrow4x-3y+2z=4*2-3*(-2)+2*1=8+6+2=16\rightarrow correct \\ \Rightarrow-4y-z=-4*(-2)-1(1)=8-1=7\rightarrow correct \\ \Rightarrow6x-y=6*2-1(-2)=12+2=14\rightarrow correct \end{gathered}[/tex]Hi how do I graph these? I don't understand how I'm supposed to graph fractions?
To plot in the plane points with fraction number coordinates (x or y) you can rewrite the fractions as decimal numbers:
[tex]\begin{gathered} \frac{-5}{2} \\ \\ -5\text{ divided into 2} \\ \\ -\frac{5}{2}=-2.5 \\ \\ \\ \\ \\ \frac{-1}{4} \\ \\ -1\text{ divided into 4:} \\ -\frac{1}{4}=-0.25 \end{gathered}[/tex]Then, you have the next coordinates;
(- 2.5, 2) and (1, -0.25)
And the next graph:
Use a line to link the points
Are y = 3x +7 and y = 3x - 8 parallel to each other?
Answer:
They are parallel to each other
Explanation:
Two lines are parallel if they have the same slope.
Additionally, in an equation with the following form:
y = mx + b
The number m beside the x, is the slope
So, in this case, both equations have a 3 besides the x, then, they are parallels
I need help for my assignment I need to submit today
The general equation of a circle is expressed as
[tex]\begin{gathered} (x-a)^2+(y-b)^2=r^2\text{ ----- equation 1} \\ \text{where} \\ (a,\text{ b)}\Rightarrow\text{ center of the circle} \\ r\Rightarrow radius\text{ of the circle} \end{gathered}[/tex]Given that a circle having equation
[tex]\begin{gathered} (x-2)^2+(y-5)^2\text{ = 16} \\ \Rightarrow(x-2)^2+(y-5)^2\text{ = }4^2 \end{gathered}[/tex]is moved up 3 units and 1 unit to the left. Thus, we have
[tex]\begin{gathered} (x-2+1)^2+(y-5-3)^2\text{ = }4^2 \\ \end{gathered}[/tex]This gives
[tex](x-1)^2+(y-8)^2\text{ = }4^2\text{ ----- equation 2}[/tex]Comparing equations 1 and 2, we have
[tex]a\text{ = 1, b = 8, r = 4}[/tex]Hence,
the center (a, b) of the circle is (1, 8),
the radius r of the circle is 4,
the equation of the circle is
[tex](x-1)^2+(y-8)^2=4^2[/tex]