(10
points
)
Let
S(t)= 1+e −t
1
. (a) Find
S ′
(t)
. (b) Which of the following equations hold true? Show why your choice is true. [Note: only one equation is true.] i.
S ′
(t)=S(t)
ii.
S ′
(t)=(S(t)) 2
iii.
S ′
(t)=S(t)(1−S(t))
iv.
S ′
(t)=−S(−t)
Answers
The derivative of S(t)= 1+e −t is S'(t) = S(t)(1 - S(t)). So, the correct answer is (iii).
To find S'(t), we can use the chain rule:
S'(t) = (d/dt) [1 + e^(-t/2)]^-2 * d/dt [1 + e^(-t/2)]
Using the chain rule again for the second derivative:
d/dt [1 + e^(-t/2)] = (-1/2)e^(-t/2)
d/dt [1 + e^(-t/2)]^-2 = -2(1 + e^(-t/2))^-3 * (-1/2)e^(-t/2) = (1/2) e^(-t/2) / (1 + e^(-t/2))^3
Substituting into the expression for S'(t), we have:
S'(t) = [(1/2) e^(-t/2) / (1 + e^(-t/2))^3] * [1 - (1/2)e^(-t/2)]
S'(t) = (1/2) e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]
S'(t) = e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]
Taking the derivative of S(t), we have:
S'(t) = e^(-t/2) / (1 + e^(-t/2))^2
Comparing this to the given choices, we can see that:
S'(t) = S(t) is not true, since S(t) = 1 + e^(-t/2) and S'(t) is a different function.
S'(t) = (S(t))^2 is not true, since (S(t))^2 = (1 + e^(-t/2))^2 is a different function from S'(t).
S'(t) = S(t)(1 - S(t)) is true, since we can substitute S(t) and S'(t) from above and simplify:
S'(t) = e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]
S(t)(1 - S(t)) = [1 + e^(-t/2)] * [1 - (1 + e^(-t/2))] = e^(-t/2) / (1 + e^(-t/2))
Therefore, S'(t) = S(t)(1 - S(t)) is true.
S'(t) = -S(-t) is not true, since S(-t) = 1 + e^(t/2) and -S(-t) is a different function from S'(t).
So the correct choice is (iii): S'(t) = S(t)(1 - S(t)).
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Related Questions
which of the following are always true? (1) the derivative of a constant is always zero. (2) the derivative of a quadratic function will always be a linear function.
Answers
Both the statements are correct here
Statement (1) is true: the derivative of a constant function is always zero. This is because the slope of a constant function is always zero, and the derivative measures the slope of the function at each point.
Statement (2) is true. A quadratic function is a function of the form f(x) = ax² + bx + c, where a, b, and c are constants. Taking the derivative of this function gives f'(x) = 2ax + b, which is a linear function as b is constant.
Therefore, both the statements are true about the derivative of a constant is always zero and the derivative of a quadratic function will always be a linear function
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What is the sales tax for a purchase of $1,000 if the tax rate is 7 percent?
Answers
How do I use PI also known as 3.14 =22/7
Answers
A rectangular patio has a length that is 4 times its width and has an area of 5000 square feet. What is the width of the patio?
Answers
\( \bf \underline{ \underline{Given : }}\)
A rectangular patio has a length that is 4 times its width.Area of rectangular patio is 5000 square feet.\( \bf \underline{ \underline{To \: be \: calculated : }}\)
Calculate the width of the patio.
\( \bf \underline{ \underline{Solution : }}\)
Let the width be a.
Then,Length = 4a
According to the question,
\( \sf \: Area \: of \: rectangle = 5000 \: ft {}^{2} \)
\( \sf \implies \: Length \times Width = 5000\)
Substituting the values of Length and width, we get..
\( \sf \implies 4a \times a = 5000\)
\( \sf \implies4 {a}^{2} = 5000\)
\( \sf\implies \: {a}^{2} = \cancel \dfrac{5000}{4} \)
\( \sf\implies \: {a}^{2} = 1250\)
\( \sf \implies \: a = \sqrt{1250} \)
\( \sf \implies \: a = 35.35 \: \: (approx)\)
\( \sf\therefore \: The \: width \: of \: rectangular \: patio \: is \: 35.35 \: ft\)
8,546.54 to nearest 1,000.00
Answers
Step by step: remember 5 or higher raise the score 4 or less let it rest. You underline the thousands place in this situation then look back to the number right to it. Since that number is 5 you must change the 8 to a 9. All the other numbers to the right if 8 is now 0.
Hi can someone please help me with this!!! I'm struggling with this!
I really appreciate it!!!
Find the value of a, m, x
Answers
Answer:
Refer to pic..........
As a construction manger, you are asked to build a new road which crosses the point (5,3). There is another road already built whose location can be expressed as y=5x-2. you are asked to build your road such that it intersects making a right angle on the blue-print. write a statement to represent the locations of the new road and the road is already built. provide evidence that algebraically justifies your responses
Answers
The product of the slopes is -1, we can conclude that the two roads intersect at a right angle at the point (10/11, 18/11).
What is slope?
In mathematics, slope refers to the steepness or incline of a line on a graph. It is a measure of how much the dependent variable changes for every unit change in the independent variable.
To build the new road such that it intersects the existing road at a right angle at the point (5,3), we can start by determining the slope of the existing road.
The slope of the existing road, given by the equation y=5x-2, is 5. This is because the equation is in slope-intercept form, y=mx+b, where m is the slope and b is the y-intercept. Therefore, we know that the slope of any line perpendicular to this road will be -1/5, since the product of the slopes of perpendicular lines is -1.
To find the equation of the new road, we can use the point-slope form of a linear equation, which is y-y1=m(x-x1). We know that the new road passes through the point (5,3) and has a slope of -1/5. So, substituting these values into the equation gives:
y - 3 = (-1/5)(x - 5)
Simplifying this equation, we get:
y = (-1/5)x + 4
Therefore, the equation of the new road is y = (-1/5)x + 4, and the equation of the existing road is y = 5x - 2. These two equations represent the locations of the new road and the existing road, respectively.
To verify that the new road intersects the existing road at a right angle, we can find the point of intersection of the two roads and then check that the product of the slopes of the two lines is -1.
To find the point of intersection, we can set the two equations equal to each other:
(-1/5)x + 4 = 5x - 2
Solving for x, we get:
x = 10/11
Substituting this value of x back into either equation gives:
y = 18/11
So the point of intersection is (10/11, 18/11).
Now, to check that the two roads intersect at a right angle, we can find the slopes of the two lines and check that their product is -1:
Slope of existing road, m1 = 5
Slope of new road, m2 = -1/5
Product of slopes, m1m2 = 5(-1/5) = -1
Since the product of the slopes is -1, we can conclude that the two roads intersect at a right angle at the point (10/11, 18/11).
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4. Calculate the Social Security and Medicare tax that would be applied to an annual salary of $56,400.
Answers
Answer:759
Step-by-step explanation:
What’s 6r+3=4r-11?????
Answers
Answer:
-7
Step-by-step explanation:
6r+3=4r-11
distributive property
6r-4r=-11-3
2r=-14
r=-7
Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Match the absolute value functions with their vertices.
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f(x)= |x-51+
f(x) = -2|2|-
-
f(x)= |x-1| +4
f(x)=x+11-4
Vertex
(-1,-4)
(5, 1)
(0, -1)
(2, 4)
f(x) = |z|-
f(2)=|z-|+{
Absolute Value Function
Answers
Answer:
\(\boxed {\left(-1, -\dfrac{3}{7}\right)} \longrightarrow \boxed{f\left(x\right)\:=\:\frac{1}{2}\left|x\:+\:1\right|-\frac{3}{7}}\)
\(\boxed{\left(5,\:\frac{2}{3}\right)} \longrightarrow \boxed{f\left(x\right)\:=\:\frac{3}{5}\left|x\:-\:5\right|+\:\frac{2}{3}}\)
\(\boxed{\left(0,\:-\frac{4}{5}\right)} \longrightarrow \boxed{f\left(x\right)\:=\frac{1}{2}\left|x\right|\:-\:\frac{4}{5}}\)
\(\boxed{\left(\frac{2}{5},\:\frac{5}{3}\right)} \longrightarrow \boxed{f\left(x\right)\:=\frac{3}{2}\left|x-\frac{2}{5}\right|+\frac{5}{3}}\)
Step-by-step explanation:
The vertex of an absolute function
y = f(x) = a|x - b| + c occurs at x - b = 0 or when x = b
Plugging this into the original equation will give the value for f(b) which will be f(b) = 0 + c which will be the y-value of the vertex
\(\text{Vertex of } f(x) = \dfrac{3}{5} |x - 5| + \dfrac{2}{3}\\\\ \longrightarrow x = 5, y = \dfrac{2}{3} \\\\\longrightarrow \left(5, \dfrac{2}{3} \right)\)
You can do the others in a similar manner.
Here it is easier because the constant in f(x) corresponds to the y-coordinate of the vertex and they are different in the answer choices
Let X and Y be discrete random variables and let a and b be constants Which of the following is. FALSE? (a) mean (X + Y) = mean (X) + mean (Y).
Answers
Mean (X + Y) = Mean (X) + Mean (Y).
The above statement is false.
Discrete Random Variable:
A discrete random variable can be defined as a type of variable whose value depends on the numerical outcome of some random phenomenon. Also called a random variable. Discrete random variables are always easily countable integers. A probability mass function is used to describe the probability distribution of a discrete random variable.
Probability Distributions of Discrete Random Variables:
Probability distributions of discrete random variables list the probabilities associated with each possible outcome. Also called probability function or probability mass function.
The probability of a discrete random variable is between 0 and 1. Also, the sum of the probabilities of a discrete random variable is equal to 1. The probability distribution of discrete random variables resembles the normal distribution.
Example:
Suppose two dice are rolled and a random variable X is used to represent the sum of the numbers. The minimum value of X goes from result 1 + 1 = 2 to 2 and the maximum value goes from result 6 + 6 = 12 to 12. Therefore, X can have any value between 2 and 12 (inclusive). If probabilities are assigned to each outcome, we can determine the probability distribution of X.
Discrete random variables should not be confused with algebraic variables. Algebraic variables represent the values of unknown quantities in computable algebraic equations. However, a discrete random variable can have a range of possible values that result from experimentation.
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Need some help with this ! Would really appreciate it.
Answers
The approximate amount of time, in minutes, for light to travel from the Sun to Mars is 3.89 × 10⁻³ minute.
What is speed?In Mathematics, speed is the distance covered by a physical object per unit of time. This ultimately implies that, speed can be measured by using miles per hour (mph).
Mathematically, the speed of any a physical object can be calculated by using this formula;
Speed = distance/time
By making time the subject of formula, we have:
Time = distance/speed
Time = (1.12 × 10⁷)/(2.88 × 10⁹)
Time = 3.89 × 10⁻³ minute.
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Homework part2 need help asap
Answers
The key features of the given quadratic functions are listed below.
What is the graph of a quadratic function?In Mathematics and Geometry, the graph of any quadratic function would always form a parabolic curve because it is a u-shaped. Based on the first quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0) i.e 3 > 0.
For the quadratic function y = 3x² - 5, the key features are as follows;
Axis of symmetry: x = 0.
Vertex: (0, -5).
Domain: [-∞, ∞]
Range: [-5, ∞]
For the quadratic function y = -2x² + 12x - 15, the key features are as follows;
Axis of symmetry: x = 3.
Vertex: (3, 3).
Domain: [-∞, ∞]
Range: [-∞, 3]
For the quadratic function y = -x² + 1, the key features are as follows;
Axis of symmetry: x = 0.
Vertex: (0, 1).
Domain: [-∞, ∞]
Range: [-∞, 1]
For the quadratic function y = 2x² - 16x + 30, the key features are as follows;
Axis of symmetry: x = 4.
Vertex: (4, -2).
Domain: [-∞, ∞]
Range: [-2, ∞]
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Frank paid $22.44 I sweatshirt had been discounted by 30% what is the regular price the sweatshirt
Answers
Answer: $32.06
Step-by-step explanation:
Let the regular price of the sweatshirt be x.
Therefore,
$22.44 = x - 0.3x
22.44 = 0.7x
x = about $32.06
Answer:
Step-by-step explanation:
22.44 represent 70% of the normal price of the sweatshirt.
Why 70%?
The reason is if you take 30% of the original price and subtract that from thirty %, you will get 70% of the original price.
Original price = 70% of x
Equation
70% x = 22.44
70/100 x = 22.44 Multiply both sides by 100
70 x = 22.44 * 100
70 x = 2244 Divide both sides by 70
70x/70 = 2244 / 70
x = 32.06
Answer: The sweatshirt originally cost 32.06
Which ordered pair maximizes the objective function p=3x+8y
(0,0)
(2,7)
(5,6)
(8,1)
Answers
Answer:
P(5,6) = 63
Step-by-step explanation:
Test each point to see which ordered pair maximizes the objective function:
(0,0): p = 3(0) + 8(0) = 0
(2,7): p = 3(2) + 8(7) = 6 + 56 = 62
(5,6): p = 3(5) + 8(6) = 15 + 48 = 63
(8,1): p = 3(8) + 8(1) = 24 + 8 = 32
Hence, (5,6) is the ordered pair that maximizes the objective function.
Which is the graph of f(x)=(2/3)x
Answers
Answer:
Here is the desmos graph
Step-by-step explanation:
super easy algebra 4 x (15- ?) =48
Answers
after this you are left with 15-?=12.
?=3
Maxwell wants to know how much space his globe occupies. Should he find its surface area or volume? Explain your answer
Answers
Since, Maxwell wants to know how much space his globes occupies he has to find its volume.
What is a volume?
Volume is the amount of space an object or substance occupies.
Every three dimensional object occupies a space.
Volume is measured in cubic units.
Therefore, the three dimensional object that has a width, length and height will have its volume as follows;
volume = lwh
where
l = lengthw = widthh = heightTherefore, the space occupied by the globe is the volume of the globe.
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2. Jamal makes $11.50 an hour and works 37.5 hours a week. Jamal's friend, Trenten,
makes $13.45 an hour and works 32 hours a week. Jamal thinks he should quit his job
and get a job with Trenten.
Does Jamal make more money than Trenten in a week?
O
O
CLEAR ALL
Yes
No
Answers
Josiah wants to ride his bicycle 42.5 miles this week. He has already ridden 18 miles.
If he rides for 5 more days, what is the average number of miles he would have to ride
each day to meet his goal?
Answers
Hello there! Allow me to help you!
|| ▼ Answer ▼ ||
4.9 miles
|| ✪ Solution ✪ ||
Step 1:
In this question, we are given the following:
Josiah wants to ride his bicycle 42.5 miles this week.
He has already ridden 18 miles. If he rides for 5 more days,
what is the average number of miles he would have to ride each day to meet his goal?
Step 2:
The details of the solution are as follows:
Total distance = 42. 5 miles
Distance ridden = 16 miles
Number of days left = 5 days
Let the number of miles that'll be ridden for each day be represented by x.
Hence, we have that,
\(5x+18=42.5\)
\(5x=42.5-18\)
\(5x=24.5\)
Divide both sides by 5 we have that,
\(x=\frac{24.5}{5}\)
\(x=4.9\)
CONCLUSION:
The average number of miles he would have to ride each day to meet his goal = 4.9 miles
Hope this helps!
If you have any questions please ask.
The required average miles for him to ride for 5 days is 4.9 miles per day.
Given that,
Josiah wants to ride his bicycle 42.5 miles this week. He has already ridden 18 miles.
he rides for 5 more days, what is the average number of miles he would have to ride each day to meet his goal is to be determined
The average of the values is the ratio of the total sum of values to the number of values.
Here,
Josiah wants to ride his bicycle for a total of 42.5 miles,
he rides for 18 miles by now,
Remaining miles to ride = 42.5 - 18 = 24.5 miles
For 5 day is must ride for average = 24.5 / 5 = 4.9 miles per day.
Thus, the required average miles for him to ride for 5 days is 4.9 miles per day.
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Select the correct answer from each drop-down menu. Graph shows a four-sided polygon is plotted on a coordinate plane at A (3, 4), B (5, 3), C (3, 2), and D (1, 3). Polygon ABCD is plotted on a coordinate plane. If the polygon translates 3 units to the left and 1 unit down, the length of diagonal A ′ C ′ ¯ is units. If the polygon translates 3 units further to the left and four units down, the length of diagonal A ′ C ′ ¯ . Reset Next
Answers
Therefore, the length of diagonal A'C' is also 2 units after the polygon translates 3 units further to the left and 4 units down.
What is a polygonal example?A polygon is any form having three sides. Although a circle is curved and devoid of sides and angles, it is not categorised as a polygon even if it is a plane figure.
If the polygon translates 3 units to the left and 1 unit down, the coordinates of the new vertices would be:
A' (0, 3)
B' (2, 2)
C' (0, 1)
D' (-2, 2)
To find the length of diagonal AC, we can use the distance formula:
AC = √[(3-3)^2 + (2-4)^2] ≈ 2 units
If the polygon translates 3 units further to the left and 4 units down, the coordinates of the new vertices would be:
A'' (-3, -1)
B'' (-1, -2)
C'' (-3, -3)
D'' (-5, -2)
To find the length of diagonal A'C', we can use the distance formula:
A'C' = √[(0-0)^2 + (1-3)^2] ≈ 2 units
Therefore, the length of diagonal A'C' is also 2 units after the polygon translates 3 units further to the left and 4 units down.
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Find the area of the rectangle on this centimetre grid.
Answers
Answer:
A = 35 cm²
Step-by-step explanation:
the area (A) of a rectangle is calculated as
A = length × width
by counting the squares on the sides
length = 7 cm and width = 5 cm , then
A = 7 × 5 = 35 cm²
A Bakery made 423 sandwiches in one day. What is that number rounded to the nearest hundred sandwiches?
Answers
Answer: 400
Step-by-step explanation:
Can someone please tell me the property of number 6 it’s urgent
Answers
Answer:
Distributive property
Step-by-step explanation:
Not sure, but I think it is distributive property because you distributive -1/5 to -5x. Hope that helps, thank you !!
Solve only in the Substitution method-3x + y = -24x - y = 4
Answers
The given equations are:
\(\begin{gathered} -3x+y=-2\text{ Eq(1)} \\ 4x-y=4\text{ Eq(2)} \end{gathered}\)Solve for y in terms of x in equation 1:
\(\begin{gathered} -3x+y=-2 \\ y=-2+3x\text{ Eq(3)} \end{gathered}\)Substitute y in equation 2:
\(\begin{gathered} 4x-(-2+3x)=4 \\ \text{Apply the distributive property} \\ 4x+2-3x=4 \\ x+2=4 \\ x=4-2 \\ x=2 \end{gathered}\)Now, substitute x in equation 3 and find y:
\(\begin{gathered} y=-2+3\cdot2 \\ y=-2+6 \\ y=4 \end{gathered}\)Thus, x=2 and y=4.
Brihana kayllie works in a chocolate factory. She gets 8.75 per hour. how much money does she earn in 8 hours? 40 hours? Show your work in the box provided.
Answers
Answer:
Briana would get $70 in 8 hours and $350 in 40 hours.
Step-by-step explanation:
Multiply 8.75 by 8 to get $70
Multiply 8.75 by 40 to get $350
Hope this helps!!
Answer:
70$ in 8 hours and 350$ in 40 hours
Step-by-step explanation:
8.75 + 0.25 = 9 X 8 = 72 - 2 = 70
0.25 X 8 = 2
8.75 + 0.25 = 9 X 40 = 360 - 10 = 350
0.25 X 40 = 10
The function f(x) = -2(4)²+1 +140 represents the number of tokens a child has x hours after arriving at an arcade.
What is the practical domain and range of the function?
Enter your answer by filling in the boxes to correctly complete the statements. If necessary, round to the nearest hundredth.
The practical domain of the situation is ?
The practical range of the situation is ?
Answers
The practical domain of the situation is all real numbers, since there is no restriction on the number of hours the child can spend at the arcade.
The practical range of the situation is [1,140], since the minimum value of the function is 1 and the maximum value is 140.
Help!!!!!! Please thanks
Answers
As you can see in the image below, the answer is graph C.
WANT POINTS JUST ANSWER THIS CORRECTLY FOR 45!!!
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How many minutes would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?
Answers
1 hr 15 mins = 75 mins
1 hr 30 mins = 90 mins
Next, calculate how many total minutes Gage has skated in the first 8 days
75(5) + 90(3) = 645 mins
Create an equation to find the average of Gage’s minutes of skating. Add up all the minutes and divide by the total amount of days and set equal to 85, the average we are trying to get.
(645 mins + x mins)/9 days = 85
Solve for x
645 + x = 765
x = 120
So, in order to have an average of an 85 minute skate time, Gage would need to skate 120 minutes on the ninth day.
Subtract 4 from 1/6 of 42 5th grade
Answers
Answer:
3
Step-by-step explanation:
subtract 4 from 1/6 of 42. In middle school my traxhers always told me that of menas multiply, and in this situation it still holds. 1/6 x 42 is 42/6 or 42 divided by 6 which is 7. So now that makes this simpler, its saying subtract 4 from 7, so 7-4 =3