Differentiation from First Principles is a formal method for determining a tangent's gradient.
What is meant by differentiation?Finding a function's derivative is the process of differentiation. It is also the process of determining how quickly a function changes in relation to its variables.
According to the Sum rule, a sum of functions' derivatives equals the sum of those functions' derivatives. The derivative of two different functions is the difference of their derivatives, according to the Difference rule.
Differentiation from First Principles is a formal method for determining a tangent's gradient. The straight line connecting any two locations on the curve that are fairly near to one another will have a gradient that is similar to that of the tangent at those places.
s(t) = ∫vdt = ∫sin(πt)dt = (-cos(πt))/π + c
substituting the values t = 3, we get
s(-3) = (-cos(-3π))/π + c = 0
simplifying the above equation, we get
(-cos(3π))/π + c = 0
1/π + c = c
c = -1/π
Therefore, the correct answer is s(t) = (-cos(πt))/π - 1/π = (-cos(πt) - 1)/π.
The complete question is:
Given the velocity v=ds/dt and the initial position of a body moving along a coordinate line, find the body's position at time t. v=sin(pi*t), s(-3)=0
To learn more about Differentiation refer to:
https://brainly.com/question/954654
#SPJ13
solve using properties of logarithm round two decimal places1. 5e^ (-0.4t) = 1.506
To solve the exercise, we can use the following property of logarithms:
[tex]\ln (e^x)=x[/tex]Then, we can solve the equation like this:
[tex]\begin{gathered} 1.5e^{-0.4t}=1.506 \\ \text{ Divide by 1.5 from both sides of the equation} \\ \frac{1.5e^{-0.4t}}{1.5}=\frac{1.506}{1.5} \\ e^{-0.4t}=1.004 \\ \text{ Apply }\ln \text{ from both sides of the equation} \\ \ln (e^{-0.4t})=\ln (1.004) \\ \text{ Apply the mentioned property of logarithms} \\ -0.4t=\ln (1.004) \\ \text{ Divide by -0.4 from both sides of the equation} \\ \frac{-0.4t}{-0.4}=\frac{\ln(1.004)}{-0.4} \\ t\approx-0.01\Rightarrow\approx\text{ it reads "approximately"} \end{gathered}[/tex]Therefore, the solution of the equation rounded to two decimal places is -0.01.
A wise man once said 200 reduce twice my age is 32.
what is his age?
Answer:84
Step-by-step explanation:
200 - 2x = 32-2x = -168x = 84
Answer:
His age is 84
find the mode of the following list of points earned on a 16-point quiz given during a finance class 7, 7, 3, 2, 7, 16, 12, 16, 12
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data
7, 7, 3, 2, 7, 16, 12, 16, 12
mode = ?
Step 02:
7, 7, 3, 2, 7, 16, 12, 16, 12
n = 9
2, 3, 7, 7, 7, 12, 12, 16, 16
The mode is the value that appears most frequently in a serie of data.
mode = 7
The answer is:
The mode is 7.
DATA ANALYSES AND STATISTICSUnderstanding likelihoodThere are two boxes containing only blue and purple pens.Box A has 6 purple pens and 2 blue pens.Box B has 12 purple pens and 8 blue pens,A pen is randomly chosen from each box,List these events from least likely to most likelyEvent 1: choosing a purple or blue pen from Box B.Event 2: choosing a blue pen from Box B.Event 3: choosing a red pen from Box A.Event 4: choosing a purple pen from Box A.Least likelyMost likelyEvent ). Event | Event | .Event
Answer
Event 3, Event 2, Event 4, Event 1.
Explanation
Given data:
Box A has 6 purple pens and 2 blue pens.
Box B has 12 purple pens and 8 blue pens.
Let X represents purple, Y represents blue, and Z represents red.
Box A: n(X) = 6, n(Y) = 2, and n(Z) = 0. Hence, n(S) = 6 + 2 + 0 = 8
Box B: n(X) = 12, n(Y) = 8, and n(Z) = 0. Hence, n(S) = 12 + 8 + 0 = 20
So if a pen is randomly chosen from each box, the probability of:
Event 1: choosing a purple or blue pen from Box B. will be:
[tex]\begin{gathered} p(X\text{ or Y})=p(X)+p(Y)=\frac{n(X)}{n(S)}+\frac{n(Y)}{n(S)}=\frac{12}{20}+\frac{8}{20}=\frac{12+8}{20}=\frac{20}{20}=1 \\ \end{gathered}[/tex]Event 2: choosing a blue pen from Box B will be:
[tex]p(Y)=\frac{8}{20}=\frac{1}{5}=0.4[/tex]Event 3: choosing a red pen from Box A will be:
[tex]p(Z)=\frac{n(Z)}{n(S)}=\frac{0}{8}=0[/tex]Event 4: choosing a purple pen from Box A wil be:
[tex]p(X)=\frac{n(X)}{n(S)}=\frac{6}{8}=\frac{3}{4}=0.75[/tex]If a pen is randomly chosen from each box, the list of the events from least likely to most likely are:
Event 3, Event 2, Event 4, Event 1.
part (a) Solve for x: 1 /x − 1 /x + 1 = 3.
part (b) Solve for t: 2 = Square root √ (1 + t) power of 2 + (1 − 2t) power of 2
part (c) Solve for t: 3/√5 = Square root √(1 + t) power of 2 + (1 − 2t)power of 2
part (d) Solve for t: 0 = Square root √ (1 + t)power of 2 + (1 − 2t)power of 2
The value of x in part (a) is [-3+√21]/6 and [-3-√21]/6, the value of t in part (b) is (1/5+√11/5) and (1/5-√11/10), the value of t in part (c) is 1/5 and the value of t in part (d) is 1/5-3/5i and 1/5+3/5i.
Part a. Solving for x,
1/x - 1/(x+1) = 3
(1-x+x)/x(x+1) = 3
1/(x²+x) = 3
1 = 3x²+3x
3x²+3x-1=0
Solving the quadratic by using quadratic formula,
x = [-b±√(b²-4ac)]/2a
Here,
a = 3
b = 3
c = -1
Putting all the values,
x= [-3±√(3²-4(3)(-1)]/6
x = [-3±√21]/6
We get,
x = [-3+√21]/6 and
x = [-3-√21]/6
Part b. Solving for t;
2 =√[(1+t)²+(1-2t)²]
Squaring both sides,
4 = (1+t)²+(1-2t)²
4 = 1+t²+2t+1+4t²-4t
2 = 5t²-2t
5t²-2t-2=0
Solving the equation by using quadratic formula,
t= [-b±√(b²-4ac)]/2a
Here,
a = 5
b = -2
c = -2
Putting all the values,
t = [2±√(4-4(5)(-2))]/10
t = [2±√44]/10
We get
t = (1/5+√11/5) and (1/5-√11/10)
Part c. Solving for t;
3/√5 = √[(1 + t)² + (1 − 2t)²]
Squaring both sides,
9/5 = (1+t)²+(1-2t)²
9/5 = 1+t²+2t+1+4t²-4t
9 = 5+5t²+10t+5+20t²-20t
25t²-10t+1=0
t = [-b±√(b²-4ac)]/2a
Here,
a = 25
b = -10
c = 1
Putting all the values,
t = [10±√(100-4(25))]/50
t= [10±0]/50
t= 1/5
Part d. Solving for t;
0 = √[(1 + t)²+(1 − 2t)²]
Squaring both sides,
(1+t)² =-(1-2t)²
1+t²+2t = -(1+4t²-4t)
1+t²+2t = -1-4t²+4t
5t²-2t+2=0
Solving the equation by using quadratic formula,
t = [-b±√(b²-4ac)]/2a
Here,
a = 5
b = -2
c = 2
Putting all the values,
t = [2±√(4-4(5)(-2)]/10
t = [2±√(-36)]/10
t = [2±6√(-1)]/10
√-1 is 'iota' or i.
t = (2±6i)/10
We get,
t = 1/5-3/5i and 1/5+3/5i
To know more about Quadratic formula, visit,
https://brainly.com/question/1214333
#SPJ1
These two trangles are scaled copies of one another. the area of the smaller triangle is 9 square
units.
The area of the larger triangle will be equal to 36 square units.
Triangle may be defined as a closed figure of three sides and three interior angles. Scaling of image may be defined as a ratio that can be used to represent the relationship between the shape and size of a figure and the corresponding dimensions of the actual figure or object. Dilation may be defined as the process for creating similar figures by changing the size. A graph can be defined as a pictorial representation or a diagram that represents data or values. According to question the area of smaller triangle is given as 9 square units. After scaling the length of scale copies become double and the area becomes four times. So, the area of bigger triangle is 4×9 = 36.
Learn more about the Scale images here:
brainly.com/question/13194929
#SPJ9
Complete Question:
These two triangles are scaled copies of one another. The area of the smaller triangle is 9 square units. What is the area of the larger triangle? Explain or show how you know.
There are 54 dogs in a pet store which is 90% of all pets. How many pets are in the pet store?
Let there are x pets in the store. So 90% of all pets means that 0.9x.
The equation for the dogs anf total pets in the store is,
[tex]0.9x=54[/tex]Simplify the equation to obtain the value of x.
[tex]\begin{gathered} 0.9x=54 \\ x=\frac{54}{0.9} \\ =60 \end{gathered}[/tex]So there are 60 pets in the pet store.
F(x)=x+2 g(x)=x-4 (f g)(x)=
Given functions are:
[tex]\begin{gathered} f(x)=x+2 \\ g(x)=x-4 \end{gathered}[/tex]So:
[tex]\begin{gathered} f(x)\times g(x)=(x+2)(x-4) \\ f(x)\times g(x)=x^2-4x+2x-8 \\ f(x)\times g(x)=x^2-2x-8 \end{gathered}[/tex]The graphs of 4 different functions are given below. Find the average rate of change for each function on the interval [1,5]. Write answers below graphs.
Using it's concept, the average rate of change for each function on the interval [1,5] is given as follows:
Function 1: -0.5.Function 2: -0.5.Function 3: -0.5.Function 4: -0.5.What is the average rate of change of a function?The average rate of change of a function is given by the change in the output divided by the change in the input. Hence, over an interval [a,b], the average rate of change is given according to the following rule:
[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]
For the each function, we have that the numeric values are given as follows:
f(1) = 4.f(5) = 2.Hence the rate is given by:
r = (2 - 4)/(5 - 1) = -0.5.
Even though they have different formats, they have the same average rate of change, as the only values mattering to calculate the rate of change are the values at x = 1 and x = 4, which are the same for each graph.
More can be learned about the average rate of change of a function at https://brainly.com/question/13930624
#SPJ1
Andy measures some pieces of yarn. He records the lengths in inches on the line plot.5, 4^1/2, 4^3/4, 4, 4^1/2, 3^3/4, 4^1/4, 4^1/2, 5what mistake does Andy make on the plot line A .Andy does not draw enough Xs at 4^3/4B Andy did not graph 3^3/4C .Andy measures 9 pieces of yarn but only graph 5 pieces D . Andy draws too many Xs for 4^1/2 inches and 5 inches
ANSWER
B. Andy did not graph 3 3/4
EXPLANATION
That measure, 3 3/4 in, is not graphed on the plot line.
I need to know the Highest, Second Highest, Middle, Second lowest, and lowest interest rate in that order.
Norman, this is the solution to the exercise:
According to the information provided, we have:
• Highest interest rate = 8.81%
,• Second highest interest rate = 8 4/5 % = 8.8%
,• Middle interest rate = 8 1/2 % = 8.5%
,• Second lowest interest rate = 8.313%
,• Lowest interest rate = 8.3%
Find: f^-1(x)=2x/2+3x make sure it is 1-1, if so find the inverse and verify by composition in both directions
The function is one-one can be dtermined by using horizontal line test. If horizontal line on the graph of function intersect the function more than once then such function is not one-one. The graph of function is,
Since horizontal lines intersect the curve of function only once so function is one-one.
Determine the inverse of the function.
[tex]y=\frac{2x}{2+3x}[/tex]Interchange x with y and y with x and simplify the obtain equation for x.
[tex]\begin{gathered} y=\frac{2x}{2+3x} \\ y\cdot(2+3x)=2x \\ 2y+3xy=2x \\ 3xy-2x=-2y \\ x(3y-2)=-2y \\ x=-\frac{2y}{3y-2} \end{gathered}[/tex]Substitute y by x for the inverse of function.
[tex]f(x)=\frac{-2x}{3x-2}[/tex]So inverse of the function is -2x/(3x - 2).
So functions f and g are inverse of each other if,
[tex]f(g(x))=g(f(x))=x[/tex]Check the obtained inverse function by composition.
[tex]\begin{gathered} f^{-1}(-\frac{2x}{3x-2})=\frac{2\cdot(-\frac{2x}{3x-2})}{2+3\cdot(-\frac{2x}{3x-2})} \\ =\frac{-\frac{4x}{3x-2}}{\frac{6x-4-6x}{3x-2}} \\ =-\frac{4x}{-4} \\ =x \end{gathered}[/tex][tex]undefined[/tex]The graph shown below expresses a radical function that can be written in the form f(x) = a(x+k)lin+c. What does the graph tell you about the value of k in this function? 1 1 15 1 1 1 1 1 1 1 1 1 1 1: 18 4 A kis greater than zero. B. It is not possible to tell whether kis greater than or less than zero. c. kis less than zero. D. kequals zero.
Given,
The function of the graph is,
[tex]undefined[/tex]Write the system of equations associated with the given augmented matrix. Do not solve
The elements in the first 3 columns are the coefficients of system of equations.
The 4th column are the constants of the system of equations.
This is a system of 3 equations.
Let's write the system using the variables x, y, and z. Shown below:
[tex]\begin{gathered} 1x+1y+0z=3 \\ 0x+4y+1z=-5 \\ 1x+0y-1z=4 \end{gathered}[/tex]Let's simplify and write the equations:
[tex]\begin{gathered} x+y=3 \\ 4y+z=-5 \\ x-z=4 \end{gathered}[/tex]Find the end behavior of the polynomial function y = 9x2 + 8x + 9.
SOLUTION
[tex]\begin{gathered} Given \\ y=9x^2+8x+9 \end{gathered}[/tex]The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
The first option is the correct answer.
para el concierto de inicio de clase Se vendieron un total de 150 boletos. el costo del boleto fue de $5.00 para estudiantes y de $8.00 a los invitados. si los ingresos totales por concepto de la venta de boletos fueron de $930.00 Entonces cuántos boletos se vendieron de cada categoría?
x + y = 150
5x + 8y = 930
y = -x + 150
5x + 8(-x + 150) = 930
5x - 8x + 1200 = 930
-3x = -270
x = 90
90 + y = 150
y = 60
90 boletas para estudiantes y 60 a los invitados vendieron
what is the value of 10-¹
The value of
[tex]10^{-1}[/tex]which can be written as:
[tex]\frac{1}{10}[/tex]is
[tex]0.1[/tex]i need help with this problem
Personal finance Funding a retirement goal. Austin Miller wishes to have $800,000 in a retirement fund 20 years from now. He can create the retirement fund by making a single lump-sum deposit today. How much would Austin need to have on deposit at retirement in order to withdraw $35,000 annually over the 15 years if the retirement fund earns 4 percent? To achieve his annual withdrawal goal of $35,000 calculated in part b, how much more than the amount calculated in part a must Austin deposit today in an investment earning 4 percent annual interest?
Answer:
$389,200
$21,200
Since,
[tex]\text{PMT}=\frac{PV}{\text{PVA}}[/tex]We are to find PV when the PMT is $35000. Since the PVA is 11.12,
[tex]PV=\text{PVA}\cdot\text{PMT}[/tex][tex]PV=(11.12)(35000)=389200[/tex]Hence, Austin would need to deposit $389,200.
For the last part, we first need to solve the PV at 4% in 20 years.
The PVIF would be:
[tex]\text{PVIF}=\frac{1}{(1+0.04)^{20}}=0.46[/tex]Then, solving for the PV:
[tex]PV=800000(0.46)=368000[/tex]Now, to know how much more should Austin deposit, we need to subtract the original PV from the PV that we got from part B.
That would be,
[tex]389200-368000=21200[/tex]Austin would need to deposit $21,200 more to achieve his withdrawal goal.
Estimate.45.86 + 54.14A.90B.99C.100D.110
We need to estimate the sum:
[tex]45.86+54.14[/tex]In order to do so, we first round each term to the nearest integer. Then we add them.
We have:
[tex]\begin{gathered} 45.86\cong46\text{ (because 0.86>0.50)} \\ \\ 54.14\cong54\text{ (because 0.14<0.50)} \end{gathered}[/tex]Then, we can estimate the sum to be:
[tex]45.86+54.14\cong46+54=100[/tex]Therefore, option C is correct.
If the first two terms of a fibonacci sequence are 5, 25 then what us the next term ?
30,60,90,
Step-by-step explanation:
Solve the following system of equations.{x=2y−3 x−y=1Write your answer as an ordered pair in the form (x,y).
Solve the system of equations:
[tex]\begin{gathered} x=2y-3\ldots\ldots\ldots\text{.}(1) \\ x-y=1\ldots\ldots\ldots\text{.}(2) \end{gathered}[/tex]step 1: Substitute the value of 2y-3 for x in equation (2) and solve for y
[tex]\begin{gathered} x-y=1\ldots\ldots\ldots\text{.}(2) \\ (2y-3)-y=1 \\ 2y-3-y=1 \\ \text{ collect like terms} \\ 2y-y=1+3 \\ y=4 \end{gathered}[/tex]step 2: Solve for x by substituting 4 for y in equation (1)
[tex]\begin{gathered} x=2y-3\ldots\ldots\ldots\text{.}(1) \\ \text{put }y=4 \\ x=2(4)-3 \\ x=8-3 \\ x=5 \end{gathered}[/tex]Therefore, the solution to the system of equations in ordered pair is
[tex](x,y)=(5,4)[/tex]HELP ME PLEASE LIKE PLEASE I DONT UNDERSTAND THIS MATH
Given that:
- A race car game takes 6 points each time the player hits a cone.
- You must find the integer that represents the change in total points if the player hits 10 cones.
It is important to remember that the Set of Integers contains the negative numbers, the positive numbers, and zero.
Then, by analyzing the data given in the exercise, you can identify two integers that must be multiplied, in order to calculate the total points if the player hits 10 cones. These are:
[tex]\begin{gathered} 10\text{ } \\ -6 \end{gathered}[/tex]Since the game takes 6 points from the player each time this hits a cone, the integer that represents the points taken from the player must be negative.
In order to multiply the integers, you need to remember the Sign Rules for Multiplication:
[tex]\begin{gathered} +\cdot+=+ \\ -\cdot-=+ \\ +\cdot-=- \\ -\cdot+=- \end{gathered}[/tex]Therefore, you get this result:
[tex](10)(-6)=-60[/tex]Hence, the answer is:
Tony is laying out the design for a concrete path. The path is to be L-shaped as shown in the plan. The measurements are in millimetres. Question prompt and response areaFill in all answer spaces Lengths of timber, called formwork, mark out the edges of the path. When ordering the timber for the formwork Tony adds an extra 10% to the measured length to allow for wastage. The timber supply company sells the formwork timber in lengths of 3600 mm. How many lengths of timber does Tony need to order?
Shape area of timber measured in millimeters is 10.8 m2.
Given:
Rectangle 1's length is 4000 mm, or 4 meters.
Rectangle 1's width is 1200 mm, or 1.2 meters.
Rectangle 2's length is [6200-1200] mm, or 5 meters.
Rectangle width 2 = 1200 mm = 1.2 m
Area of the form,
Shape's area is equal to the sum of its two rectangles.
Shape area equals [(4)(1.2)] plus [(5)(1.2)].
Shape's area equals 4.8 + 6.
Length area of the shape is 10.8 m2.
Learn more about area at
https://brainly.com/question/22684014
#SPJ9
Someone help me ASAP!!!plsss
Answer:
See my photo bro. hope it can help
I need help with question #17. This a homework question!
Answer:
Explanation:
Here, we want to get the amount of money it costs each of the friends to bowl
Let us call this cost x
For the 9, the cost will be:
[tex]9\text{ }\times\text{ x = \$9x}[/tex]To rent a bowling shoe, it costs $3, the cost for all will be :
[tex]3\text{ }\times\text{ 9 = \$27}[/tex]The sum of the bowling cost and the shoe rental cost is a total $162
Mathematically, we have it that:
[tex]\begin{gathered} 9x\text{ + 27 = 162} \\ 9x\text{ = 162-27} \\ 9x=\text{ 135} \\ x\text{ = }\frac{135}{9} \\ \text{ x= 15} \end{gathered}[/tex]It costs each of the friends $15 to bowl excluding the shoe rental cost
Isotope: 239Pu
Half-life
(years): 24,100
Initial
Quantity:
Amount After
4000 Years:
3.7g
The mass of isotope 239-Pu left after 4000 years is approximately 3.3g
RadioactivityRadioactivity is the phenomenon of the spontaneous disintegration of unstable atomic nuclei to atomic nuclei to form more energetically stable atomic nuclei. Radioactive decay is a highly exoergic, statistically random, first-order process that occurs with a small amount of mass being converted to energy.
In solving the disintegration constant, we can use the half-life formula.
The half-life of a sample is given as;
[tex]T_\frac{1}{2} = \frac{ln2}{\lambda}[/tex]
Substituting the values into the equation;
[tex]T_\frac{1}{2} = \frac{ln2}{\lambda}\\24100 = \frac{ln2}{\lambda} \\\lambda = \frac{ln2}{24100} \\\lambda = 0.00002876years^-^1[/tex]
The formula of radioactivity is given as
[tex]N = N_oe^(^-^\lambda ^t)[/tex]
No = 3.7gt = 4000We can plug in the values and solve.
[tex]N = N_oe^(^-^\lambda ^t)\\N = 3.7e^-^(^0^.^0^0^0^0^2^8^7^6 ^*^4^0^0^0^)\\N = 3.297g\\N = 3.3g[/tex]
The amount left after 4000 years is 3.3g.
Learn more on radioactivity here;
https://brainly.com/question/26626062
#SPJ1
Quick
Check
Match the key aspect of a function's graph with its meaning.
f(x) <0
x-intercept
f(x) > 0
ching he Meaning of Key Features of a Graph
y-intercept
intervals of the domain where the
graph is below the x-axis
location on graph where output is
zero
location on graph where input is zero
intervals of the domain where the
graph is above the x-axis
Answer:
ramila ate four ninth of orange before lunch and two-ninths of orange after lunch.how much of the orange did she eat at All?
There are same number of packets of sweets and biscuits. There are 5 sweets
in a packet. There are 3 biscuits in a packet. Faizal counted 120 sweets and
biscuits altogether. How many packets of biscuits are there?
Y.
Answer:
There are 45 packets of biscuits
Step-by-step explanation:
Ratio Actual
5 Sweets
3 Biscuits
8 120 Total
Make a proportion:
[tex]\frac{Bicuit}{total}[/tex] = [tex]\frac{biscuits}{total}[/tex] ratio on the left and actual on the right
[tex]\frac{3}{8}[/tex] = [tex]\frac{x}{120}[/tex] x equals the unknown actual biscuit count.
8 x 15 =120, so solve for x by multiplying 3 by 15
x = 45
Answer:
15
Step-by-step explanation:
equation=
[tex]5x+3x=120\\equals..\\8x=120\\x= 15[/tex]
There are 15 packets each of both sweets and biscuits.
Check!
[tex]5(15)=75[/tex] sweets
[tex]3(15)=45[/tex] biscuits
[tex]75+45=120[/tex] sweets and biscuits altogether
Hence, there are 15 packets of biscuits.
Can you help me Find x