Answers
(a) Differentiate [tex]s(t)[/tex] to get the velocity.
[tex]s(t) = \cos(t^2 - 1)[/tex]
Use the chain rule.
[tex]u = t^2 - 1 \implies \dfrac{du}{dt} = 2t[/tex]
[tex]s(u) = \cos(u) \implies \dfrac{ds}{du} = -\sin(u)[/tex]
[tex]\implies \dfrac{ds}{dt} = \dfrac{ds}{du} \dfrac{du}{dt} = \boxed{-2t\sin(t^2-1)}[/tex]
(b) Evaluate the derivative from (a) at [tex]t=0[/tex].
[tex]\dfrac{ds}{dt}\bigg|_{t=0} = -2\cdot0\cdot\sin(0^2-1) = \boxed{0}[/tex]
(c) The object is stationary when the derivative is zero. This happens for
[tex]-2t \sin(t^2 - 1) = 0[/tex]
[tex]-2t = 0 \text{ or } \sin(t^2 - 1) = 0[/tex]
[tex]t = 0 \text{ or } t^2 - 1 = n\pi[/tex]
(where [tex]n[/tex] is any integer)
[tex]t = 0 \text{ or } t = \pm\sqrt{n\pi + 1}[/tex]
We omit the first case. In the second case, we must have [tex]n\ge0[/tex] for the square root to be defined. Then in the given interval, we have two solutions when [tex]n=\in\{0,1\}[/tex], so the times are
[tex]t_1 = \sqrt{1} = \boxed{1}[/tex]
[tex]t_2 = \boxed{\sqrt{\pi + 1}} \approx 2.035[/tex]
(d) A picture is worth a thousand words. See the attached plot. If you're looking for a verbal description, you can list as many features of the plot as are relevant, such as
• intercepts (solve [tex]s(t)=0[/tex] to find [tex]t[/tex]-intercepts and evaluate [tex]s(0)[/tex] to find the [tex]s[/tex]-intercept)
• intervals where [tex]s(t)[/tex] is increasing or decreasing (first derivative test)
• intervals where [tex]s(t)[/tex] is concave upward or concave downward (second derivative test; at the same time you can determine any local extrema of [tex]s(t)[/tex], which you can see in the plot agrees with the critical points found in (c))
Related Questions
Find all the integer solutions of each system of inequalities {y>=0 {72-y>=4
Answers
The integer solutions of each of the system of inequalities are 0, 1, 2, 3, .....65, 66, 67, 68. There are total 69 integer values that y can take.
In the question, we are given the inequalities:-
y ≥ 0 and 72 - y ≥ 4
We have to find the integer solutions of each of the system of inequalities.
From 72 - y ≥ 4, we can write,
72-4 ≥ y
68 ≥ y
Hence, from 68 ≥ y and y ≥ 0, we can write,
68 ≥ y ≥ 0
Hence, y can take 69 integer values from 0 to 68, both included.
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-(16 - 5x) simplified
Answers
Answer:
5x-16
Step-by-step explanation:
-(16 - 5x)
Use distributive property to open parenthesis.
-(16 - 5x)=
=(-1*16)+(-1*-5x)=
=(-16)+(5x)=
=5x-16
Finding x.
5x-16=0
5x=16
x=3.2
Hope this helps!
Please mark as brainliest if correct!
Find the linear function represented by the graph.
The slope of the line is
.
The y-intercept of the line is at
.
What linear function is represented by the graph?
Answers
Step-by-step explanation:
photo please ......................
Answer:
Step-by-step explanation:
need to see the graph
Find the sum of (8.6 x 106) and (3.7 x 105). Write the final answer in scientific notation.
8.97 x 105
8.97 x 106
1.23 x 1010
12.3 x 1011
Answers
Answer:
8.97 x [tex]10^{6}[/tex]
Step-by-step explanation:
To add in scientific notation, the exponents need to be the same for the bases of 10. You can either make them both to the exponent of 6 or 5. It does not matter. I made them both to the power of 5
(8.6 x [tex]10^{6}[/tex]) + (3.7 x [tex]10^{5}[/tex])
(86 x [tex]10^{5}[/tex]) + (3.7x[tex]10^{5}[/tex])
(86 x 3.7) x [tex]10^{5}[/tex]
89.7 x[tex]10^{5}[/tex] Rewrite in scientific notation
8.97 x [tex]10^{6}[/tex]
Which number is greater,
23
×
10
3
or
11
×
10
4
? How do you know? Show your work and explain your steps.
Answers
Answer:
Answer: 11x10^4 is greater. Step-by-step explanation: 23x10^3 = 23x10x10x10 = 23000. 11x10^4 = 11x10x10x10x10 = 110000. 110000 is greater than 23000.
Step-by-step explanation:
cos
The midpoint of AB is M (0, -4). If the coordinates of A are (2, -1), what are the
coordinates of B?
Answers
Answer: B(-2,-7)
Step-by-step explanation:
[tex]\displaystyle\\M(x_M,y_M)=(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}) \\\\x_A=2\ \ \ \ x_M=0\ \ \ \ y_A=-1\ \ \ \ y_M=-4\\\\x_M=\frac{x_A+x_B}{2} \\\\Multiply\ both\ parts\ of \ the \ equation\ by\ 2:\\2x_M=x_A+x_B\\\\\Rightarrow\ x_B=2x_M-x_A\\\Rightarrow \ x_B=2(0)-2\\\Rightarrow\ x_B=0-2\\\Rightarrow\ x_B=-2\\Similarly,\\y_B=2y_M-y_A\\y_B=2(-4)-(-1)1\\y_B=-8+1\\y_B=-7\\Thus,\\B(-2,-7)[/tex]
A startup sports public relations company is looking to grow. They currently represent 2 clients and plan on representing 3 additional clients each year. The following graph can be used to help determine the number of clients, y, the company will have after x years.
ased on the context, what is the range of the function?
[0, ∞)
[2, ∞)
[−7, 7]
ℝ
Answers
Based on the context, the range of this function is given by: C. [−7, 7]
What is a range?A range can be defined as the set of all real numbers that connects with the elements of a domain. This ultimately implies that, a range refers to the set of all possible output numerical values, which are shown on the y-axis of a graph.
How to identify the range of this graph?The vertical extent of a graph represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.
By critically observing the graph of function f(x) shown, we can infer and logically deduce the following:
Range = [−7, 7]
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. Use Structure is the product the same
when multiplying 22 x (-5) and multiplying
(-5) x 22? Explain.
Answers
Answer:
Yes.
Step-by-step explanation:
Let's say a=22 and b=-5
According to the Commutative Property of Multiplication, a*b=b*a. Hence, 22*(-5)=(-5)*22.
22*(-5)=
-(22*5)=-110
(-5)*22=
-(5*22)=-110
-110=-110 √
i don’t understand this i already tried but got 1.25 bad i’m pretty sure it’s wrong
Answers
Answer:
Smallest integer=14
Step-by-step explanation:
(x+x+2+x+4+x+6)/4=17
(4x+12)/4=17
4x+12=68
4x=68-12
4x=56
x=56/4
x=14
numbers are: 14, 16, 18, 20
Bombonica s-a gândit la un număr, i-a adăugat 74, rezultatul obţinut l-a mărit de 10 ori, iar din
noul rezultat a luat de 7 ori câte 86 şi a obţinut 318, număr care reprezenta numărul zilelor de când
fusese răpit împăratul. La ce număr s-a gândit?
Answers
[tex]x + 74 \cdot 10 - (7 \cdot 86) = 318[/tex]
[tex]x+74\cdot \:10-7\cdot \:86=318[/tex]
[tex]x+740-7\cdot \:86=318[/tex]
[tex]x+740-602=318[/tex]
[tex]x+138=318[/tex]
[tex]x = 318 - 138 = 80[/tex]
Imparatul s-a gandit la numarul 80.
Please Help!!
Complete the conjecture based on the pattern you observe in the examples below
1) Conjecture: The sum of any two odd numbers is…
1+1=2
1+5=6
7+9=16
9+11=20
13+21=34
101+103=204
2) Conjecture: The product of any two odd numbers is…
1*1=1
1*5=6
7*9=16
9*11=20
13*21=34
101*103=204
3) Conjecture: The product of a number (x-1) and the number (x+1) is always equal to…
3*5=4^2-1
4*6=5^2-1
5*7=6^2-1
7*9=8^2-1
8*10=9^2-1
Answers
Answer:
Hello,
Step-by-step explanation:
1) Conjecture: The sum of any two odd numbers is even
(2a+1)+(2b+1)=2a+2b+2=2(a+b+1) is even
2) Conjecture: The product of any two odd numbers is ODD
(2a+1)*(2b+1)=4ab+2b+2a+1=2(2ab+a+b)+1 is ODD
1*1=1
1*5=5
7*9=63
9*11=99
13*21=273
3) Conjecture: The product of a number (x-1) and the number (x+1) is always equal to the difference of the square of x between 1
(x-1)(x+1)=x²-x+x-1=x²-1
how do i do part (iii). the answer is 2 but i got the answer based on guessing. how do i do this w/o guessing? thank you.
Answers
Answer:
(i) 2² · 3 · 7
(ii) 3024
(iii) k = 16
(iv) 12 cm
Step-by-step explanation:
Part (i)The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, ...
To find which prime numbers multiply together to make 84, start by dividing 84 by the first prime number, 2:
⇒ 84 ÷ 2 = 42
As 42 is not a prime number, we need to divide again:
⇒ 42 ÷ 2 = 21
As 21 is not a prime number, we need to divide again.
21 is not divisible by 2, so let's try dividing by the next prime number, 3:
⇒ 21 ÷ 3 = 7
As 7 is a prime number, we can stop.
Therefore, 84 is the product of:
⇒ 84 = 2 · 2 · 3 · 7
As 2 appears two times, we can write this using exponents:
⇒ 84 = 2² · 3 · 7
Part (ii)Lowest Common Multiple (LCM): The lowest multiple shared by two or more numbers.
Prime Factors Method of finding LCM
Step 1
Find the prime factorization of each number.
From part (i): 84 = 2² · 3 · 7
Given in part (ii): 432 = 2⁴ · 3³
Step 2
Write each number as a product of primes, matching primes vertically where possible:
[tex]\begin{array}{ r c c c c c c c ccccccccc}84&=& 2 & \cdot & 2 & \cdot & & & & & 3& \cdot & & & && 7\\\\432 &=& 2 & \cdot & 2 &\cdot & 2& \cdot & 2& \cdot & 3& \cdot & 3& \cdot & 3\\\\\cline{1-17} \\\sf LCM & = & 2 & \cdot & 2 &\cdot & 2& \cdot & 2& \cdot & 3& \cdot & 3& \cdot & 3 &\cdot & 7\end{array}[/tex]
Step 3
Bring down the primes in each column then multiply the factors to get the LCM:
⇒ LCM = 2 · 2 · 2 · 2 · 3 · 3 · 3 · 7
⇒ LCM = 2⁴ · 3³ · 7
⇒ LCM = 3024
Part (iii)From part (ii) we know that the prime factorization of 432 is:
432 = 2⁴ · 3³
A perfect cube is a number multiplied by itself three times.
One of the factors of 432 is a perfect cube → 3³.
Therefore,
[tex]\begin{aligned}432 & = 2^4 \cdot 3^3\\\\\implies \dfrac{432}{2^4} & = 3^3\\\\\dfrac{432}{16} & = 3^3\end{aligned}[/tex]
So we need to divide 432 by 2⁴ to get the perfect cube 3³.
Therefore,
⇒ k = 2⁴
⇒ k = 2 · 2 · 2 · 2
⇒ k = 16
Part (iv)To find the largest possible length of the side of the square, we need to find the greatest common factor of 432 and 84.
Prime Factors Method of finding GCF
Step 1
Find the prime factorization of each number.
From part (i): 84 = 2² · 3 · 7
Given in part (ii): 432 = 2⁴ · 3³
Step 2
Write each number as a product of primes, matching primes vertically where possible:
[tex]\begin{array}{ r c c c c c c c ccccccccc}84&=& 2 & \cdot & 2 & \cdot & & & & & 3& \cdot & & & && 7\\\\432 &=& 2 & \cdot & 2 &\cdot & 2& \cdot & 2& \cdot & 3& \cdot & 3& \cdot & 3\\\\\cline{1-17} \\\sf GCF & = & 2 & \cdot & 2 & \cdot & & & & & 3& & & & && \end{array}[/tex]
Step 3
Bring down the primes in the columns where both numbers have the factor, then multiply the factors to get the GCF:
⇒ GCF = 2 · 2 · 3
⇒ GCF = 12
Therefore, the largest possible length of the side of the square is 12 cm.
HELP! (3x4)+ 52 MARKING BRAINLY!
Answers
Answer:
64
Step-by-step explanation:
[tex]3 \times 4\\= 12\\= 12 + 52\\= 64[/tex]
Answer:
64
Step-by-step explanation:
first 3 x 4 = 12
12+52=64
If f (x) = 3x - 1 and g (x) = x² + 1, find the following.
a) g(-4) + f(-2) =
b) f(1) - g(0) =
Will give brainliest and 5 star rating. Thank you. :)
Answers
The value of the functions are:
a. g(-4) + f(-2) = 24
b.. f(1) - g(0) = 1
What is a function?A function can be defined as an expression showing the relationship between two variables.
Given the functions;
f (x) = 3x - 1 g (x) = x² + 1g(-4); substitute the value of x as - 4 in g(x)
g(-4) = (-4)² + 1
g(-4) = 16 + 1
g(-4) = 17
f(-2); substitute the value of x as - 2 in f(x)
f(-2) = 3(-2) - 1
f(-2 = -7
a. g(-4) + f(-2) = 17 -(-7)
g(-4) + f(-2) = = 24
f(1) = 3(1) - 1
f(1) = 2
g(0) = (0)² + 1
g(0) = 1
b. f(1) - g(0) = 2 - 1
f(1) - g(0) = 1
Thus, the value of g(-4) + f(-2) and f(1) - g(0) are 24 and 1 respectively.
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Givien h(x) = -x- 5, find h(-6)
Answers
Answer:
h(-6) = 1
Step-by-step explanation:
Hello!
Substitute -6 for x in the equation.
Evaluateh(x) = -x - 5h(-6) = -(-6) - 5h(-6) = 6 - 5h(-6) = 1h(-6) is 1.
(c) Solve the equation 7x³ +28x² = 0
Answers
Answer:
x = - 4 , x = 0
Step-by-step explanation:
7x³ + 28x² = 0 ← factor out 7x² from each term on the left side
7x²(x + 4) = 0
equate each factor to zero and solve for x
7x² = 0 ⇒ x² = 0 ⇒ x = 0
x + 4 = 0 ⇒ x = - 4
How do I simplify this?
Answers
1- 3/4 = 0.25 which is equivalent to 1/4
the problem now looks like 7/4 / 1/4
you cancel out both 4
this problem simplified looks like 7/1
which is equivalent to 7 if you divide
Pigeonhole suppose that every student in a discrete mathematics class of 25 students is a sophomore, a junior, or a senior. is the following statement true or false:
Answers
Suppose that every student in a discrete mathematics class of 25 students is a sophomore, a junior, or a senior: FALSE
What is a sophomore?A sophomore in the United States is a student in their second year at an educational institution, usually a secondary school or a college or university, but also other types of post-secondary educational institutions. A sophomore in high school is equivalent to a tenth-grade or Class-10 student.In sports, a sophomore is a professional athlete in their second season.As in the description, it is given that a sophomore in high school is equivalent to a tenth-grade or Class-10 student.
So, the above-given statement becomes false as it says that every student is a sophomore but the class has juniors and seniors.
Therefore, the statement "suppose that every student in a discrete mathematics class of 25 students is a sophomore, a junior, or a senior" is FALSE.
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The complete question is given below:
Suppose that every student in a discrete mathematics class of 25 students is a sophomore, a junior, or a senior. Is the following statement true or false:
"The course must have at least five sophomores, or at least 20 juniors, or at least 10 seniors."
Question 2(Multiple Choice Worth 1 points)
(01.01 LC)
Evaluate the expression 5(7-4)2 = 3 + 11.
03
14
026
33
Answers
26 is the right answer .
Given expression 5(7 - 4)² ÷ 3 + 11.
BODMAS rule is an acronym that is used to remember the order of operations to be followed while solving expressions in mathematics. It stands for B - Brackets, O - Order of powers or roots, D - Division, M - Multiplication, A - Addition, and S - Subtraction. It means that expressions having multiple operators need to be simplified from left to right in this order only. First, we solve brackets, then powers or roots, then division or multiplication (whichever comes first from the left side of the expression), and then finally, subtraction or addition, whichever comes on the left side.
Using the BODMAS rule, we solve bracket first,
5(3)² ÷ 3 + 11
45 ÷ 3 + 11
= 15 + 11
= 26
after evaluate the expression we get 26 as answer.
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find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continuesm8, 0, 8, 0,
Answers
The formula for the general term and of the sequence is ±8.
What do we mean by sequence?A sequence in mathematics is an enumerated collection of objects in which repetitions are permitted and order is important. It, like a set, has members (also called elements, or terms). The length of the sequence is defined as the number of elements (which could be infinite). Unlike a set, the same elements can appear multiple times in a sequence at different positions, and the order does matter. Formally, a sequence can be defined as a function from natural numbers (the sequence's positions) to the elements at each position. The concept of a sequence can be expanded to include an indexed family, which is defined as a function from an index set that may or may not is numbered to another set of elements.To find the formula for the given sequence:
Given sequence: 8, 0, 8, 0, So, the formula cen be ±8.+8 to get the terms of odd numbers and -8 to get the terms of even numbers.Therefore, the formula for the general term and of the sequence is ±8.
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Will give brainliest.
Answers
Answer:
Option 3
Step-by-step explanation:
First tke lcm i.e. 6 then multiply the dinominator to make six.
Multiply tje above no. With the same digit the sub. The numerators
Answer:
c) 7/6
Step-by-step explanation:
The difference will be,
→ (5/3) - (1/2)
→ (10/6) - (3/6)
→ (10 - 3)/6
→ 7/6
So, the answer is 7/6.
Which shows ray C A?
Answers
If real gdp in a particular year is $80 billion and nominal gdp is $240 billion, the gdp price index for that year is 100. 300. 240. 200.
Answers
Answer:
If real GDP in a particular year is $80 billion and nominal GDP is $240 billion, the GDP price index for that year is: 300.
Step-by-step explanation:
SOLVE ALL OF THESE AND NUMBER THEM ON WHAT QUESTION THE ANSWER IS FOR
Answers
Which exponential function bests models the graph below?
A.
B.
C.
D.
D.
Answers
=. What does the expression (x+3)(9) represent? Why?
Answers
I hope this helps!! :)
Help I need some help due tomorrow
Answers
Answer:
Angle 1: 141 degrees
Angle 2: 39 degrees
Angle 3: 141 degrees
Step-by-step explanation:
Angle 2 and Angle 4 are equal since they are vertical angles. This means that Angle 2 is equal to 39 degrees.
Angle 1 and Angle 4 add up to 180 degrees. We call these supplementary angles. This means that Angle 1 is equal to 180-39=141 degrees.
Angle 1 and Angle 3 are vertical angles, so Angle 3 is also equal to 141 degrees.
misha is the 50th best as well as the 50th worst student in her grade. how many students are in misha's grade?
Answers
There are 99 students in Misha's grade.
What do we mean by Venn diagrams?A Venn diagram is a popular diagram style that shows the logical relationship between sets. It was popularized in the 1880s by John Venn (1834-1923). The diagrams are used in probability, logic, statistics, linguistics, and computer science to teach elementary set theory and to illustrate simple set relationships. To represent sets, a Venn diagram employs simple closed curves drawn on a plane. These curves are frequently circles or ellipses.So,
We'll make a Venn diagram (consider the diagram given below).Misha is common in both groups.According to the diagram, 49 + 49 + 1 = 99.Therefore, there are 99 students in Misha's grade.
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(-1)^{3}]^{6} find the value of
Answers
The value of Exponent [tex](-1^{3}) ^{6}[/tex] is 1
Exponent:
A number's exponent indicates how many times we multiply a number by itself. For example, [tex]3^{4}[/tex] indicates that we have multiplied 3 four times. The expanded form is 3×3×3×3. An exponent is also known as a number's power. It can be an integer, a fraction, a negative number, or decimals.
Law of Power of a Power: [tex](x^{a}) ^{b}[/tex]=[tex]x^{ab}[/tex] {Here a,b can be an integer, a fraction, a negative number, or decimals}
Given question is [tex](-1^{3}) ^{6}[/tex]
⇒[tex](-1^{3}) ^{6}[/tex]=[tex](-1)^{6.3}[/tex]{ using Law of Power of a Power}
⇒[tex](-1)^{18}[/tex]=1 { -1 raised any even number= positive one}
Therefore,The value of Exponent [tex](-1^{3}) ^{6}[/tex] is 1
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(x+2, y+3)
help please
Answers
Answer:
is that the equation ? is that all they gave you
im going to answer but do you mean
(x +2y+3)
Step-by-step explanation: