Answer:
(a) [tex]y'= 2cos(2x - \pi)[/tex]
(b) [tex]y=2x - \pi + 1[/tex]
(c) [tex]y=-\frac{x}{2} +\frac{\pi + 4}{4}[/tex]
Step-by-step explanation:
[tex]y=sin(2x-\pi)+1[/tex]
Part (a)Find the derivative of this function by using the chain rule and the power rule.
We know that the derivative of sinx = cosx. Find the derivative of this entire function first, [tex]sin(2x-\pi)+1[/tex], then multiply this by the derivative of the inside function, [tex]2x-\pi[/tex].
[tex]\frac{d}{dx}(sin(2x-\pi)+1)[/tex]Use the chain rule to find the derivative of sin(2x - π) + 1, which is cos(2x - π), then multiply this by the derivative of (2x - π). The derivative of π is 0, because it is a constant. The derivative of 2x is 2 based on the Power Rule.
[tex]cos(2x-\pi) \times 2[/tex]Simplify this expression.
[tex]2cos(2x - \pi)[/tex]This is the derivative of [tex]y=sin(2x-\pi)+1[/tex]; therefore, we can write:
[tex]y'= 2cos(2x - \pi)[/tex] Part (b)In order to find the equation of the tangent line at [tex]x=\frac{\pi}{2}[/tex], we will need to find the slope of the tangent line and the x- and y- coordinates (we already know the x- cord).
The steps to finding the equation of the tangent line at a certain are:
Plug into y' to find the slope of the tangent line.Plug into y to find the (x, y) coordinates.Use point-slope to write our equation in slope-intercept form.We know that y' = 2cos(2x - π). Let's plug x = π/2 into this equation for x to find the slope of the tangent line.
[tex]y'(\frac{\pi}{2} ) = 2cos(2(\frac{\pi}{2})-\pi)[/tex]Simplify inside the parentheses.
[tex]y'(\frac{\pi}{2} ) = 2cos(\frac{2\pi}{2}-\pi)[/tex] [tex]y'(\frac{\pi}{2} ) = 2cos(\pi - \pi)[/tex] [tex]y'(\frac{\pi}{2} ) = 2cos(0)[/tex] [tex]y'(\frac{\pi}{2} ) = 2[/tex]Now we know that the slope of the tangent line is 2.
Let's plug x = π/2 into the original function, y.
[tex]y(\frac{\pi}{2})=sin(2(\frac{\pi}{2})-\pi)+1[/tex]Simplify inside the parentheses.
[tex]y(\frac{\pi}{2})=sin(0)+1[/tex] [tex]y(\frac{\pi}{2})= 0+1[/tex] [tex]y(\frac{\pi}{2})=1[/tex]This tells us that the y-value, when x = π/2, equals 1. Our coordinates that we can use are (π/2, 1).
Now we can use point-slope form to write an equation for the tangent line to y at x = π/2.
Point-slope equation:
[tex]y-y_1=m(x-x_1)[/tex]We have [tex](x_1, \ y_1)[/tex], which are the x- and y- coordinates, and [tex]m[/tex], which is the slope of the tangent line.
Substitute these values into the equation:
[tex]y-(1)=2(x-(\frac{\pi}{2}))[/tex]Distribute 2 inside the parentheses.
[tex]y-1=2x-\frac{2 \pi}{2}[/tex]Add 1 to both sides of the equation.
[tex]y=2x-\frac{2\pi}{2} + 1[/tex] [tex]y=2x - \pi + 1[/tex]This is the equation of the tangent line of [tex]y=sin(2x-\pi)+1[/tex] at [tex]x=\frac{\pi}{2}[/tex].
Part (c)In order to find the equation of the normal line at x = π/2, we can use the information that the tangent line is perpendicular to the normal line.
This information is helpful because this means that their slopes are opposite reciprocals.
Let's use the point-slope equation again, but instead of m = 2, m will be the opposite reciprocal of 2 ⇒ -1/2. We will still use the same coordinate points.
[tex]m=-\frac{1}{2} \ \ \ \ \ \ (\frac{\pi}{2}, \ 1)[/tex] [tex]y-(1) = -\frac{1}{2}(x - (\frac{\pi}{2} ))[/tex]Distribute -1/2 inside the parentheses.
[tex]y-1=-\frac{1}{2}x + \frac{\pi}{4}[/tex]Add 1 to both sides of the equation.
[tex]y=-\frac{1}{2}x + \frac{\pi}{4}+1[/tex] [tex]y=-\frac{1}{2}x + \frac{\pi}{4} + \frac{4}{4}[/tex] [tex]y=-\frac{1}{2}x + \frac{\pi+4}{4}[/tex]You can leave it written as this, or write it as:
[tex]y=-\frac{x}{2} +\frac{\pi + 4}{4}[/tex]This is the equation of the normal line of [tex]y=sin(2x-\pi)+1[/tex] at [tex]x=\frac{\pi}{2}[/tex].
1. Determine the intervals on which each function is increasing or
decreasing
(a) f(x) = x^3 - 11/2x^2 - 4x
Answer:
[tex]\text{$f(x)$ is increasing for $(-\infty, -\frac{1}{3})$ and $(4, \infty)$} \\ \text{$f(x)$ is decreasing for $(-\frac{1}{3}, 4)$}[/tex]
Step-by-step explanation:
We have the function:
[tex]f(x)=x^3-\frac{11}{2}x^2-4x[/tex]
And we want to determine the intervals for which the function is increasing or decreasing.
We will need to first find the critical points of the function. Note that our original function is continuous across the entire x-axis.
To find the critical points, we need to find the first derivative and then solve for the x. So:
[tex]f^\prime(x)=\frac{d}{dx}[x^3-\frac{11}{2}x^2-4x][/tex]
Differentiate:
[tex]f^\prime(x)=(3x^2)-\frac{11}{2}(2x)-(4) \\ f^\prime(x)=3x^2-11x-4[/tex]
Set the derivative equal to 0 and solve for x:
[tex]0=3x^2-11x-4[/tex]
Factor:
[tex]0=3x^2-12x+x-4 \\ 0=3x(x-4)+(x-4) \\ 0=(3x+1)(x-4)[/tex]
Zero Product Property:
[tex]3x+1=0\text{ or } x-4=0 \\ x=-\frac{1}{3}\text{ or } x=4[/tex]
Therefore, our critical points are -1/3 and 4.
We can thus sketch the following number line:
<————-(-1/3)——————————————(4)—————>
Now, let’s test values for the three intervals: less than -1/3, between -1/3 and 4, and greater than 4.
For less then -1/3, we can use -1. So, substitute -1 for x for our first derivative and see which we get:
[tex]f^\prime(-1)=3(-1)^2-11(-1)-4=3+11-4=10>0[/tex]
The result is positive.
Therefore, for all numbers less than -1/3,f(x) is increasing (since its derivative is positive).
For between -1/3 and 4, we can use 0. Substitute 0 for our derivative:
[tex]f^\prime(0)=3(0)^2-11(0)-4=-4<0[/tex]
The result is negative.
Therefore, for all numbers between -1/3 and 4, f(x) is decreasing (since its derivative is negative).
And finally, for greater than 4, we can use 5:
[tex]f^\prime(5)=3(5)^2-11(5)-4=16>0[/tex]
The result is positive.
Therefore, for all numbers greater than 4, f(x) is increasing (since its derivative is positive.
Therefore:
[tex]\text{$f(x)$ is increasing for $x<-\frac{1}{3}$ and $x>4$}\\ \text{ $f(x)$ is decreasing for $-\frac{1}{3}<x<4$}[/tex]
In interval notation:
[tex]\text{$f(x)$ is increasing for $(-\infty, -\frac{1}{3})$ and $(4, \infty)$} \\ \text{$f(x)$ is decreasing for $(-\frac{1}{3}, 4)$}[/tex]
Theo wants to find the surface area and volume of a present he received
Answer:
So whats the question then ????
Need help asap
Graph f(x) = 3x + 4 and h(x) = f(x) +1
The coordinate points to plot on the graph are (0, 5) and (1, 8). The graph for the given functions are plotted below.
The given functions are f(x) = 3x + 4 and h(x) = f(x) +1.
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
Here, h(x)=3x+5
The graph of f(x) = 3x + 4 is as follows:
Graph the line using the slope and y-intercept, or two points.
Slope: 3
y-intercept: (0, 4)
The coordinate points to plot on the graph are (0, 4) and (1, 7)
The graph of h(x)=3x+5 is as follows:
Graph the line using the slope and y-intercept, or two points.
Slope: 3
y-intercept: (0, 5)
The coordinate points to plot on the graph are (0, 5) and (1, 8)
The coordinate points to plot on the graph are (0, 5) and (1, 8). The graph for the given functions are plotted below.
To learn more about the function visit:
https://brainly.com/question/28303908.
#SPJ2
What’s the answer
5 3/7 - 2 1/5
Answer:3.22
Step-by-step explanation:
Hope this helps
See attachment for math work and answer.
A panel of 12 jurors needs to be selected from a group of 28 people. How many different compositions for the jury are possible?A. 479,001, 600.B. 18, 564.C. 41, 430, 393, 164, 160,000.D. 86, 493, 225.
Complete Question
A panel of 12 jurors needs to be selected from a group of 30 people. How many different compositions for the jury are possible?
A. 479,001, 600.
B. 18, 564.
C. 41, 430, 393, 164, 160,000.
D. 86, 493, 225.
Answer:
The correct option is D
Step-by-step explanation:
From the question we are told that
The number of jurors to be selected is q = 12
The total number of people available n = 30
Generally the number of different compositions for the jury that is possible is mathematically represented as
[tex]N = ^nC_q[/tex]
Here C stands for combination hence we are going to be making use of the combination function in our calculators
[tex]N = ^{30}C_{12}[/tex]
=> [tex]N = 86,493,225[/tex]
x - 2y = - 7:(.3), (-9, )
Answer:
{x,y}={1,−3}
Step-by-step explanation:
Simplify: –6(2x – 5)
–12x – 30
–12x + 30
12x – 30
12x + 30
Answer: -12x + 30
Step-by-step explanation:
Answer:
the answer is b
Step-by-step explanation:
-6(2x-5)
-6(2x+5)
-6(2x) + 6(5)
-12x +30
help please : (
convert 10.0 cm into in
Answer:
3.93700787 Inch
Step-by-step explanation:
:)
Answer:
3.93701
Step-by-step explanation:
In order to convert cm into inches you divide the cm by 2.54.
3
3. Write an expression that is equivalent to
5
(3y + 15). (4 points)
O 3
-y + 9
5
03
-y + 15
5
09
5
Y + 9
09
- y + 15
5
Answer:
9/5y+15
Step-by-step explanation:
Use separation of variables to find, if possible, product solutions for the given partial differential equation. (Use the separation constant −λ ≠ 0. If not possible, enter IMPOSSIBLE.)
k∂^2u/∂x^2 - u = ∂u/∂t, k>0
Answer:
Following are the solution to this question:
Step-by-step explanation:
differentail equation:
[tex]yu_{xy} + u = 0..........(1)[/tex]
when [tex]u(x,y) = X(x)Y(y)[/tex]
[tex]u_x = \frac{\partial u}{\partial x} X'Y' \\\\ u_{xy} = \frac{\partial^2 u }{ \partial x\ \partial y}= X'Y'[/tex]
by substituting the value we get:
[tex]\to \int \frac{Y'}{Y} = -\lambda \int \frac{1}{y} dy\\\\[/tex]
by solving the value:
[tex]\to Y=c_3 y^{-\lambda}\\\\\therefore\ \ c_4=c_2C_3\\\\\to u(x,y) = c_4 e^{\frac{x}{\lambda}} y^{-\lambda}[/tex]
The solution of the given partial differential equation will be
[tex]u(x,y)=c_4e^\frac{x}{\lambda}y^{-\lambda}[/tex]
What is partial differential equation?The partial derivative is a way to find the slope in either the x or y direction, at the point indicated.
[tex]yu_{xy}+u=0..............................(1)[/tex]
Now if [tex]u(x,y)=X(x)Y(y)[/tex]
[tex]u_x=\dfrac{\partial u}{\partial x} X'Y'[/tex]
[tex]u_{xy}=\dfrac{\partial^2u}{\partial x\ \partial y}=X'Y'[/tex]
by substituting the value, we get:
[tex]\int \dfrac{Y'}{Y}=-\lambda\int\dfrac{1}{y}dy[/tex]
by solving the value:
[tex]Y=c_3y^{-\lambda}[/tex]
[tex]c_4=c_2c_3[/tex]
[tex]u(x,y)=c_4e^\frac{x}{\lambda}y^{-\lambda}[/tex]
Hence the solution of the given partial differential equation will be
[tex]u(x,y)=c_4e^\frac{x}{\lambda}y^{-\lambda}[/tex]
To know more about partial differential equation follow
https://brainly.com/question/2293382
Pls help me out I really need help
Answer: -0.8
Step-by-step explanation:
If start fraction 1 over 3 end fraction is equivalent to 33start fraction 1 over 3 end fraction%, what percent is equivalent to two-thirds? A. 33two-thirds% B. 150% C. 66two-thirds% D. 65%
Answer:
The answer is C 66%
Step-by-step explanation:
In a salad recipe, the ratio of carrots to cucumbers must remain constant. The table below shows some possible combinations of carrots and cucumbers.
Salad Ingredients
Carrots
Cucumbers
3
9
4
12
6
18
7
21
If only whole vegetables can be used, what is the fewest number of vegetables that can be used to make this salad?
9514 1404 393
Answer:
4
Step-by-step explanation:
The ratio is ...
carrots : cucumbers = 3 : 9 = 1 : 3
The fewest number of vegetables that can be used is 1 carrot and 3 cucumbers, a total of 4 vegetables.
Answer:
4
Step-by-step explanation:
+87 points and guaranteed brainliest if you get it right :)
If 14x + 6 / 3 = -8 + 4x, what is the value of -3x^2 - 7? TYSM
What would be the answee
Solve this equation for y.
3x - 5y = 12
Answer: y= 3/5x -12/5
help plz!!!! its urgent
2,3
1,1.5
maybe, hope this helps
What is the fourth term in a sequence defined by f(n) = 2n?
Answer:
8
Step-by-step explanation:
When we look at sequences, we have the function and we have the sequence, the sequence is f(x)... f(n)... or just f with any variable within a parenthesis right next to it.
What the variable signifies is the value that we will be replaced within the sequence, which is 2n in this case. So by your question, it we are looking for the fourth term, that means f(4) = 2n. Since n is replaced with 4, we do the same with the sequence.
2(4)
Multiply
8
What must be true in order for AABC = AEDC by the SAS Congruence Postulate?
PLEASE HELP <3
Answer:
B
Step-by-step explanation:
Find the surface area of the square pyramid shown below.
9514 1404 393
Answer:
160 units^2
Step-by-step explanation:
Each of the four triangular faces has an area of ...
A = (1/2)bh = (1/2)(8)(6) = 24 . . . square units
The square base has an area of ...
A = s^2 = 8^2 = 64
Then the total surface area is ...
A = base area + 4 · triangle area
A = 64 + 4(24) = 160 . . . square units
At a school party there are 40 cookies. The ratio of chocolate cookies to oatmeal cookies is shown in the tape diagram.
There is no tape diagram edit and put it in so I can answer, thank you.
If ∠ + ∠ = 180° and ∠ ≅ ∠, what can you conclude about ∠ and ∠?
Answer:
both ∠=90
Step-by-step explanation:
180/2=90
what is 1 plus 1000000000
Answer:
1000000001
Step-by-step explanation:
Thanks for some free answer points! :)
Determine the expression
trinomial and i believe its a degree of 4
13,84,85 are the last ones please help
Answer:
49,36,85
Step-by-step explanation:
c in step 4 comes from step 3... 49+36
a and b in step 3 come from step 2... in the same (you've got a - b and then 7 squared - 6 squared, so a is 49 and b is 36
Fill in the table using this function rule.
y=3x-1
Answer:
Step-by-step explanation:
y= 3(1)-1 --> y=2 (1,2)
y=3(5)-1 --> y=14 (5,14)
y=3(6)-1 --> y=17 (6,17)
y=3(8)-1 --> y=23 (8,23)
What is the answer to
5 + 3 x 3x -2 X=2
Answer:
x = 1/6
Step-by-step explanation:
x+1/x-4 times 5x/x+1
Answer:
Step-by-step explanation:
Q(9,10) R(-5,2) S(-8,-2) T(-1,2)
Answer:
its perpendicular
Step-by-step explanation:
Help I don’t understand this
Answer:
y = 12.5
Step-by-step explanation:
Notice these are similar triangles since they have angles of the same measure (congruent angles), then there must be a proportion among their sides such that:
[tex]\frac{y+5}{5} =\frac{14}{4}[/tex]
therefore by cross-multiplying, we get:
4(y + 5) = 14 * 5
4 y + 20 = 70
4 y = 70 - 20
4 y = 50
y = 12.5
