Find the equation of the line tangent to y = sin(x) going through х = pi/4​

Answers

Answer 1

Answer:

[tex]\displaystyle y - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \bigg( x - \frac{\pi}{4} \bigg)[/tex]

General Formulas and Concepts:

Algebra I

Coordinates (x, y)

Functions

Function Notation

Point-Slope Form: y - y₁ = m(x - x₁)

x₁ - x coordinate y₁ - y coordinate m - slope

Pre-Calculus

Unit Circle

Calculus

Derivatives

The definition of a derivative is the slope of the tangent line

Derivative Notation

Trig Derivative:                                                                                                          [tex]\displaystyle \frac{d}{dx}[sin(u)] = u'cos(u)[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle y = sin(x)[/tex]

[tex]\displaystyle x = \frac{\pi}{4}[/tex]

Step 2: Differentiate

Trig Derivative:                                                                                                 [tex]\displaystyle y' = cos(x)[/tex]

Step 3: Find Tangent Slope

Substitute in x [Derivative]:                                                                              [tex]\displaystyle y' \bigg( \frac{\pi}{4} \bigg) = cos \bigg( \frac{\pi}{4} \bigg)[/tex]Evaluate [Unit Circle]:                                                                                       [tex]\displaystyle y' \bigg( \frac{\pi}{4} \bigg) = \frac{\sqrt{2}}{2}[/tex]

Step 4: Find Tangent Equation

Substitute in x [Function y]:                                                                             [tex]\displaystyle y \bigg( \frac{\pi}{4} \bigg) = sin \bigg( \frac{\pi}{4} \bigg)[/tex]Evaluate [Unit Circle]:                                                                                       [tex]\displaystyle y \bigg( \frac{\pi}{4} \bigg) = \frac{\sqrt{2}}{2}[/tex]Substitute in variables [Point-Slope Form]:                                                     [tex]\displaystyle y - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \bigg( x - \frac{\pi}{4} \bigg)[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Derivatives

Book: College Calculus 10e


Related Questions

what is the decimal equivalent of 7/20

Answers

7/20 x 5= 35/100
35/100= .35

I need to know the answer please

Answers

Focusing on the center point of f(x) (0,0), we can see that it has moved to the left 4 units and up 3 units.

g(x) = [tex](\sqrt[3]{x + 4}) + 3[/tex]

Option C

Hope this helps!

What is the distance between (-5,-5) and (-9,-2)

Answers

Answer:

A (5)

Step-by-step explanation:

The distance is the slope/gradientIn the pythogaras theorem [tex]c^{2} = a^{2} + b^{2}[/tex],

        c represents the slope and a and b represent the two shorter sides of the right angled triangle ( x,y)

x = -9 - (-5 ) = -9 +5 = -4y = -2 - (-5) = -2 +5 = 3

       [tex]c^{2}[/tex] = [tex]-4^{2} + 3^{2}[/tex]

            = 16 + 9

             = 25,

therefore [tex]\sqrt{c^{2} }[/tex] = [tex]\sqrt{25}[/tex]

                   c = 5

                                                                                     

Find TAN
Instructions: Find the value of the trigonometric ratio. Make
sure to simplify the fraction if needed.

Answers

tan Z =opposite/adjacenttan Z=50/14tan Z=25/7

please mark this answer as brainlist

Which of the following can be used as "reasons" in a two-column proof?

Answers

Answer:

A definition and a theorem can be used as a reason in a two-column proof. A two column proof is assembled into statement and reason columns, where each statement should have verified reason.

Step-by-step explanation:

Answer:

Definitions and algebraic properties

Step-by-step explanation:

an expression equivalent to 12+ 21 is

Answers

Answer:

33

Step-by-step explanation:

The answer is 1628. Because 16 x 21 = 28 x 12 = 336

Hope this helps

Vehicles that get more than 40 miles per gallon can cross one county’s new bridge for free. Which graph shows the fuel-use rate of vehicles that have to pay to cross the bridge?

Answers

Answer:

D

Step-by-step explanation:

More than 40 miles per gallon

Open circle at 40 and line goes to the left

Answer: the answer is C

Step-by-step explanation:

because it is a open circle going to the left

3^x= 3*2^x
solve this equation

Answers

Answer:

[tex] {3}^{x} = 3 \times {2}^{x} \\ x = \frac{ln(3)}{ln(3) -ln(2) } = \frac{1.09}{1.09 - 0.69} \\ x = 2.7[/tex]

I hope I helped you^_^

Choose the expression that represents “divide 0.04 by n.”

Answers

Answer:

.04/n=4/100n or 4÷100n

Is interquartile range a measure of center or a measure of variation?

Answers

Answer:

The interquartile range is the middle half of the data that is in between the upper and lower quartiles. ... The interquartile range is a robust measure of variability in a similar manner that the median is a robust measure of central tendency.

Three numbers are in the ratio of 1:2:4. If 3 is added to the first and 8 is subtracted from the third, the new numbers will be the first and third terms of an A.P., whose second term is the second number. Find the original numbers.​

Answers

9514 1404 393

Answer:

  5, 10, 20

Step-by-step explanation:

Suppose the three numbers are x, 2x, and 4x. Then they have the required ratios. After the transformation, we have ...

  ((x+3) +(4x -8))/2 = 2x . . . . . 2nd term is average of 1st and 3rd

  5x -5 = 4x   ⇒   x = 5

The original numbers are 5, 10, 20.

_____

After the adjustment, the arithmetic sequence is 8, 10, 12.

4
5
7
11
19
?

a. 41
b. 35
c. 23
d. 29

Answers

Answer:

35

Step-by-step explanation:

The pattern is adding powers of 2.

4+1=5 (exception)

5+2=7

7+4=11

11+8=19

19+16=35

Answer:

35

Step-by-step explanation:

4 + 1 = 5

5 + (1 × 2) = 5 +2 =7

7 + (2×2) = 7 + 4 = 11

11 +(4×2) = 11 + 8 = 19

19 + (8×2) = 19 + 16 = 35

Find the greatest common factor of 65a3b4 and 39a4b5.

Answers

Step-by-step explanation: Let's begin by finding the

greatest common factor for the numbers 65 and 39.

I would make a factor tree and break up 65 and 39.

So 65 is 13 x 5 and 39 is 13 x 3.

Since the 13's match up, the greatest

common factor between 65 and 39 is 13.

For the variables, we use the smallest power on each of them.

So we use a^3 and b^4 to get 13a^3b^4 as our GCF.

what’s the value of x? and what’s the measure of angel JHK?

Answers

Answer:

x = 14

JHK =  21

Step-by-step explanation:

The angles are vertical angles and vertical angles are equal

3x-21 = x+7

Subtract x from each side

3x-x -21 = x+7-x

2x-21 = 7

Add 21 to each side

2x-21+21 = 7+21

2x = 28

Divide by 2

2x/2 =28/2

x = 14

JHK = 3x-21 = 3(14) -21 = 42-21 = 21

Answer:

Because ∠GHI and ∠JHK are vertical angles, they're congruent. Therefore, set their angle measures equal to each other & solve for x.

[tex]x+7=3x-21\\x-3x=-7-21\\-2x=-28\\x=\frac{-28}{-2} =14\°[/tex]

Substitute in the value of x to find ∠JHK:

[tex]3x-21=3(14)-21=42-21=21\°[/tex]

Write the quadratic equation whose roots are 2 and -4 and whose leading coefficient is 2

Answers

Answer:

2x^2+4x-16

Step-by-step explanation:

The quadratic can be written as

f(x) = a(x-z1)(x-z2) where z1 and z2 are the roots

f(x) = a (x-2)(x- -4)

a is the leading coefficient

f(x) = 2(x-2)(x+4)

     = 2(x^2 -2x+4x-8)

     = 2(x^2 +2x-8)

     = 2x^2 +4x-16

Oscar has 1/5 of a jar of mustard. He puts equal amounts of the mustard onto 7 sandwiches and uses all of the mustard. What fraction of a jar of mustard does each sandwich have?

Answers

Answer:

1/35 jar of mustard yuh yuh

A triangle has base of 7 1/8 feet and height 6 1/4 feet. Find the area of a triangle as a mixed number.

Answers

Answer: The area is 22 17/64.

Step-by-step explanation:

base = 7 1/8 = 57/8

height = 6 1/4 = 25/4

area = 1/2*b*h

= 1/2*57/8*25/4

= 1425/64

= 22 17/64

what is an example of a quintic bionomial?

Answers

3x^5+2

quantic means with the degree of 5 and binomial means two terms.

could anyone help me solve this? I’ve had several questions like this and I don’t understand how to solve it. I’ll give brainliest:)

Answers

Answer:

-2, - 1, - 2 and - 3

Step-by-step explanation:

As the graph depicts an odd function, it will follow the rule f(-x) = - f(x)

If a and b are positive numbers, find the maximum value of f(x) = x^a(2 − x)^b on the interval 0 ≤ x ≤ 2.

Answers

Answer:

The maximum value of f(x) occurs at:

[tex]\displaystyle x = \frac{2a}{a+b}[/tex]

And is given by:

[tex]\displaystyle f_{\text{max}}(x) = \left(\frac{2a}{a+b}\right)^a\left(\frac{2b}{a+b}\right)^b[/tex]

Step-by-step explanation:

Answer:

Step-by-step explanation:

We are given the function:

[tex]\displaystyle f(x) = x^a (2-x)^b \text{ where } a, b >0[/tex]

And we want to find the maximum value of f(x) on the interval [0, 2].

First, let's evaluate the endpoints of the interval:

[tex]\displaystyle f(0) = (0)^a(2-(0))^b = 0[/tex]

And:

[tex]\displaystyle f(2) = (2)^a(2-(2))^b = 0[/tex]

Recall that extrema occurs at a function's critical points. The critical points of a function at the points where its derivative is either zero or undefined. Thus, find the derivative of the function:

[tex]\displaystyle f'(x) = \frac{d}{dx} \left[ x^a\left(2-x\right)^b\right][/tex]

By the Product Rule:

[tex]\displaystyle \begin{aligned} f'(x) &= \frac{d}{dx}\left[x^a\right] (2-x)^b + x^a\frac{d}{dx}\left[(2-x)^b\right]\\ \\ &=\left(ax^{a-1}\right)\left(2-x\right)^b + x^a\left(b(2-x)^{b-1}\cdot -1\right) \\ \\ &= x^a\left(2-x\right)^b \left[\frac{a}{x} - \frac{b}{2-x}\right] \end{aligned}[/tex]

Set the derivative equal to zero and solve for x:

[tex]\displaystyle 0= x^a\left(2-x\right)^b \left[\frac{a}{x} - \frac{b}{2-x}\right][/tex]

By the Zero Product Property:

[tex]\displaystyle x^a (2-x)^b = 0\text{ or } \frac{a}{x} - \frac{b}{2-x} = 0[/tex]

The solutions to the first equation are x = 0 and x = 2.

First, for the second equation, note that it is undefined when x = 0 and x = 2.

To solve for x, we can multiply both sides by the denominators.

[tex]\displaystyle\left( \frac{a}{x} - \frac{b}{2-x} \right)\left((x(2-x)\right) = 0(x(2-x))[/tex]

Simplify:

[tex]\displaystyle a(2-x) - b(x) = 0[/tex]

And solve for x:

[tex]\displaystyle \begin{aligned} 2a-ax-bx &= 0 \\ 2a &= ax+bx \\ 2a&= x(a+b) \\ \frac{2a}{a+b} &= x \end{aligned}[/tex]

So, our critical points are:

[tex]\displaystyle x = 0 , 2 , \text{ and } \frac{2a}{a+b}[/tex]

We already know that f(0) = f(2) = 0.

For the third point, we can see that:

[tex]\displaystyle f\left(\frac{2a}{a+b}\right) = \left(\frac{2a}{a+b}\right)^a\left(2- \frac{2a}{a+b}\right)^b[/tex]

This can be simplified to:

[tex]\displaystyle f\left(\frac{2a}{a+b}\right) = \left(\frac{2a}{a+b}\right)^a\left(\frac{2b}{a+b}\right)^b[/tex]

Since a and b > 0, both factors must be positive. Thus, f(2a / (a + b)) > 0. So, this must be the maximum value.

To confirm that this is indeed a maximum, we can select values to test. Let a = 2 and b = 3. Then:

[tex]\displaystyle f'(x) = x^2(2-x)^3\left(\frac{2}{x} - \frac{3}{2-x}\right)[/tex]

The critical point will be at:

[tex]\displaystyle x= \frac{2(2)}{(2)+(3)} = \frac{4}{5}=0.8[/tex]

Testing x = 0.5 and x = 1 yields that:

[tex]\displaystyle f'(0.5) >0\text{ and } f'(1) <0[/tex]

Since the derivative is positive and then negative, we can conclude that the point is indeed a maximum.

Therefore, the maximum value of f(x) occurs at:

[tex]\displaystyle x = \frac{2a}{a+b}[/tex]

And is given by:

[tex]\displaystyle f_{\text{max}}(x) = \left(\frac{2a}{a+b}\right)^a\left(\frac{2b}{a+b}\right)^b[/tex]

Only the top runners in the world can finish a marathon in close to 2 hours. This graph
shows the pace that one such runner followed in a race. What is the unit rate?
Marathon Pace
N
28
24
20
Distance (mi)
10
12
B
8
A
4
0
0 20 40 60 80 100120140
Time (min)

Answers

Answer:

the unit rate is 4mi per minute.

Step-by-step explanation:

if you find the slope between point A and point B, you see that the line rises at 4 mi (distance) per minute (time). thus the unit rate is 4mi per minute. please correct me if I'm wrong, I hope this answer helps.

Will give brainliest answer

Answers

It is A) Tamara’s work is correct

When graphed you can see that the function is even

The number of patients treated at Dr. Frank's dentist office each day was recorded for nine days: 18, 19, 19, 4, 14, 8, 22, 3, 1. Using the given data, find the mean for this sample.

Answers

Answer:

12

Step-by-step explanation:

To find the mean, add up all the numbers

18+ 19+ 19+ 4+14+ 8+ 22+ 3+ 1

108

Then divide by the number of terms

108/9 =

12

18+19+19+4+14+8+22+3+1 = 108
108 divided by 9 is 12
your mean is 12

Help me with this question plz

Answers

9514 1404 393

Answer:

  17

Step-by-step explanation:

The points at the ends of the interval are ...

  (0, f(0)) = (0, 0)

  (7, f(7)) = (7, 119)

The average rate of change is given by the slope formula:

  m = (y2 -y1)/(x2 -x1)

  m = (119 -0)/(7 -0) = 119/7 = 17

What is the 21st term of the sequence with a1 = -6 and d = 4?

a) 70
b) 74
c) -40
d) 78

Answers

Answer:

74

Step-by-step explanation:

for this question you have to use the formula of the nth term which is Tn=a+(n-1)d,

T21=-6+(21-1)4

=-6+(20)4

=-6+80

=74

I hope this helps

Mrs. Gomez has two kinds of flowers in her garden. The ratio of lilies to daisies in the garden is 5:2

If there are 20 lilies, what is the total number of flowers in her garden?

Answers

Answer:

28

Step-by-step explanation:

5 : 2

since this is a simplified ratio, they have a common factor. let's say it is 'x'

so now :

5x : 2x

we know that 5x is lilies, and we also know that she has 20 lilies, so:

5x = 20

x = 4

the daisies would be 2x so 2*4 = 8

total flowers is 20 + 8

28

Scientists have steadily increased the amount of grain that farms can produce each year. The yield for farms in France is given by y=−2.73x2+11000x−11000000 where x is the year and y is the grain yield in kilograms per hectare (kg/ha).

What does the y-intercept of this function represent?

Answers

9514 1404 393

Answer:

  the yield in year 0

Step-by-step explanation:

The y-value is the yield for farms in France in year x. The y-value when x=0 is the yield for farms in France in year 0.

_____

Additional comment

The reasonable domain for this function is approximately 1843 ≤ x ≤ 2186. The function is effectively undefined for values of x outside this domain, so the y-intercept is meaningless by itself.

Help me out!! Anyone

Answers

Answer:

4:10

Step-by-step explanation:

if they have to wait for plane B and it arrives every 10 mins then 4:10 is the anser

The average of three numbers is 16 if one of the numbers is 18 what is the sum of the other two numbers

Answers

Answer:

sum of two numbers is 30

Step-by-step explanation:

let three numbers are x,y,z

average =x+y+z/3

x+y+z/3=16

x+y+z=48......(1)

The sum of two numbers is 18.

according to condition:

let x=18

subtitute x=18 in (1)

18+y+z=48

y+z=48-18

y+z=30

Been stuck on this since yesterday !!?!?

Answers

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

I have uploaded a graph for you. The x axis is the number of years. The y axis is the salary multiplied by 1000. I should have made the multiplication factor 10000 but 1000 will do.

The 5 given points are plotted in red. The blue line is the function.

The function is y = 5x + 35. That means for every year you add 5 times the year onto the salary.

No years is 35000

1 year is 1 * 5000 + 35000

2 years is 2 * 5000 + 35000 = 45000

6 years is 5 * 5000 * 35000 = 65000

and so on.

The point you want is x = 12

12 years is 12 * 5000 + 35000 = 95000

Forgot the graph

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