Mathematics, 08.11.2020 14:00, carog24

Given y = sin(2x - π) + 1, find the (a) derivative, (b) equation of the tangent line at x = π/2, (c) equation of the normal line at x = π/2.

Answers: 1

Show answers

Answers

answered: dee6991

(a) $y'= 2cos(2x - \pi)$

(b) $y=2x - \pi + 1$

Step-by-step explanation:

$y=sin(2x-\pi)+1$

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, $sin(2x-\pi)+1$ , then multiply this by the derivative of the inside function, $2x-\pi$ .

$\frac{d}{dx}(sin(2x-\pi)+1)$

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.

$cos(2x-\pi) \times 2$

Simplify this expression.

$2cos(2x - \pi)$

This is the derivative of $y=sin(2x-\pi)+1$ ; therefore, we can write:

$y'= 2cos(2x - \pi)$ Part (b)

In order to find the equation of the tangent line at $x=\frac{\pi}{2}$ , 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.

$y'(\frac{\pi}{2} ) = 2cos(2(\frac{\pi}{2})-\pi)$

Simplify inside the parentheses.

$y'(\frac{\pi}{2} ) = 2cos(\frac{2\pi}{2}-\pi)$ $y'(\frac{\pi}{2} ) = 2cos(\pi - \pi)$ $y'(\frac{\pi}{2} ) = 2cos(0)$ $y'(\frac{\pi}{2} ) = 2$

Now we know that the slope of the tangent line is 2.

Let's plug x = π/2 into the original function, y.

$y(\frac{\pi}{2})=sin(2(\frac{\pi}{2})-\pi)+1$

Simplify inside the parentheses.

$y(\frac{\pi}{2})=sin(0)+1$ $y(\frac{\pi}{2})= 0+1$ $y(\frac{\pi}{2})=1$

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:

We have $(x_1, \ y_1)$ , which are the x- and y- coordinates, and , which is the slope of the tangent line.

Substitute these values into the equation:

$y-(1)=2(x-(\frac{\pi}{2}))$

Distribute 2 inside the parentheses.

$y-1=2x-\frac{2 \pi}{2}$

Add 1 to both sides of the equation.

$y=2x-\frac{2\pi}{2} + 1$ $y=2x - \pi + 1$

This is the equation of the tangent line of $y=sin(2x-\pi)+1$ at $x=\frac{\pi}{2}$ .

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.

$m=-\frac{1}{2} \ \ \ \ \ \ (\frac{\pi}{2}, \ 1)$ $y-(1) = -\frac{1}{2}(x - (\frac{\pi}{2} ))$

Distribute -1/2 inside the parentheses.

$y-1=-\frac{1}{2}x + \frac{\pi}{4}$

Add 1 to both sides of the equation.

$y=-\frac{1}{2}x + \frac{\pi}{4}+1$ $y=-\frac{1}{2}x + \frac{\pi}{4} + \frac{4}{4}$ $y=-\frac{1}{2}x + \frac{\pi+4}{4}$

You can leave it written as this, or write it as:

$y=-\frac{x}{2} +\frac{\pi + 4}{4}$

This is the equation of the normal line of $y=sin(2x-\pi)+1$ at $x=\frac{\pi}{2}$ .

answered: Guest

idk

step-by-step explanation:

do it yourself, cause that made me do it myself today, in english!

answered: Guest

your question is incomplete, write all the numbers in the sequence that it tells you.

step-by-step explanation:

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 16:30, rhineharttori

Acircle has a circumference of 2 pi cm. which statement about the circumference and area is true? a comparison of the area and circumference is not possible since the area cannot be determined. the numerical values of the circumference and area of the circle are equal. the numerical value of the circumference is greater than the numerical value of the area. the numerical value of the circumference is less than the numerical value of the area.

Answers: 3

continue

Mathematics, 21.06.2019 19:40, jennifercastillo1233

If w = 6 units, x = 3 units, and y = 5 units, what is the surface area of the figure? 168 units2 240 units2 144 units2 96 units2

Answers: 1

continue

Mathematics, 21.06.2019 23:30, kathlynskare06

Side xy of triangle xyz is extended to point w, creating a linear pair with ∠wyz and ∠xyz. measure of an exterior angle what is the value of x? 64 80 100 180

Answers: 2

continue

Mathematics, 21.06.2019 23:30, Thefiles

In stepmber, daniel and justin opened a business. during their first three months of business, their average monthly profit was $51. if during their first month, they lost $29, and during their second month, they gained $115, what was their profit or loss for their third plzzz i will give you 299 points

Answers: 1

continue

Do you know the correct answer?

Given y = sin(2x - π) + 1, find the (a) derivative, (b) equation of the tangent line at x = π/2, (c)...