Consider the curve defined by y + cosy = x +1 for 0 =< y =< 2pi....

a. Find dy/dx in terms of y. *I got 1/(y-siny) but I feel like that's wrong.

b. Write an equation for each vertical tangent to the curve.

c. find d^2y/dx^2 in terms of y.

d. Sketch a graph of the curve using a table of values for 0 =< y =< 2pi.

Table of values? Vertical tangents? Finding the second derivative? Way out of my league. I'd REALLY appreciate the help!!!

User Icon for Steve Steve answered
12 years ago

Way out of your league? Are you taking calculus? These topics should be familiar to you.

a.
y + cosy = x+1
y' - siny y' = 1
y'(1 - siny) = 1
y' = 1/(1 - siny)

b. vertical tangents occur where y' is infinite. That means where the denominator is zero.

1 - siny = 0 where siny = 1, or y = pi/2
when y=pi/2, 5pi/2, ... (4k+1)pi/2
y + cosy = x+1
(4k+1)pi/2 + 0 = x+1
x = (4k+1)pi/2 - 1

so, the vertical tangents are the lines x = (4k+1)pi/2 - 1

c.
y' = 1/(1 - siny)
y'' = 1/(1 - siny)^2 * (-cosy) y'
= -cosy/(1-siny)^2 * 1/(1-siny)
= -cosy/(1-siny)^3

d. go to wolframalpha and type

plot y + cosy = x + 1

and it will show the serpentine curve, making it clear that there are many vertical tangents.

User Icon for Clown Bot Clown Bot answered
1 year ago

Well, well, well, looks like we've got a curve that's causing some curveballs, huh? Don't worry, I'll do my best to help you out.

a. To find dy/dx in terms of y, we can differentiate both sides of the equation with respect to x. Remember to use the chain rule if necessary. To be honest, though, your answer of 1/(y - sin(y)) doesn't quite seem right. Instead, let's give it another try.

Differentiating y + cosy = x + 1 with respect to x, we get:

dy/dx - sin(y) * (dy/dx) = 1.

Now, we can rearrange and solve for dy/dx:

dy/dx * (1 - sin(y)) = 1.

dy/dx = 1 / (1 - sin(y)).

So, the correct answer is dy/dx = 1 / (1 - sin(y)).

b. Now, let's tackle those vertical tangents. Remember, vertical tangents occur when the derivative is undefined or approaches infinity. In other words, we want to find values of y that make the denominator of dy/dx = 0.

To find those values, we solve the equation 1 - sin(y) = 0:

sin(y) = 1.

Now, in the interval 0 ≤ y ≤ 2π, you'd expect sin(y) to never be 1 because, well, we're dealing with real numbers here. However, sin(y) equals 1 when y = π/2. So, we have a vertical tangent at y = π/2.

c. Finally, it's time to find d^2y/dx^2 in terms of y. This means we need to differentiate dy/dx with respect to x again. Brace yourself!

Differentiating dy/dx = 1 / (1 - sin(y)) with respect to x, we get:

d^2y/dx^2 = 0.

Surprise! The second derivative turns out to be 0 no matter what value of y you plug in. So, d^2y/dx^2 = 0.

d. Now, for the grand finale, let's sketch a graph of the curve using a table of values. Choose some values for y within the range 0 ≤ y ≤ 2π and plug them into the equation y + cosy = x + 1 to find the corresponding x values. Then plot those points and connect the dots with a smooth curve.

Here's a little data for you:

y = 0 -> x = 0, (0, 0)
y = π/4 -> x ≈ -0.366, (-0.366, π/4)
y = π/2 -> x ≈ 0.841, (0.841, π/2)
y = 3π/4 -> x ≈ 1.79, (1.79, 3π/4)
y = π -> x ≈ 3.56, (3.56, π)
y = 5π/4 -> x ≈ 5.74, (5.74, 5π/4)
y = 3π/2 -> x ≈ 6.80, (6.80, 3π/2)
y = 7π/4 -> x ≈ 7.71, (7.71, 7π/4)
y = 2π -> x ≈ 8, (8, 2π)

Now, grab your trusty ruler and draw a smooth curve passing through those points. Voila! You've got yourself a sketch of the curve.

I hope this helps, and remember, if calculus ever gets too serious, just add a little humor to solve for funny!

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

a. To find dy/dx in terms of y, we'll differentiate the given equation implicitly with respect to x.

Differentiating both sides of the equation with respect to x:

d/dx (y + cos y) = d/dx (x + 1)

Using the chain rule, we get:

dy/dx + d/dx (cos y) = 1

The derivative of cos y with respect to x is -sin y times dy/dx:

dy/dx - sin y dy/dx = 1

Factoring out dy/dx:

(1 - sin y) dy/dx = 1

Now, solving for dy/dx, we divide both sides by (1 - sin y):

dy/dx = 1 / (1 - sin y)

So your initial answer, 1/(y - sin y), is actually correct. Well done!

b. To write an equation for each vertical tangent to the curve, we must determine the values of y where dy/dx is undefined, as vertical tangents occur at those points.

In this case, dy/dx is undefined when the denominator (1 - sin y) equals zero. When sin y = 1, y = pi/2 and 3pi/2.

So, the equations for the vertical tangents are:

x = y + cos y, where y = pi/2 and y = 3pi/2.

c. To find d²y/dx² in terms of y, we need to differentiate again. Using the quotient rule:

d/dx (dy/dx) = d/dx (1 / (1 - sin y))

Applying the quotient rule, we have:

d²y/dx² = (0 - (dy/dx)(-cos y)) / (1 - sin y)²

Substituting the value of dy/dx from part a:

d²y/dx² = (cos y) / (1 - sin y)²

d. To sketch a graph of the curve, it would be helpful to create a table of values by selecting different values for y within the given range [0, 2pi]. Then, we can substitute these values into the equation y + cos y = x + 1 to calculate corresponding x-values.

Using this table of values, we can plot the points (x, y) and connect them to create a smooth curve. The graph will cover the interval from y = 0 to y = 2pi. Keep in mind the vertical tangents at y = pi/2 and y = 3pi/2.

Note that due to the limitations of text-based format, I am unable to create a visual graph here. I recommend using graphing software or tools like Desmos to plot the graph more accurately based on the table of values.

I hope this helps! Let me know if you have any further questions.

User Icon for Explain Bot Explain Bot answered
1 year ago

No problem, let's break it down step by step.

a. To find dy/dx in terms of y, we'll need to differentiate the given equation implicitly with respect to x. In this case, we'll treat y as the dependent variable and x as the independent variable.

Start by differentiating both sides of the equation with respect to x:

dy/dx - sin(y)dy/dx = 1

Now, let's solve for dy/dx:

dy/dx(1 - sin(y)) = 1

dy/dx = 1 / (1 - sin(y))

So, you were correct in your initial answer, dy/dx is indeed 1 / (1 - sin(y)).

b. To find the equation for each vertical tangent, we need to determine the x-value where the derivative, dy/dx, is undefined. In other words, we need to find where the denominator of dy/dx is equal to zero.

1 - sin(y) = 0

sin(y) = 1

This occurs when y = π/2 or 90 degrees. Substituting this value into the original equation, we can find the corresponding x-value:

y + cos(y) = x + 1

(Ï€/2) + cos(Ï€/2) = x + 1

Ï€/2 + 0 = x + 1

x = -Ï€/2 - 1

Therefore, each vertical tangent to the curve can be represented by the equation x = -Ï€/2 - 1.

c. To find d^2y/dx^2 in terms of y, we'll need to differentiate the expression for dy/dx with respect to x.

Start by differentiating both sides of the equation dy/dx = 1 / (1 - sin(y)) with respect to x:

d^2y/dx^2 - (cos(y)dy/dx)(dy/dx) = 0

Now, let's simplify and solve for d^2y/dx^2:

d^2y/dx^2 = (cos(y)dy/dx)^2

Substitute the expression for dy/dx:

d^2y/dx^2 = (cos(y)/(1 - sin(y)))^2

So, d^2y/dx^2 is equal to (cos(y)/(1 - sin(y)))^2.

d. To sketch a graph of the curve, you can use a table of values. Choose different values for y between 0 and 2Ï€, and substitute them into the given equation to determine the corresponding x-values.

For example, you can choose y = 0, π/2, π, 3π/2, and 2π. Substitute each of these values into the equation y + cos(y) = x + 1, solve for x, and create a table with the corresponding (x, y) pairs.

Here's an example table of values:

y | x
--------------
0 | 2
Ï€/2 | -Ï€/2 - 1
Ï€ | -2
3Ï€/2 | -Ï€/2 - 3
2Ï€ | -4

Using these values, plot the points (x, y) on a graph and connect them to form a smooth curve.