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.
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!!!
1 answer
1 answer