Math 161

Sample Midterm 2

Problem 6

Dr. Wilson

Find any x and y intercepts, asymptotes, horizontal, vertical, or otherwise, find where the function assumes positive values and where it assumes negative values, the places where the first derivative is either 0 or does not exists, where the function is increasing and decreasing, find the places where the second derivative is 0 or doesn't exist, and where the function is concave up and concave down. Find the local maxima and minima and points of inflection, and sketch the graph of

y - intercept: let x = 0

sin(0) = 0 so

y = 0

x - intercept: let y = 0

Set the factors = 0

never happens

x = n/4

n is any integer

Vertical asymptotes. Since the exponential function and the sine are defined for all real x, y is defined for all real x, so there are no vertical asymptotes.

Horizontal asymptotes: Since the exponential function has the x - axis as a horizontal asymptote, and the sine function is bounded between 1 and -1, this function will have the x-axis as a horizontal asymptote. However, since the function crosses the x - axis 4 times between each integer, this is an example of a function which will cross its asymptote an infinite number of times.

To see where the function assumes positive and negative values, note that the exponential factor is always positive, so the function will be positive when the sine function is positive, and negative when the sine function is negative. Since it is well known that


when

and


when

if

then

y > 0 when

n/2 < x < (2n + 1)/4

and

y < 0 when

(2n -1)/4 < x < n/2

n any integer

Let us use the first derivative to find the max's and min's.

Since our function is given by a product, we must use the product rule to find its derivative

y' = 0 when

Factor

and set the factors = 0

never happens

n an integer

We are used to seeing the sine have its max's and min's when the cosine is 0. Since the tangent is the sine over the cosine, that happens when the tangent has its vertical asymptotes. Since the exponential factor moves the graphs towards the x - axis as it moves to the right, it starts back up or down a little before the sine function assumes its max or min. 20 pi is a fairly large number, and the tangent is well on its way toward its vertical asymptote when it assumes a value that large, but it will still be a little before the cosine is 0. Note the periodicity.

The max's occur when n is even and the min's occur when n is odd.

As a result, the function is increasing between points where n is odd and n is even, and the function is decreasing between points where n is even and n is odd.

Now we consider the second derivative to find points of inflection and to determine concavity. The first derivative is a sum of products, so we use the product rule on each term.

This can be cleaned up to

So y" = 0 when

Set the factors = 0

never happens

n an integer

We are used to seeing the sine function have its points of inflection when the sine is 0 which is at integer multiples of Pi.The number inside the parentheses after the inverse tangent is a very small number, so we are adding a very small number to the roots of the sine function which we found above. Since the number inside the parentheses is negative, we get the points of inflection for this function just before the graph crosses the x - axis. Since the exponential function is concave up, there is a force acting on a particle following the graph of the function away from the x - axis. It overcomes the force pulling the particle toward the x - axis given by Hooke's Law and the sine function when the latter force is very samll which happens close to the x - axis. While the exponential decay factor causes the max's, min's and points of inflection to happen just a little before they would if the exponential decay factor wasn't there, these points all come at the same point in any cycle.

These are all points of inflection. The function is concave down between even n and odd n and it is concave up between odd n and even n.

If we let

and

Then we can make up the following table to graph this function through a few cycles.

x

y

type of point

c

-0.00253

-0.032

point of inflectionr

o

0

0

root

m

0.1237

0.9756

max

c + 1/4

0.24747

0.0302

point of inflection

1/4

0.25

0

root

m + 1/4

0.3737

-0.9279

min

c + 1/2

0.49747

-0.0288

point of inflection

1/2

0.5

0

root

m + 1/2

0.6237

0.8826

max

c + 3/4

0.74747

0.0274

point of inflection

3/4

0.75

0

root

m + 3/4

0.8737

-0.8396

min

c +1

0.99747

-0.0261

point of inflection

1

1.0

0

root

m + 1

1.1237

0.7986

max

c + 5/4

1.24747

0.0248

point of inflection

5/4

1.25

0

root

m + 5/4

1.3737

-0.7597

min

Note how the x coordinates of the max's and mins come just a little bit before the midpoint between the roots, and how the amplitude at the max's and min's keeps decreasing. You can also note that they decrease exponentially. Also note how close the points of inflections are to the roots, and how they always come just before the roots as you follow the graph as it moves to the right. If we add a couple of cycles to the graph, it looks like

This is a formula for damped harmonic motion. If you have a weight on the end of a spring, displace it and let it go, the weight will bounce up and down. Hooke's Law says that the force acting on the weight is proportional to how far you stretch the spring. You will see that if that is the only force acting on the weight, the weight will bob up and down according to a sine wave, which will go on forever without diminishing. However, in practice there are other forces besides the force from Hooke's Law. There are frictional forces which will slow the weight down until it comes to rest. If the frictional forces are proportional to the velocity of the weight and are acting in the opposite direction to the motion, then the height of the weight as a function of time will be a function like the one we are plotting here. Note how the weight bobs up and down but the amplitude is exponentially decreasing as time goes on. While this function will never come to rest, it will get to the point where the displacement of the weight is too small to measure, and it will appear that the weight will have come to rest. The coefficient on the variable in the exponential function is proportional to the frictional forces, and the coefficient on the variable in the sine function is proportional to the frequency with which the weight is bobbing up and down. In our example we have a somewhat high frequency, and a low frictional force, the result of which is that the function bobs up and down quite a bit before its motion dies down.